Jean-Pierre Aubin
Helene Frankowska
Set-Valued Analysis
With 11 Illustrations
1990
Birkhauser
Boston - Basel · Berlin

Jean-Pierre Aubin and Helene Frankowska
CEREMADE
(Centre de Recherche de Mathematiques de la Decision)
Universite de Paris-Dauphine and CNRS
F-75775 Paris Cedex 16
France
AMS Subject Classification (1985): 26A27, 26A51, 26E25, 28B20, 28D05, 34A60,
49A52, 54C60, 54C65, 58C06, 58C07, 58C30, 93C15, 93C30.
Library of Congress Cataloging-in-Publication Data
Aubin, Jean-Pierre.
Set-valued analysis / Jean-Pierre Aubin, Helene Frankowska.
p. cm.—(Systems & control ; v. 2)
Includes bibliographical references.
ISBN 0-8176-3478-9 (alk. paper)
1. Set-valued maps. I. Frankowska, Helene. II. Title.
ΙΠ. Series.
QA611.3.A93 1990
514'.32—dc20 89-48854
Printed on acid-free paper.
© Birkhauser Boston, 1990
All rights reserved. No part of this publication may be reproduced, stored in a retrieval
system, or transmitted, in any form or by any means, electronic, mechanical,
photocopying, recording or otherwise, without prior permission of the copyright owner.
ISBN 0-8176-3478-9
ISBN 3-7643-3478-9
Photocomposed text prepared using the authors' TeX files.
Printed and bound by R.R. Donnelley & Sons, Harrisonburg, Virginia.
Printed in the U.S.A.
987654321

This book is dedicated to C. Olech
and Polish Mathematicians1
who contributed so much to set-valued analysis.
1 "It was a common belief that cultivation of science and the growth of its
potential would somehow guarantee the maintenance of the nation" wrote Kura-
towski about the'situation of Poland before 1918.

I will share all [my results] with you
whenever you wish and do so without any
ambition, from which I am more exempt and more
distant than any man in the world.
Pierre de Fermata
Fermat was one of the most important
"innovators in the history of mathematics. Newton
himself recognized explicitly that he got the
hint of the differential calculus from Fermat's
method of building tangents devised half a
century earlier.
This same method is used in our book to build
a differential calculus for set-valued maps.
Fermat was also the one who discovered that
the derivative of a (polynomial) function
vanishes when it reaches an extremum. (This is
Fermat's Rule, which remains the main
strategy for obtaining necessary conditions of op-
timality, from mathematical programming to
calculus of variations to optimal control).
Fermat also was the first to discover the
"principle of least time" in optics, the prototype of
the variational principles governing so many
physical and mechanical laws. He shared
independently with Descartes the invention of
analytic geometry and with Pascal the creation
of the mathematical theory of probability. His
achievements in number theory overshadowed
his other contributions, and the Last Fermat
Theorem remains a challenge. He was on top
of that a poet, a linguist, ... and made his
living as a lawyer!
ain his answer to a first letter from Mersenne
inviting him to share his findings with the Parisian
mathematicians, which put an end to Fermat's isolation in
Toulouse, in 1636.

- "^ЗНХЖ
Li ЬЛЬиШ
P. DE FERMAT

Epigraph
Who needs set-valued analysis?
Everyone, we are tempted to say, and we shall state our case.
This strong conviction — born out of accumulated experience
in using it in control theory and differential games, mathematical
economics and game theory, biomathematics, qualitative physics and
viability theory — led us to devote time and effort to share some basic
material which is used over and over.
One can no longer afford the luxury of studying only well posed
problems in Hadamard's sense: ΠΙ posed problems, inverse problems
and many other unorthodox problems under other names are
popping up in every domain of activity, whenever the existence of a
solution may fail for some data, whenever uniqueness of the solution
is at stake. Requiring that maps should be always single-valued, and
even bijective, is too costly an attitude, above all in many applied
fields, where we are not free to make such assumptions. This was
indeed recognized during the three first decades of this century by the
founders of "Functional Calculus": Painleve, Hausdorff, Bouligand,
Kuratowski to quote only a few. In his important book Topologie,
Kuratowski gave set-valued maps their proper status.
Set-valued maps were abandoned by the authors of Bourbaki's
volume Topologie Generale, who chose to restrict their study to
single-valued maps, regarding set-valued maps as single-valued maps
from a set to the power set of another set, or factorizing single-valued
maps to make them bijective.
This is not always the solution, for, by so doing, many important
structural properties may be unfortunately lost; others are useless
artifacts, making life more difficult rather than more simple. These
points of view, which were widely disseminated all over the world

χ
after World War II, misled many of us into unnecessary detours (often
towards culs de sac), encouraging the perception that direct routes
were too arduous, or worse, that they did not exist.
Hence, set-valued analysis inherited the undeserved image of
being something difficult and mysterious and, consequently, was
regarded as a mathematical curiosity, to be left in the hands of
mathematicians who like to generalize for the sake of generalizing, without
proper motivations.
In contrast, as it turned, out, the need for set-valued analysis
in solving problems arising in other fields of knowledge — control
theory, economics and management, biology and systems sciences,
artificial intelligence, etc. — was pressing enough to help
mathematicians overcome the kind of recalcitrance felt towards set-valued
analysis.
In view of such a wide variety of motivating applications, it is
fortunate that most of the basic results of the chapters of "single-valued"
analysis can be adapted to the set-valued realm. These include:
β Limits and Continuity
β Linear Functional Analysis
β Nonlinear Functional Analysis (existence and approximation of
solutions to equations and inclusions)
β Tangents and Normals
β Differentiation of Maps
β Gradients of Functions and the Fermat Rule
β Convergence of Maps
β Measures and Integration
β Differential Equations
The set-valued version of this list is nothing other than the outline
of this book.
Our account of set-valued analysis is by no means exhaustive: it
is just an introduction. The choice of the material has been dictated

XI
by our experience in applying these results in control and viability
theory. However, we tried very hard to make this presentation as
clear as possible, to help the reader become familiar with the main
tools.
We did not restrict our exposition to the framework of finite
dimensional vector-spaces, since set-valued analysis is also useful for
solving problems involving partial differential equations or
inclusions. But whenever the proofs of the finite-dimensional and infinite-
dimensional statements are quite different, two proofs are provided.
This book presents only the tools, without mentioning
applications. Some applications (to control theory and viability theory in
particular) are presented in companion texts2.
Jean-Pierre Aubin
Helene Frankbwska
Paris
September 21, 1989
2Viability Theory [50] by Aubin, and the forthcoming books Set-Valued
Analysis and Control Theory [195] by Frankowska and Set-Valued
Analysis and Subdifferential Calculus [355] by Rockafellar and Wets.

Acknowledgments
We are indebted to. CEREMADE (Centre de Recherches de
Mathematiques de la Decision), Universite de Paris-Dauphin.e,
the Scuola Normale Dl Pisa3 and the Systems and Decision
Sciences4 of IIASA (International Institute for Applied Systems
Analysis) and their members5, who provided us with the intellectual
environment which made possible the writing of this book6.
We are also grateful to C. Hess, M. Valadier, 0. Dordan, R. Du-
luc, M. Quincampoix and C. Vigneron for their numerous suggestions
for improving the manuscript7.
We are happy to publish this monograph in the new series
Systems and Control: Foundations and Applications of Birkhauser.
3and in particular, G. da Prato.
especially, its head, A. Kurzhanski, as well as the dedicated and competent
librarians of IIASA.
5C. Byrnes, R. Rockafellar and R. Wets among many other colleagues
6 "...[a mathematician] does need a proper atmosphere; this proper atmosphere
can be created only by the cultivation of common topics." Janiszewski wrote in
1918 in On the Needs of Mathematics in Poland, a program which in our mind
can still be used all over the world.
7Fmally, as it is customary, each author is grateful to her/his coauthor for
his/her wonderful typing of the manuscript.

Contents
Introduction 1
1 Continuity of Set-Valued Maps 15
1.1 Limits of Sets 16
1.1.1 Definitions 16
1.1.2 The Compactness Theorem 23
1.1.3 The Duality Theorem 24
1.1.4 Convex Hull of Limits 26
1.2 Calculus of Limits 27
1.2.1 Direct Images 28
1.2.2 Inverse Images 30
1.3 Set-Valued Maps 33
Ί.4 Continuity of Set-Valued maps 38
1.4.1 Definitions 38
1.4.2 Generic Continuity 44
1.4.3 Example: Parametrized Set-Valued Maps ... 46
1.4.4 Marginal Maps 48
1.5 Lower Semi-Continuity Criteria 49
2 Closed Convex Processes 55
2.1 Definitions 56
2.2 Open Mapping and Closed Graph Theorems 57
2.3 Uniform Boundedness Theorem 61
2.4 The Bipolar Theorem 62
2.5 Transposition of Closed Convex Process 67
2.6 Upper Hemicontinuous Maps 74

xiv
3 Existence and Stability of an Equilibrium 77
3.1 Ky Fan's Inequality 80
3.2 Equilibrium and Fixed Point Theorems 83
3.2.1 The Equilibrium Theorem 83
3.2.2 Fixed Point Theorems 86
3.2.3 The Leray-Schauder Theorem 89
3.3 Ekeland's Variational Principle 91
3.4 Constrained Inverse Function Theorem 93
3.4.1 Derivatives of Single-Valued Maps 93
3.4.2 Constrained Inverse Function Theorems .... 94
3.4.3 Pointwise Stability Conditions 101
3.4.4 Local Uniqueness 103
3.5 Monotone and Maximal Monotone Maps 104
3.5.1 Monotone Maps 104
3.5.2 Maximal Monotone Maps 106
3.5.3 Yosida Approximations Ill
3.6 Eigenvectors of Closed Convex Processes 114
4 Tangent Cones 117
4.1 Tangent Cones to a Subset 121
4.1.1 Contingent Cones 121
4.1.2 Elementary Properties of Contingent Cones . . 125
4.1.3 Adjacent and Clarke Tangent Cones 126
4.1.4 Sleek Subsets 130
4.1.5 Limits of Contingent Cones; Finite Dimensional
Case 130
4.1.6 Limits of Contingent Cones; Infinite
Dimensional Case 132
4.2 Tangent Cones to Convex Sets 138
4.3 Calculus of Tangent Cones 146
4.3.1 Intersection and Inverse Image 146
4.3.2 Example: Tangent cones to subsets defined by
equality and inequality constraints 150
4.3.3 Direct Image 153
4.4 Normal Cones 156
4.5 Other Tangent Cones 159
4.5.1 Convex Kernel of a Cone 159
4.5.2 Paratingent Cones 160

XV
4.5.3 Hypertangent Cones 164
4.5.4 A Menagerie of Tangent Cones 165
4.6 Tangent Cones to Sequences of Sets 166
4.7 Higher Order Tangent Sets 171
5 Derivatives of Set-Valued Maps 179
5.1 Contingent Derivatives 181
5.2 Adjacent and Circatangent Derivatives 189
5.2.1 Definitions and Elementary Properties 189
5.2.2 Limits of Differential Quotients 191
5.2.3 Derivatives of monotone operators 194
5.3 Chain Rules 196
5.4 Inverse Set-Valued Map Theorem 203
5.4.1 Stability and Approximation of Inclusions . . . 203
5.4.2 Localization of Inverse Images 206
5.4.3 The Equilibrium Map 207
5.4.4 Local Injectivity 209
5.5 Qualitative Solutions 210
5.6 Higher Order Derivatives 215
6 Epiderivatives of Extended Functions 219
6.1 Contingent Epiderivatives 222
6.1.1 Extended Functions and their Epigraphs .... 222
6.1.2 Contingent Epiderivatives 224
6.1.3 Fermat and Ekeland Rules 232
6.1.4 Elementary Properties 234
6.2 Other Epiderivatives 236
6.2.1 Adjacent and Circatangent Epiderivatives . . . 236
6.2.2 Other Convex Epiderivatives 240
6.3 Epidifferential Calculus 242
6.4 Generalized Gradient 248
6.4.1 Subdifferentials and Generalized Gradients . . 248
6.4.2 Limits of Subdifferentials and Gradients . . . .251
6.4.3 Local Subdifferentials and Superdifferentials . . 252
6.4.4 Remarks 254
6.5 Convex Functions 255
6.6 Higher Order Epiderivatives 259

6.6.1 Second Order Epiderivatives of Moreau-Yosida
Approximations 263
Graphical & Epigraphical Convergence 265
7.1 Graphical Limits 267
7.1.1 Definitions 267
7.1.2 Graphical Convergence of Closed Convex
Processes 269
7.1.3 Monotone and Maximal Monotone Maps .... 270
7.2 Convergence Theorems 270
7.3 EpiUmits 274
7.3.1 Definitions and Elementary Properties 274
7.3.2 Convergence of Infima and Minimizers 281
7.3.3 Variational Systems 284
7.4 EpiUmits of Sums and Composition Products 286
7.5 Conjugate Functions of Epilimits 289
7.6 Graphical Convergence of Gradients 294
7.6.1 Convergence of Gradients of Smooth Functions 295
7.6.2 Convergence of Subdifferentials of Convex
Functions 297
7.7 Asymptotic Epiderivatives 300
Measurability and Integration of Set-Valued Maps 303
8.1 Measurable Set-Valued Maps 306
8.2 Calculus of Measurable Maps 310
8.3 Proof of the Characterization Theorem 319
8.4 Limits of Measurable Maps and Selections 322
8.5 Tangent Cones in Lebesgue Spaces 324
8.6 Integral of Set-Valued Maps 326
8.7 Proofs of the Convexity of the Integral 333
8.7.1 Finite dimensional case 333
8.7.2 Infinite Dimensional Case 340
8.8 The Bang-Bang Principle 343
8.9 Invariant Measures & Poincare's Recurrence Theorem 346
8.9.1 Linear Extension of Set-Valued Maps 346
8.9.2 Invariant Measures 350

xvii
9 Selections and Parametrization 353
9.1 Case of lower semicontinuous maps 355
9.2 Case of upper semicontinuous maps 358
9.3 Minimal Selection 360
9.4 The Steiner Selection 364
9.4.1 Steiner Points of Convex Compact Sets .... 365
9.4.2 The Intersection Lemma 369
9.4.3 Lipschitz Selections of Lipschitz Maps 372
9.5 Selections of Caratheodory maps - 373
9.6 Caratheodory Parametrization 376
9.7 Measurable/Lipschitz Parametrization 379
10 Differential Inclusions 383
10.1 The Viability Theorem 387
10.1.1 Solutions to Differential Inclusions 388
10.1.2 Statements of the Viability Theorems 389
10.1.3 Viability Kernels 392
10.1.4 Viability and Equilibria 393
10.2 Applications of the Viability Theorem 394
10.2.1 Linear Differential Inclusions 395
10.2.2 Lyapunov Functions 395
10.2.3 Tracking a Differential Inclusion 398
10.3 Nonlinear Semi-Groups 399
10.4 Filippov's Theorem 400
10.5 Derivatives of the Solution Map 403
Bibliographical Comments 411
Bibliography 421
Index 457

XIX
List of Figures
1.1 Example of Upper and Lower Limits of Sets 18
1.2 Graph of a Set-Valued Map and of its Inverse 35
1.3 Semicontinuous and Noncontinuous Maps 40
3.1 Example of Monotone and Maximal Monotone Maps . 104
4.1 Contingent Cone at a Boundary Point may be the
Whole Space 122
4.2 The Graph of T[afi](·) 123
4.3 Counterexample: Tangent Cone to the Intersection . . 142
4.4 The Menagerie of Tangent Cones: they may be all
different 161
5.1 Graph of the Contingent Derivative 183
6.1 Epigraph of the Contingent Derivative 227
9.1 Approximate Selection of an Upper Semicontinuous Map359
9.2 Illustration of the proof 370
List of Tables
2.1 Properties of Support Functions 66
4.1 Properties of Contingent Cones 124
4.2 Properties of Adjacent Tangent Cones 129
4.3 Properties of Tangent Cones to Convex Sets 141
4.4 % Properties of Contingent Cones to Derivable Sets in
Finite Dimensional Spaces 152
4.5 Properties of Normal Cones In Finite Dimensional
Vector Spaces 158
4.6 Properties of mi/l-order Contingent Sets 172
4.7 Properties of mi/l-order Adjacent Sets 174

It is a fact that in mathematical sciences there has been a
reluctance to deal with sequences of sets and set-valued maps. Despite
the emergence of exciting new vistas for the applications of
mathematics, our long familiarity with sequences (of elements) and with
(single-valued) maps has perhaps been so deeply rooted in
traditional mathematical conceptualizations that it has appeared easier
to sacrifice the breadth of some problems or some simple underlying
structure in order to avoid set-valued maps.
For this reason, we begin by providing examples of natural and/or
general problems involving set-valued maps before giving a rough
description of the results presented in the pages that follow.
Examples of Set-Valued Maps
1. First, we encounter set-valued maps each time we face ill-posed
problems or inverse problems, i.e., problems for which either
the existence of a solution or its uniqueness is not guaranteed
for some data: Set-valued maps allow us to get away from the
restriction that a map is bijective when we want to solve an
equation.
Indeed, the first natural instance when set-valued maps occur
is the inverse /-1 of a single-valued map / from X to Y. We
always can define /-1 as a set-valued map which associates
with any data у the (possibly empty) set of solutions
Г Чу) ■= {x€X\f(x)=y}
to the equation f(x) = y.
Of the three commandments of Hadamard's tablets, existence,
uniqueness and stability, we shall only retain the stability
requirements, which can be encapsulated in adequate definitions
of continuity of jf"1: This is one of the topics of the first
chapter.

2
Introduction
2. Taking into account uncertainties, disturbances, modeling
errors, etc., leads naturally to set-valued maps and inclusions.
They also arise when we wish to treat a problem qualitatively,
by looking for solutions common to a set of data, sharing the
same (qualitative) properties. Set-valued analysis should play
an important role in the new field of qualitative physics, a
rapidly growing branch of Artificial Intelligence. ■
3. Problems with constraints also yield specific set-valued maps:
Solving the equation f(x) = y, where the solution χ is
required to belong to a subset K, amounts to solving the
inclusion /|к(%) = У where f\x, the restriction of / to K, is
regarded as the set-valued map associating with χ the point
f(x) when χ £ К and the empty set when χ is not in K.
4. Unilateral problems in mechanics were formulated in the
framework of variational inequalities (also called "generalized
equations" by some authors), which are again inclusions in disguise.
Their solution by Stampacchia and J.-L. Lions in the sixties
gave a new impetus to set-valued maps, with a different
vocabulary.
5. Set-valued maps provide a useful framework for control theory,
since the early contributions of Wazewski and Filippov in the
beginning of the sixties.
Such set-valued maps, called parametrized maps, are associated
with a family of maps χ ι-> f(x, и) from X to Υ when и ranges
over a set U(x) of parameters.
The (single-valued) map / describes the dynamics of the
system: It associates with the state χ of the system and the control
и the velocity f(x, u) of the system. The set-valued map U
describes a feedback map assigning to the state χ the subset U(x)
of admissible controls.
Hence the map F which associates with each state χ the subset
F(x) of feasible velocities is defined by:
F(x) := f(x,U(x)) = {f(x,u)}ueU{x)

Examples of Set-Valued Maps
3
So, the control system governed by the family of parametrized
differential equations
x'(t) = f(x(t),u(t)) where u(t)eU(x(t))
is actually governed by the differential inclusion
x'(t) e F(x(t))
6. Optimization provides examples of problems where uniqueness
of the solution is naturally lacking:
Let ЖЬеа function from Χ χ Υ to R. We consider the family
of minimization problems
VyGY; V(y) := ЫМх,у)
parametrized by parameters y.
The function V is called the marginal (or performance or value)
function. For every у G Y, let
G(y) := {x€X\W(x,y) = V(y)}
be the subset of solutions to our minimization problem.
One of the main issues of optimization theory is to study the
set-valued map G (nonvacuity, continuity and differentiability
in a suitable sense, and so on.) We shall call G the marginal
map. It is no wonder that game theory and mathematical
economics use set-valued maps in a natural way.
7. Another source of strong motivations came from optimization
and mathematical programming, when necessary conditions
(the Fermat rule1, stating that the derivative of a function
vanishes at points where it achieves an extremum) were needed to
replace optimization problems by the resolution of equations.
1 "Je desire seulement qu'il [Descartes] sache que nos questions de Maximis et
Minimis et de Tangentibus linearum curvarum sont parfaites depuis huit ou dix
ans et que plusieurs personnes qui les ont vues depuis cinq ou six ans le peuvent
temoigner", Fermat wrote in 1638 when Descartes accused that the Methodus de
Maxima et Minima was just due to luck and trial and error! ("a tatons et par
rencontre".)

4
Introduction
The Fermat rule is indeed one idea which was revisited and
enhanced again and again under different names:
— The Euler-Lagrange equations, when dealing with
problems of the calculus of variations;
— Lagrange and Kuhn-Tucker multipliers, when state
constraints were added to optimization problems;
— The Pontriagin principle when dealing with optimal
control- problems.
After the advances of Functional Analysis, it was time to
uncover the common fact behind all these results. It is still and
always the Fermat rule, provided we are able to
"differentiate" larger and larger classes of functions beyond differentiable
functions.
The crucial revolution in the history of the concept of gradients
happened in the sixties when J.- J. Moreau and R. T. Rock-
afellar proposed in the framework of convex analysis the notion
of subdifferential of a convex function, which is no longer an
element, but a set of "subgradients".
8. The use of set-valued maps in mathematical economics and
game theory started when von Neumann asked for an
extension of the Brouwer Fixed Point Theorem to set-valued maps,
which was needed for finding noncooperative equilibria for n-
person games, for instance. This was achieved with the famous
Kakutani Fixed-Point Theorem, in the forties. It has been
used by Arrow and Debreu in the early fifties to provide the
long-expected proof of the existence of a Walrasian equilibrium
price.
While this achievement made set-valued maps popular among
mathematical economists, it was not until the challenges raised
by optimization, control theory and unilateral problems in
mechanics at the beginning of the sixties that renewed motivations
arose to study set-valued maps, as an important subject in its
own rite.
This was the time when Zarantonello introduced monotone
maps, which cover many important nonlinear single-valued or

Properties of Set-Valued Maps
5
set-valued maps of the Calculus of Variations.
Properties of Set-Valued Maps
Having briefly indicated the importance of set-valued maps in a
wide spectrum of applications and of fundamental mathematics, we
paint a broad picture of their properties. We follow, in doing so, the
outline of the book.
о Limits and Continuity
Limits of sets were introduced by Painleve in the first years of
this century, just after Prechet axiomatized in 1906 the concept
of ^-spaces (on which a notion of limit is defined2.) Studying
limits of sets together with limits of elements may have been
very natural in this context.
The topological ideas are, indeed, quite simple and
straightforward. In the same way that topological concepts are based
on the notions of limits and cluster points of sequences of
elements, their set-valued analogues are rooted in the concepts
of lower and upper limits of sequences of sets, which are, so to
speak, "thick" limits and cluster points respectively: The lower
limit of a sequence of subsets Kn is the set of limits of sequences
of elements xn G Kn, and the upper limit is the set of cluster
points of such sequences.
We mentioned already that stability is the only requirement
that we retain to study ill posed or inverse problems. Stability
is a catch word which means that the set of solutions depends
continuously upon the data.
How can we proceed to define continuity of set-valued maps?
If we try to adapt to the set-valued case the two equivalent
definitions of continuity of single-valued maps, we obtain two
notions which are no longer equivalent!
2In his famous thesis, judged at the time far too much abstract; it was
published in the Rendiconti Cir. Mat. di Palermo!
Lebesgue wrote: "....set theory was placed outside the pale of mathematics
by the high priests of analytic functions... Set theory, which developed from the
theory of analytical functions, could prove useful to its elder sister and could show
people of good will its qualities and richness".

Introduction
This unfortunate situation led to two concepts of semiconti-
nuity of set-valued maps, introduced at the beginning of the
thirties by Bouligand and Kuratowski: Lower and upper semi-
continuity. These issues were developed in the monographs of
Hausdorff and Kuratowski and thoroughly investigated at this
time by many (but not exclusively) Polish and French
mathematicians.
For some reason, just a while later, set-valued maps yielded
the way to single-valued maps: A set-valued map was viewed
at the time as a single-valued map from a set to the power
set of another set. However, as it turned out, the structures
exported to power sets were too poor, and specific information
was indeed wasted by doing so.
For instance, when we regard a set-valued map as a single-
valued map from one set to the power set of the other
(supplied with any one of the topologies we can think of), we arrive
at continuity concepts which are stronger than both lower and
upper semicontinuity, introducing parasitic artifacts. For
example, using such topologies to differentiate set-valued maps,
leads to such strong requirements, that most set-valued maps
would become nondifferentiable.
This is the reason why we shall start this book with the study
of limits and leave aside the examination of topologies on power
sets.
Furthermore, we shall renew history, by regarding a map not
... as a map, but as a graph (a subset of the product of the
departure and the arrival sets), reestablishing some symmetry
by putting these two sets on the same footing. This brings us
back to the source of analytical geometry, at the time of F.
Viete, P. de Fermat and R. Descartes, before the concept of
function and map evolved from the one of curves and graphs.
To regard a map as a graph is our constant and basic point of
view throughout this book (which has been called the graphical
approach.)
For instance, closed maps, that is maps with closed graph, shall
play a starring role in this book. It is a weaker property than

Properties of Set-Valued Maps
7
continuity or even, upper semicontinuity, very familiar and thus
easy to check, common to both set-valued maps and their
inverses.
© Linear Functional Analysis
What is the set-valued version of a continuous linear operator?
Remembering that the graph of a continuous linear operator
is a closed vector subspace, we are tempted to single out the
maps whose graphs are closed linear subspaces (called linear
processes.)
This generalization is not bold enough, since dealing exclusively
with closed vector subspaces is still too restrictive: We need to
use the notion of closed convex cone, which is a kind of vector
subspace in which it is forbidden to use subtraction. These
cones enjoy many properties of the vector subspaces.
For this reason, we select the closed convex processes, i.e., the
maps whose graphs are closed convex cones, as the candidates
to play the part of set-valued linear maps.
We shall see later that derivatives of some set-valued maps
are closed convex processes, which is a desirable property for
a derivative3. Indeed, the two basic theorems on continuous
linear operators due to Banach, the Closed Graph Theorem
(equivalent to the Open Mapping Principle) and the Banach-
Steinhaus Theorem, can be adapted to closed convex processes.
The first one states that a closed convex process defined on the
whole space is continuous, and the second states that pointwise
bounded families of closed convex processes are bounded —
a prerequisite for studying the convergence of closed convex
processes. But most important of all, one can transpose closed
convex processes and use the benefits of duality theory, based
on the Bipolar Theorem.
Closed convex processes also possess eigenvectors and invariant
spaces and cones. They truly deserve the status of linear set-
valued maps.
3keeping us in line with Hadamard's linearity bill enforced by Frechet that a
derivative should be linear with respect to the increment.

8
Introduction
© Nonlinear Functional Analysis
We are convinced that many problems can be regarded as
inclusions
given F : Χ -^ Υ and у €Y, find χ € X such that F(x) Э у
Most theorems on existence of solutions to nonlinear equations
can be extended to the case of inclusions.
For example, this is the case for the Brouwer Fixed Point
Theorem, whose generalization to set-valued maps is the famous
Kakutani Fixed-Point Theorem. We shall prove an equivalent
statement, called the Equilibrium Theorem, which provides the
existence of an equilibrium of a set-valued map, a solution to
the inclusion F(x) Э 0.
Of course, for applications, we need not only to solve such a
problem, but also to approximate its solutions by solutions to
approximate problems:
given Fn : Xn -^ Yn, yn € Yn, find xn € Xn with Fn(xn) Э yn
where Xn and Yn are subspaces of X and Y.
The famous Lax's principle of numerical analysis states that
convergence of the data yn to y, consistency of Fn to F and
stability of the Fn's imply the convergence of approximate
solutions. As an important special case, when the spaces Xn = X,
Yn = Y and the maps Fn = F are constant, this principle boils
down to the statement of the Inverse-Function Theorem.
Convergence, consistency and stability, which were originally
defined in the framework of linear equations, can be extended
to this general case by introducing adequate notions of
convergence and derivatives of set-valued maps. For instance, the
concept of consistency is nothing other than the fact that the
graph of F is the lower limit of the graphs of the approximate
maps Fn, while stability is the boundedness of the inverses of
the derivatives of the maps Fn.
This provides a first motivation for devising a set-valued
differential calculus. For that purpose, we need to begin with the
study of tangents.

Properties of Set-Valued Maps
9
β Tangents and Normals
The concept of tangency has been overshadowed in some sense
by the requirement that the space of tangent vectors must be a
vector space, so that the original idea became concealed after
its formal implementation in differential geometry.
If we come back to the idea underlying the notion of tangency
to a subset К at some point χ G K, we are tempted to form
"thick" differential quotients
K-x
h
and to take (in various ways) their limits when h > 0 goes to
0.
We obtain in this way a variety of closed cones made of what we
call tangent vectors. The most popular of these tangent cones
is for the time the contingent cone introduced in the thirties by
Bouligand, (which is the upper limit of these differential
quotients.) Some of these tangent cones are closed convex cones,
and they enjoy a property which is the natural extension of
linearity (without subtraction.)
These tangent cones possess a rich calculus which justifies their
use in many questions, mainly in problems with state con-
' straints: They are involved in the sufficient conditions for the
existence of an equilibrium and for the stability of solutions
to equations with constraints. They also appear in the
formulation of necessary conditions in optimization problems with
constraints and play a key role in viability theory.
In order to define space of normals, which in differential
geometry consists of vectors othogonal to the tangent vector space,
we are led to introduce the dual concept of normal cones to
any subset.
© Differentiation of Maps
We already mentioned that the concept of stability in the
Inverse-Function Theorem requires the notion of derivative of a
set-valued map, leading to the question: How do we formulate
this concept?

10
Introduction
The idea is very simple and goes back to the prehistory of the
differential calculus, when Pierre de Fermat introduced in the
first half of the seventeenth century the concept of tangent to
the graph of a function: The tangent space to the graph of a
function fata point (x, y) of its graph is the line of slope f'(x),
i.e., the graph of the linear function и ι-> f'(x)u.
It is possible to implement this idea for any set-valued map F,
since we have introduced a way to implement the tangency for
any subset of a normed space. Therefore, in the framework of
a given problem, we can regard a tangent cone to the graph of
the set-valued map F at some point (x, y) of its graph as the
graph of the associated "derivative" of F at this point (x,y).
Derivatives built in this way from the various choices of
tangent cones are called graphical derivatives and the calculus of
tangent cones can be transferred to a set-valued differential
calculus, including chain rules.
With such derivatives of set-valued maps in our hands, we can
linearize set-valued problems for approximating them by
linearized ones. The latter involving closed convex processes, this
strategy provides ways for transferring some properties of linear
set-valued maps to nonlinear maps.
β Gradients of Functions and the Fermat Rule
The particular case of real-valued functions deserves a study
by itself for taking into account the order relation of real
numbers. We are led to do so whenever we look for a minimizer
of a function or when we study the monotone behavior of a
function along a solution to a differential equation or inclusion
(Lyapunov property.)
The set-valued approach indicates the route: We associate with
a function V the set-valued map Vj defined by
VT(s) := [V(x),+oo[
whose graph is the epigraph of V.
The graphs of the derivatives of such set-valued maps Vj are
the epigraphs of functions which are called epiderivatives. We

Properties of Set- Valued Maps
11
discover that they are close relatives of the directional
derivatives introduced by Dini, who was among the first to revolt
against the rigidity imposed by the heirs of Cauchy4.
As far as optimization is concerned, the Fermat Rule can be
extended to any function by using these epiderivatives. Since
they enjoy a rich calculus, we obtain in this way many necessary
conditions for a minimum. This can be done by transferring
the set-valued differential calculus to what can be called an
epidifferential calculus.
By duality, we associate with each of the epiderivatives a
concept of generalized gradient: It is in general a subset of
elements, reduced to the usual gradient whenever the function is
differentiable in the usual way. In this framework, the Fermat
Rule becomes: // a point achieves the minimum of a function,
then it is an equilibrium of the generalized gradient, i.e., the
generalized gradient at an optimal point contains 0.
© Convergence of Maps
What about the convergence of a sequence of set-valued maps
The first idea which comes to mind is to extend the various
notions of uniform convergence of single-valued maps, regarded
as a map from one space to another.
Since we know how to deal with limits of sets, it is again natural
to use the graphical approach and to study the upper and lower
limits of graphs.
We follow this hint and study graphical upper and lower limits
of a sequence of set-valued maps as the maps whose graphs are
the upper and lower limits of the graphs.
4Cauchy, however, had the merit to formalize the concept of limits, continuity
and differentiability. His definitions have been canonized ever since: A function
was allowed to be differentiated only if the differential quotients were converging
to the derivative for the pointwise convergence topology. The need to use nondif-
ferentiable functions has been felt several times. By Bouligand, with the notions
of contingent and paratingent, by Dini who also broke Hadamard's linearity law,
by L. Schwartz and S. Sobolev, with the discovery of weak derivatives of functions
and distributions. But each of these extensions was devised for specific purposes
(solving partial differential equations, for instance.)

12
Introduction
When we deal with real-valued functions, we are led to use
the set-valued maps νπ| associated with functions Vn, whose
graphs are epigraphs of the functions Vn. The graphical limits
of the set-valued maps νπ| induce what we call epigmphical
limits of the functions Vn. This concept is closely related to G-
convergence introduced by de Giorgi and has been extensively
used in the study of stability of optimization problems5.
An interesting question arises: What are the connections
between the (epigraphical) convergence of a sequence of functions
and the (graphical) convergence of their gradients? We shall
answer such questions.
® Measures and Integration
We encounter measurable maps whenever we deal with models
of systems having measurable data, and in particular when we
deal with random set-valued variables (an issue we shall not
address in this book.)
Another important instance where measurable set-valued maps
do arise is in the linearization of a control system (or a
differential inclusion) along a solution.
Hence, we cannot escape the burden of studying measurable
maps, which are the maps whose graphs are measurable, and
checking in particular that all the standard operations preserve
measurability.
We also need measurability for defining integrals of set-valued
maps. Integrals of set-valued maps are involved in many con-
vexification (also called relaxation) problems, since roughly
speaking the integral of a measurable set-valued map is always
convex.
This property was in fact the original motivation to introduce
the integral of set-valued maps in mathematical economics and
game theory (with a continuum of players.)
We shall also address the basic questions of ergodic theory,
extending to set-valued maps the Poincare Recurrence Theorem
5See the forthcoming book Set-Valued Analysis and Subdifferential
Calculus [355] by Rockafellar and Wets.

Properties of Set-Valued Maps
13
and the existence of invariant measures.
β Differential Inclusions
Control Theory on one hand, and the evolution of macro-
systems under uncertainty on the other hand, constitute very
strong motivations for extending differential and partial
differential equations to differential and partial differential
inclusions.
We shall provide only an introductory survey of some basic
results, since covering this material requires books by themselves6.
We shall state some existence theorems, show that the set of
solutions depends continuously upon the initial data, describe
some properties of Lyapunov functions (which, thanks to the
concept of epiderivatives, can even be taken lower semicontinu-
ous), relate the graphical derivatives of the solution map to the
solution map of variational inclusions (which are linearizations
of the differential inclusion along a solution) and state some
applications of the Viability Theorem.
β Selections and Parametrization
We cannot escape in fine answering two natural questions: Can
we find selections of set-valued maps inheriting their regularity
properties? Are set-valued maps parumetrizablel
We shall be able to show that measurable set-valued maps do
have measurable selections and that continuous (Caratheodory,
Lipschitz) maps do have continuous (Caratheodory, Lipschitz)
selections under severe restrictions: The images of the set-
valued map must be convex.
Actually, a Lipschitz set-valued map F with closed convex
images is parametrizable in the sense that there exists a "control
space" U and a Lipschitz map / : X x U ι-> Χ such that
Vx, F(x) = {f(x,u)}ueu
6See for instances the books Differential Inclusions [33] by Aubin & Cel-
lina, Viability Theory [50] by Aubin and Set-Valued Analysis and
Control Theory [195] by Frankowska.

14
Introduction
Therefore, this limited class of set-valued maps is in essence
made of families of single-valued maps.

Chapter 1
Continuity of Set-Valued
Maps
Introduction
This chapter is devoted to the elementary topological properties
of sequences of sets and set-valued maps.
We begin the first section by extending concepts of limits and
cluster points of sequences of elements to sequences of sets. These
set-valued analogues have been introduced by Painleve in 1902 under
the names of upper and lower limits of sets. They were then
popularized by Kuratowski in his famous and influential book Topologie
and thus, often Christianized Kuratowski lower and upper limits of
sequences of sets1. We shall call them simply lower and upper limits:
The lower limit of a sequence of subsets Kn is the set of limits
of sequences of elements xn G Kn and the upper limit is the set of
cluster points of such sequences.
Some elementary properties of lower and upper limits are
investigated in Section 1, whereas their calculus is presented in Section 2.
Set-valued maps can be regarded as "continuous sequences." But
before adapting the above concepts to this case in Section 4, we
gather in Section 3 the main definitions concerning set-valued maps.
lHad they been called Painlevd instead of Kuratowski, a easier name to
pronounce indeed, these concepts could have been more popular.... This at least is
the opinion of one of the authors.
15

16
1- Continuity of Set-Valued Maps
Upper and lower limits of set-valued maps are naturally closely
related to continuity concepts. Recall that for single-valued maps,
continuity amounts to mapping converging sequences io converging
sequences or, equivalently, to the celebrated incantation:
Ve>0, 3£>0,....
Unfortunately, this characterization is no longer true for set-
valued maps:
There are two distinct ways to extend the concept of
continuity: The first one, encompassing the above "convergence" idea, is
called lower semicontinuity. The second, extending the "V e, 3 δ —
definition," leads to the so-called upper semicontinuity of set-valued
maps2. Kuratowski showed however that semicontinuous maps are
generically continuous.
Lower semicontinuous maps are closely related to lower limits,
but the situation is further complicated by the fact that upper limits
characterize the maps with closed graph and are not always upper
semicontinuous maps. These subtleties are explained in Section 4.
It happens that set-valued maps with closed graph taking their
values in a compact set are upper semicontinuous. Hence we have
an easy way to verify upper semicontinuity of a map.
Lower semicontinuity is not as simple to check: the last section
is devoted to several useful lower semicontinuity criteria.
The definitions introduced in this chapter are basic and will be
used throughout the book.
1.1 Limits of Sets
1.1.1 Definitions
Let X be a metric space supplied with a distance d. When К is a
subset of X, we denote by
άχ(χ) := d(x,K) := inf d(x,y)
2These concepts should not be confused with the upper and lower
semicontinuity of real-valued functions.

1.1. Limits of Sets
17
the distance from χ to K, where we set d(x,ty) := +00. The ball of
radius r > 0 around К in X is denoted by
Bx(K,r) := {x€X \ d(x,K)<r}
When there is no ambiguity, we set
B(K,r) := Bx(K,r)
When X is a Banach space whose unit ball is denoted by В (or Β χ
if the space must be mentioned), we observe that
Bx(K,r) = K + rBx
The balls B(K, r) are neighborhoods of K. When К is compact,
each neighborhood of К contains such a ball around K.
Limits of sets have been introduced by Painleve3 in 1902, as it
is reported by his student Zoretti4. They have been popularized by
Kuratowski in his famous book Topologie and thus, often called
Kuratowski lower and upper limits of sequences of sets. '
Definition 1.1.1 Let (-Kn)neN be a sequence of subsets of a metric
space X. We say that the subset
LimsuPj^ooATn := ix&X | liminf d(x,Kn) = 0 \
is the upper limit of the sequence Kn and that the subset
LiminfTl_too.Kn := {x € X \ limn^00d(x, Kn) = 0}
is its lower limit. A subset К is said to be the limit or the set limit
of the sequence Kn if
К = Limmfn-^ooUTn = LimsuPn^^Kn =: Ь1тп_юоЛГп
3"Je dirai qu'un point a appartient a I'ensemble limite de Ea si, quelque petits
que soient les deux nombres r et e, sous la condition \a — Qo| < e, le cercle de
centre α et de rayon r renferme des points appartenant a certains des Ea."
who wrote: "Je ne veux pas terminer ce travail sans adresser tous mes remer-
ciements a tous mes maitres de 1'Ecole Normale dont la sollicitude a mon egard ne
s'est jamais ralentie, et dont les encouragements m'ont ete si precieux. D'ailleurs,
dans ce milieu normalien, tout ce qui vous entoure est un enseignement, ...."

18 1- Continuity of Set-Valued Maps
Figure 1.1: Example of Upper and Lower Limits of Sets
.
ff»:={£}x[0,l]
if η is even
υ
Я»:=фх[-1,0]
if η is odd
и
LimimV+oo-Κπ = {0} & Limswpn^^Kn = [-1
.+1]
Lower and upper limits are obviously closed. We also see at once
that
Liminfn^ooUTn с Limsup^oo-Kn
and that the upper limits and lower limits of the subsets Kn and of
their closures Kn do coincide, since d(x,Kn) = d(x,Kn).
Any decreasing sequence of subsets Kn has a limit, which is the
intersection of their closures:
if Kn с Km when η >m, then Limn-юоКп = Q Kn
7l>0
An upper limit may be empty (no subsequence of elements xn G Kn
has a cluster point.)
Concerning sequences of singleta {xn}, the set limit, when it
exists, is either empty (the sequence of elements xn is not converging),
or is a singleton made of the limit of the sequence.
It is easy to check that:
Proposition 1.1.2 If (Kn)ne^ is α sequence of subsets of α metric
space, then Linimfn->oo-Kn is the set of limits of sequences xn G Kn
and LimsuPj^oQ-ftTn is the set of cluster points of sequences xn G Kn,

1.1. Limits of Sets
19
i.e., of limits of subsequences xn/ G Kn/. The upper limit is also equal
to the subset of cluster points of "approximate" sequences satisfying:
V e > О, Э Ν(ε) such that V η > Ν(ε), χη € B(Kn, e)
Remark — Replacing the balls of a metric space by neighborhoods
and the sequences of a metric space by generalized sequences, we can extend
the concepts of upper and lower limits to generalized sequences of subsets
of a topological space X.
We recall that a set Μ supplied with a preorder У is directed if every
finite subset has an upper bound. A generalized sequence is a map
μ G Μ t-> χμ G X
An element χ G X is the limit of (χμ)μ£Μ if; f°r every neighborhood V
of x: there exists μο G Μ such that χμ belongs to V for all μ У μ0. An
element χ is said to be a cluster point of this generalized sequence if, for
every neighborhood V of a; and every μ G M, there exists ν У_ μ such that
xv G V.
We recall that a single-valued map / from a topological space X to
another Υ is continuous at χ if it maps any generalized sequence converging
to a; to a generalized sequence converging to fix) and that a subset К С X
is compact if and only if every generalized sequence of elements of К has
a cluster point. (See for instance [148, Section 1.7].) The main difference
with sequences of a metric space is that a converging generalized sequence
is not necessarily bounded. (We can also replace generalized sequences by
"filters.")
- The upper limit of a generalized sequence of subsets Κμ where μ ranges
over a directed subset M. is the set of cluster points of generalized sequences
χμ of elements of Κμ and the lower limit is the set of limits of such
generalized sequences.
We may need the above extension when dealing with weak topologies of
a Banach space X and of its dual denoted by X*. We say that the bilinear
map < ·, · >
(i>,x) G X* x X ь-+< p,x > := p(x)
is the duality pairing.
We recall that the weakened topology σ(Χ, Χ*) of X is defined by the
semi-norms
Pm [x) ■= sup | < q, χ > |
q&M
when Μ := {qi,... ,qi} ranges over the finite subsets of X*. The weak —*·
topology σ{Χ*ι Χ) of the dual X* is defined by the semi-norms
pN{q) := sup | <q,y>\
yeN

20
1 - Continuity of Set-Valued Maps
when N := {x\,..., xn} ranges over the finite subsets of X.
In other words, a generalized sequence of elements χμ e X converges
weakly_to χ e X if and only if for any q e X*,'< ς,χμ > converges to
< q, χ > and a sequence ρμ e X* converges weakly to ρ if and only if for
any у e -X", < ρμ, у > converges to < p, у >.
The key facts that can be recalled now are that X* is still the dual of
X supplied with the weakened topology, that X is the dual of X* supplied
with the weak-* topology, and that the bounded subsets of the dual X* are
weakly relatively compact.
However, in general X is a closed subspace of the bidual X** of X,
which is the dual of the Banach space X*, endowed with the norm
||p||* := sup | <p,x> |
IMI<i
The space X is called reflexive if X = X**. In this case, it enjoys
both the properties of a Banach space and of the dual of a Banach space,
including the weak compactness of the unit ball. Π
We could for instance use the fact that the weakly compact
subsets of the dual X* of a separable5 Banach space are metrizable (for
the weak-* topology) (see [I48, Theorem 5.6.3]) to avoid generalized
sequences.
Instead, for the sake of simplicity, we shall use the concept of
sequentially weak upper limit:
Definition 1.1.3 Let us consider a sequence of subsets Kn of the
dual of a Banach space. We shall say that the subset
σ - Limsup^eoATn
of weak-* limits of subsequences of elements xn G Kn is the
sequentially weak upper limit of the subsets Kn.
In this way, we can present lower and upper limits in the framework
of metric spaces or of (countable) sequences, leaving to the
interested reader the task of checking some extensions to the case of non
metrizable topological spaces.
5This means that there is countable basis of open subsets, or, equivalently, that
there exists a countable basis spanning a dense vector space. The separability
concept goes back to Frechet.

1.1. Limits of Sets
21
We also point out the quite impressive equivalent formulation of
upper and lower limits which follows from Proposition 1.1.2:
Limsup^oolfn = Π U Kn = Π Π U Β(Κη,ε)
Ν>0η>Ν ε>0Ν>0η>Ν
and
Limmin^Kn = Π U Π Β(Κη,ε)
ε>0Ν>0η>Ν
and single out the following property of upper limits:
Theorem 1.1.4 Let К be a subset of a metric space X satisfying
the following property:
for any neighborhood U of K, 3 N such that V η > Ν, Kn С U
Then Limsupjj tooKn С К.
Conversely, if X is compact, then the upper limit lAmsw£>n^OQKn
enjoys the above property (and thus, is the smallest closed subset
satisfying it J
Proof — The first statement is obvious. The second one is a
consequence of the following more general result:
Proposition 1.1.5 Let us consider sequences of subsets Ln and Mn
of a metric space and assume that there exists a compact subset Μ
satisfying the following property:
for any neighborhood W of M, 3 N such that V η > iV, Mn С W
Then, for any neighborhood U of Μ Π (LiiaswPn^^Ln), there exists
an integer N such that Ln Π Mn С U whenever n> N.
Proof — If the neighborhood U contains M, the result
follows from the assumption on M. Otherwise, by taking an open
neighborhood U, the subset К := M\U is not empty, disjoint of
Limsupj^ooLn and is compact by assumption.
Let у belong to K. Since у does not belong to LimsuPj^ooLn,
there exist ey > 0 and Ny such that, for all η > Ny,y does not belong
to B(Ln,ey). The subset К being compact, it can be covered by ρ

22
1 - Continuity of Set-Valued Maps
balls B(yi,eyi). This implies that for all η > No := maxj^i,...^iV^
and. . .-:
V := \jB(yi,eyi)
the intersections Lni~)V are empty.
On the,other hand, W := U U V being a neighborhood of M, we
deduce from the assumption that there exists JVi such that
Vn>JVi, M„ С WUV
Therefore LnnMncU for all η > max(iV0) iVi). □
Remark — If Μ is not compact, but just closed, the
conclusion of the proposition remains true for any neighborhood U of
Μ Π (LimsupTl_HX1LTl) whose complement in Μ is compact. □
We also provide a useful technical lemma:
Lemma 1.1.6 Let us consider α sequence of subsets Ln С Ζ of α metric
space Ζ and a sequence of subsets Mn С Υ of a compact metric space Y.
Let ψ : Ζ χ Υ t-> R be an upper semicontinuous function. We denote by
M$ the upper limit of the Mn and Lb the lower limit of the Ln. Then
umsuPn-^oo ( SUP inf <P{z,y)) < SUP ίηίμ <Р(*,У)
\уемп zeb" / Уеми zeu·
Proof— Let у belong to M". Since φ is upper semicontinuous, we
know that for any ε > 0 and any ζ £ Lb, there exist neighborhoods N(z)
of ζ and N(y) of у such that
Уг'ёВД, Vy'eiV(y), φμ,ν') < <p{z,y) + e/2
In particular, by taking zy £ Lb such that
<f{Zy,y) < inf tp{z,y)+e/2
zEL*
and approximating zy by elements z™ £ Ln, we infer that there exists
Ny > 0 satisfying
Vn>iVy, \fy'£N(y), φ{ζ",υ') < <p(zy,y) + e/2
and thus, that for any у £ Μ",
V η > Ny, V у' £ N {у), inf φ(ζ, у') < inf p(z, у) + ε
zELn zELb

1.1. Limits of Sets
23
On the other hand, the compact set M" can be covered by η neighborhoods
N(yi) so that, by Theorem 1.1.4, there exists an integer NQ such that
Vn>JV0, Mn С (J N(yi)
i=l,...,n
Set N :=■ maxi=o,...,n Nyi. Then, for all η > N and у € Mn, у belongs to
some N(yi), so that,
inf <p{z,y) < inf ip(z,yi)+e < sup inf ¥>(z,2/)+e
^ebn zgbb Уем> zeib
Hence we have proved that for any ε > 0, there exists Ν > Ό such that
\/n>N, sup inf ip(z,y) < sup inf </?(z, ί/) + ε □
уемп zeL" уеми геьь
1.1.2 The Compactness Theorem
The Bolzano-Weierstrass Compactness Theorem was adapted in 1927
to the set-valued framework:
Theorem 1.1.7 (Zarankiewicz) Every sequence of subsets Kn of
a separable metric space X contains a subsequence which has a
(possibly empty) limit.
Proof — Since X is separable, there exists a countable family
of open subsets Um satisfying the following property:
V open subset U, V χ G U, 3 Um such that χ e Um С U
Let us consider a sequence of subsets Kn. We shall construct a
sequence of subsequences (K„ )n>o by induction.
For τη = 0, we set Kn := Kn. Assume that the m — 1 first
subsequences , 0 < ρ < m — 1 have been constructed.
Consider the mth open subset Um. Then either for every
subsequence Tlj,
Um П (Limsup^i^-1)) φ 0
in which case we set Kj := Kj , or there exists a subsequence
rij such that
Um П (Limsup^i^-1)) = 0

24
1 - Continuity of Set-Valued Maps
in which case we set Kj := Кщ '. (The choice of such a
subsequence does not matter.)
These sequences [Kn ) being constructed, we extract the
\ / n>0
diagonal subsequence Dn := Kn . We claim that it has a set limit.
If not, there would exist
x0 € Limsup^oo-Dn h x0 φ LiminfTl_too.DTl
The latter condition means that there exists an open neighborhood
U of xq and a subsequence Dn. such that U Π Dnj = 0 for any j. Let
us fix an open subset Um such that xq € Um С U. We thus deduce
that
Um Π (himswpj^^Dn.) = 0
Since for rij > m, Dnj := Кщ' = Щт~ ' for some pj, we observe
that Dn, is a subsequence of the sequence [Knm~~ ') , the upper
3 \ In>0
limit of which is disjoint from Um.
By the very construction of [Kn ) , we infer that
\ / n>0
к^ = лг^-1)
and consequently, that
Um П (Limsup^i^H) = Um П (Limsup^i^-1)) = 0
Since Dn := Knn' = Kp^' for some pn, we deduce that the sequence
(Dn)n>m is a subsequence of the sequence (Ют ). . Thus
xq € Limsup^oo-Dn С Limsupj^^Kj С X \ Um
which contradicts the fact that xq belongs to Um- n
1.1.3 The Duality Theorem
For closed convex cones Kn, upper limits and lower limits can be
exchanged by duality.

1.1. Limits of Sets
25
We introduce the (negative) polar cones to subsets К с X and
L С X* defined by
K~ := {pGX* \ V χ G Κ, < ρ, χ > < 0}
ала
L~ := {ж € A" | Vp€L, <p,x>< 0}
Let σ — LimsuPj^ooAT" denote the sequentially weak upper limit
of the polar cones K~.
Theorem 1.1.8 Let (Kn)ne^ be a sequence of closed convex cones
of a Banach space X. Then
Limmfn^ooATn = (σ - Limsupn_+00iir~)~
Proof — Inclusion
Liminfn-юоЛГп С (σ -Limsup^^Ar")""
is obvious: If χ G Liminin^coKn is the limit of a sequence of elements
xn G Kn and
ρ G σ - Limsupn^oolf"
is the weak-* limit of a subsequence pni G K~,, inequalities < ρηι, xn> >
< 0 imply that < p,x >< 0.
Conversely, assume that some
χ G (a-Limsup^^UT-)-
does not belong to the lower limit Limmin^ooKn. Then there exists
e > 0 and a subsequence (again denoted (ΑΓπ)πεΝ) such that
V π > 0, (χ + ε Β) Π Κη = 0
The Separation Theorem implies the existence of elements pn G X*
of norm equal to one such that
σκη(ρη) <<Pn,x> -ε||ρη|| = < Ρη,χ > -ε
The subsets Kn being cones, we deduce that pn G K~ and that
σκη(Ρη) = 0. Then a subsequence (again denoted) pn converges
weakly—* to some p, which thus belongs to σ — lAmsn^^^K^, so
that < pyx >< 0. Inequalities
0 < (ρη,Χη) -ε
imply that e < 0, a contradiction. □

26
1 - Continuity of Set-Valued Maps _
1.1.4 Convex Hull of Limits
Since the distance function to a subset of a normed space is convex
if and only if this subset is convex, we infer that the lower limit of а
sequence of convex subsets is closed and convex.
It is useful to have a characterization of the closed convex hull of
upper limit. We denote by co(K) the convex hull oiK and by co(K)
its closed convex hull.
Lemma 1.1.9 Let us consider a sequence of subsets Kn contained
in a bounded subset of a finite dimensional vector space X. Then
со (Limsup^^-Kn) = f] со MJ Kn
N>0 \n>N
Proof — The closed convex hull of the upper limit is obviously
contained in the closed convex subset
A :=
We have to prove that it is equal to it when the dimension of X is
finite and the subsets Kn are contained in a bounded set.
Since an element ж of Л is the limit of a subsequence of convex
combinations ν ν of elements of (Jn>N Kn and since the dimension of
X is an integer p, Caratheodory's Theorem allows us to write that
ρ
ΣΝ
3=0
where
ρ
Nj > N, of > 0, Σα? = 1
j=o
and a;^ belong to K^y The vector aN of p + 1 components a?
contains a subsequence (again denoted by) aN. which converges to some
nonnegative vector a oi p + 1 components aj such that J2^=o аз = 1-
The subsets Kn being contained in a given compact subset, we
can extract successively subsequences (again denoted by) х$[.
converging to elements Xj, which belong to the upper limit of the subsets
Kn. Hence χ is equal to the convex combination YTj=q ajXj, and the
lemma is proved. □

1.2. Calculus of Limits
27
1.2 Calculus of Limits
We begin by pointing out the following obvious properties:
Proposition 1.2.1 Let Kn, Ln, Кгп, (i = Ι,.,.,ρ) be sequences of
subsets of α metric space. Then
i) Limsupn^oo(iin П Ln) С Limsupn_>00iin Π Limsupn_>00Ln
ii) Liminfn-yooC-Kn HLn) С Liminfn^oouTn П Liminin^ooLn
Hi) Limsup^^(Kn U Ln) - Limswpn^coKn U Limsupn_tooLn
гг;) Liminfn_>oo (-^n U L„) э Liminfn_>00l£n U Limmfn^ooLr,
υ) Limsup^oo EELi ^n С Π?=ι Limsup^^ii^
i) Liminfn-чоо Ш=1 •K'n = Π?=ι Liminfn-»oo^
We need also to relate direct and inverse images of upper and
lower limits of a sequence of subsets to the upper and lower limits
of their direct and inverse images. We mention now the obvious
relations and postpone the proofs of criteria which transform the
following inclusions to equalities.
Proposition 1.2.2 Let Kn be a sequence of subsets of a metric space
X, Mn be a sequence of subsets of a metric space Υ and f : X ι—> У
be a (single-valued) continuous map. Then
' i) fCLimsuPn^Kn) с UxDawprtr^OQf(Kn)
ii) /(Limmfn^ooATn) с Limmin^oo f(Kn)
Hi) Limsupjj^QQ
iv) Liminfn-^o/^Mn) С jf"1 (Liminf^ooMn)
The question arises of providing converse results under adequate
assumptions.

28
1 - Continuity of Set-Valued Maps
1.2.1 Direct Images
We begin with the direct images of upper limits: We easily obtain
equalities when / is proper. We recall that a continuous single-valued
map from a metric space X to a metric space Υ is proper if and only
if one of the following equivalent statements
If f(xn) converges in Y, then xn has a cluster point
or
i) f maps closed subsets to closed subsets
ii) V compact Μ С Υ, f~1(M) is compact
holds true.
Proposition 1.2.3 We posit the assumptions of Proposition 1.2.2.
Let us assume that f is proper, then
/(Limsupn_tooi;!rn) = Ьш1зирп_>00/(ЛГП)
Furthermore, if f is surjective, we obtain
f1 (LimsuPn^ooMn) = Limsupn_>0o/~1(M„)
The proof is a simple consequence of definitions and is omitted.
In the case of continuous linear operators, we obtain a more
specific criterion using the polar set
K°:={p€X* | \/x € К, <р,х><1}
of a subset К of X.
When X and Υ are Banach spaces and / is a continuous linear
operator A G C(X, Y), we obtain the following result:
Theorem 1.2.4 LetX and Υ be Banach spaces, (-Kn)neN be a
sequence of subsets of X and A G £(X, Y) be a continuous linear
operator satisfying
0 € Int (lm(A*) + (J Π κ) (1.1)
V Ν>0η>Ν )

1.2. Calculus of Limits
29
— If X is α finite dimensional space, then
Limsupn_>0oA(.K'n) = A(UmS}xpn_,00Kn)
— If X is reflexive, then
σ - UmsnVn^^AiKn) = Α(σ - Limsupn^00Kn)
Proof — Observe that the first statement follows from the
second one and Proposition 1.2.2 i). Hence we have to prove only
the second claim.
Let us consider a sequence xn G Kn such that a subsequence
of elements A(xn) (again denoted A(xn)) converges weakly to some
у in Y. We shall check that (xn)neN has a weak cluster point,
by showing that it is weakly bounded, and thus, weakly relatively
compact. Assumption (l.l) implies that the set
ЩА*) + (J Π ΚΙ
Ν>0η>Ν
contains a ball around zero of radius 7 > 0. Then any ρ G X*, ||p||* <
7 can be written,
ρ := A*q + r, where q G Y*, r € \J f] K°n
N>0n>N
Therefore, there exists N > 0 such that r € Ππ>;ν·^π ап^
consequently,
suVn>N <p,xn> = supn>^(< q, Axn > + < r,xn >)
< < 8vpn>N(\\q\\\\Axn\\+a\xpxeKn <r,x>)
< swpn>N \\q\\\\Axn\\ + 1 < +00
since the weakly converging sequence Axn is bounded.
Then xn remains in a bounded subset, which is weakly
relatively compact. Hence it has a cluster point χ which belongs to
σ — lAmsxvon^00Kn. Since A is continuous from X to Y, when they
are supplied with their weak topologies, Ax is a weak cluster point
of the sequence of elements Axn, which converges weakly to у = Ax.
Hence
σ - lAmsnVn^^AiKn) С Α(σ - Limsupn_tooi;!rn)
The opposite inclusion being obvious, the proof ensues. □

30
1- Continuity of Set-Valued Maps
1.2.2 Inverse Images
We study next the case of inverse images of limits.
We begin first with a simple criterion which holds true when the
subsets Mn are convex and the map / is a continuous linear operator.
Proposition 1.2.5 Let us consider two Banach spaces X and Υ,
a continuous linear operator A G jC(X,Y) and a sequence of
convex subsets Mn С Υ. We assume that the "constraint qualification
assumption"
3 7 > 0, с > 0 such that yB С сА(Вх) - Mn
holds true for large enough η 's. Then
Liminfn_tooJ4"1(Mn) = ^"^Liminfn-^oo-Wn)
<
w Limsup^ooA-^Mn) = ^(Limsup^^Mn)
This statement is a consequence of a more precise result:
Proposition 1.2.6 Let us consider two Banach spaces X and Υ, a
continuous linear operator A G C(X,Y) and two sequences of convex
subsets Ln С X and Mn С Υ. Assume that there exist 7 > 0, с >
0, N > 0 such that
V η > Ν, jB С A(Ln Π cBx) - Mn
Then
Liminfn_too(LnnJ4"1(Mn)) = (Liminfn_t0obn)nJ4"1(Liminfn_tooMn)
Limsupn_too(LnnA-1(Mn)) D (Limsupn_tooLn)nJ4-1(Liminfn_tooMn)
and
Limswpn^00(LnnA~1(Mn)) D (Liminfn_tooLn)nA"1(Limsupn_tooMn)
Proof — The inclusion
Liminfn_too(LnnA-1(Mn)) с (Liminfn_tooLn)nA-1(Liminfn_tooMn)

1.2. Calculus of Limits
31
being obvious, let us prove the other one, by checking that any χ in
the intersection
Liminfn—y oo Ln
is the limit of a sequence of elements xn belonging to Ln such that
A(xn) belongs to Mn.
We know that χ can be approximated by elements un e Ln and
that A(x) can be approximated by elements vn € Mn. Then
en := ||A(?xn) -vn\\
converges to 0 and
θη := 7/(7 +en) €]0,1[
converges to 1 and satisfies θηεη = (l — #n)7- Therefore,
Θη(νη-Α(ηη)) e θηεηΒ = {ΐ-θη)ΊΒ С (l-9n)(A(LnncBx)-Mn)
Consequently, there exist elements u'n £ LnD cBx and v'n G Mn such
that
A {9nun + (1 - 9n)u'n) = θηυη + (1 - θη)ν'η
If we set
xn := 9nun + (1 - вп)и'п
we observe that xn belongs to Ln and that A(xn) belongs to Mn, for
these subsets are convex.
Furthermore,
||яп-ип|| = (1 - 9n)\\un - u'n\\
converges to 0 since the sequences un and u'n are bounded, the first
one because it is convergent to и and the second because it is bounded
by с by assumption. Hence xn converges to x. The proof of the
second and third statements is analogous and left as an exercise» □
Remark — Actually, Proposition 1.2.5 is equivalent to
Proposition 1.2.6: The former used with 1 x A and Ln x Mn implies
obviously the latter. □

32
1 - Continuity of Set-Valued Maps
We can prove a similar result for non convex subsets and
nonlinear maps, which is based on extensions of Graves' Inverse Function
Theorem 3.4.2 proved in Chapter 3. It states that if f is α
continuous map from an open subset of a Banach space X to a Banach
space Y, continuously differentiable at xq, the surjectivity of f'(xo)
implies the existence of a constant I > 0 such that, whenever у lies
in a neighborhood of f(xo), the equation f(x) = у has a solution
satisfying
||s-soil < l\\f(xo)-y\\ .(1.2)"
The simplest extension reads:
Proposition 1.2.7 Let us assume thatX andY are Banach spaces,
that the map f : X ι—> У is continuously differentiable at some point
χ of f~l (Liminfn^ooiWn), and that f'(x) is surjective.
Then χ belongs to the lower limit LiminfTl_too/_1(MTl), and a
similar statement holds true for upper limits.
Proof — Set у = f(x) and consider a sequence of elements yn G
Mn converging to y. Then (1.2) implies the existence of a solution xn
to the equation f(xn) = Уп satisfying inequality \\x — xn\\ < l\\y — yn\\
for η large enough. Hence χ being the limit of xn € /_1(Μπ) belongs
to the lower limit of the inverse images of the subsets Mn. The proof
of the second statement is analogous. □
We shall generalize the above inverse function theorem in
Chapter 3 and thus be able to weaken the surjectivity assumption of f'(x)
by assuming a transversality condition bearing on / and the
subsets Mn. These conditions involve the concept of contingent cone
Тмп(Уп) to a subset Mn at some point yn € Mn, which will be
studied in Chapter 4. The contingent cone Тм{у) is defined by
ν €= TM(y) <=> limine+ M = 0
/i->o+ h
However, this is a natural place to state this important extension
despite the technical complexity of this transversality assumption:
Theorem 1.2.8 Let X and Υ be two Banach spaces. We consider
a sequence of closed subsets Mn С Υ and a map f : Χ ι—► У
continuously differentiable at some point
χ € jT1 (Liminfn-юоМп)

1.3. Set-Valued Maps
33
Let us assume that there exist constants с > 0 and η > 0 such that
V yn € B(f(x),V) П МП1 Бу с с/'(а:)(ВД - ZWn(yn)
JTien χ belongs to Limmin->oo f~1(Mn) and a similar statement holds
true for upper limits.
This result is contained in the following, more general,
Theorem 1.2.9 Let X and Υ be two Banach spaces. We consider
a sequence of closed subsets Mn С Υ and a map f : Χ η-» Υ dif-
ferentiable on a neighborhood U of f~1(Limmin->coMn) (respectively
/_1(LimsupTl_HX1MTl)) such that the derivatives f'(x) are uniformly
bounded on Ы. Let us assume that there exist constants с > 0 and
η > 0 such that for every χ G U,
V yn € B(f(x),V) П Mn, By С cf(x)(Bx) - TMn{yn)
Then
Liminfn-t0o/"1(Mn) = /^(LimhuV^Mn)
respectively
Limsupn^oo/'H^fn) = /-1(LimsuPn->oo-Mn)
We postpone the proof of the last theorem to Chapter 3.
1.3 Set-Valued Maps
Sequences of subsets can be regarded as set-valued maps defined on
the set N of integers.
Naturally, we can replace N by a metric (or even, topological)
space X, and sequences of subsets n ~> Kn by set-valued maps χ ~~*
F(x) (which associates with a point χ a subset F(x)) and adapt the
definition of upper and lower limits to this case, called the continuous
case.
Before proceeding further, we recall in this section the basic
definitions dealing with set-valued maps, also called multifunctions,
multivalued functions, point to set maps or correspondences.

34
1- Continuity of Set-Valued Maps
Definition 1.3.1 Let X andY be metric spaces. A set-valued map
F from X to Υ is characterized by its graph Graph(F), the subset
of the product space Χ Χ Υ defined by
Graph(-F) := {(x,y) £ X xY \ y£ F(x)}
We shall say that F(x) is the image or the value of F at x.
A set-valued map is said to be nontrivial if its graph is not empty,
i.e., if there exists at least an element χ £ X such that F(x) is not
empty.
We say that F is strict if all images F(x) are hot empty. The
domain of F is the subset of elements χ £ X such that F(x) is not
empty:
Dom(F) := {x£X \ F(x) φ 0}
The image of F is the union of the images (or values) F(x), when χ
ranges over X:
lm(F) := (J F(x)
xeX
The inverse F~1 of F is the set-valued map from Υ to X defined by
xeF'^iy) <=> y€F(x) <=> (x, y) € Graph(F)
The domain of F is thus the image of -F-1 and coincides with the
projection of the graph onto the space X and, in a symmetric way,
the image of F is equal to the domain of -F"1 and to the projection
of the graph of F onto the space Υ.
If К is a subset of X, we denote by F\k the restriction of F to
K, defined by
F\K(x) ■= I F(X) 'lixGK
*lK{X) ' \ 0 if x i К
Let V be a property of a subset (for instance, closed, convex,
etc..) Since we shall emphasize the symmetric interpretation of a
set-valued map as a graph (instead of a map from a set to another
one), we shall say as a general rule that a set-valued map satisfies
property V if and only if its graph satisfies it.
For instance, a set-valued map is said to be closed (respectively
convex, closed convex, measurable, monotone, maximal monotone)

1.3. Set-Valued Maps
35
Figure 1.2: Graph of a Set-Valued Map and of its Inverse
Dom(F) 0
Graph and Domain of a Set-Valued Map
Graph and Image of its Inverse

36
1 — Continuity of Set-Valued Maps
if and only if its graph is closed (respectively convex, closed convex,
measurable, monotone, maximal monotone.)
If the images of a set-valued map F are closed, convex, bounded,
compact, and so on, we say that F is closed-valued, convex-valued,
bounded-valued, compact-valued, and so on.
When * denotes an operation on subsets, we use the same
notation for the operation on set-valued maps, which is defined by
Fi*F2: x ~> Fi(x)*F2(.x)
We define in that way Fi Π F2, F\ U F2, F\ + F2 (in vector spaces),
etc. Similarly, if α is a map from the subsets of Υ to the subsets of
Y, we define
a(F) ·. χ -^ a(F(x))
For instance, we shall use F, co(F), etc., to denote the set-valued
maps χ ~> F(x), χ ~> co(F(x)), etc.
We shall write
F С G <i=^ Graph(F) С Graph(G)
and say that G is an extension of F.
Let us mention the following elementary properties:
' г) Р(КгиК2) = FiKJUFfa)
ii) F(KinK2) С ^(ΛΓι)η^(ΛΓ2)
Hi) F(X\K) D lm(F)\F(K)
iv) КгСК2 =» F{K{) С F(K2)
There are two manners to define the inverse image by a set-valued
map F of a subset M:
' i) F-\M) := {x | F(x)DM фЩ
<
^ ii) F+1(M) := {x \ F(x) С М}
The subset F_1(M) is called the inverse image of Μ by F and
F+1(M) is called the core of Μ by F.

1.3. Set-Valued Maps 37
They naturally coincide when F is single-valued.
We observe at once that
F+1(V\M) = XXF-^M) L· F-^YXM) = X\F+1(M)
One can conceive as well two dual ways for defining
composition products of set-valued maps (which coincide when the maps are
single-valued):
Definition 1.3.2 Let Χ, Υ, Ζ be metric spaces and
G:X ~> Υ L· H:Y ~> Ζ
be set-valued maps.
1 — The usual composition product (called simply the product,)
Η о G : X ~> Ζ of Η and G at χ is defined by
(HoG)(x) := (J Η (у)
yeG{x)
2 — The square product HUG : X ~> Ζ of Η and G at χ is
defined by
{HUG){x) := Π Η (у)
yeG{x)
Let 1 denote the identity map from one set to itself. We deduce the
following formulas
' Graph(tfoG) = (G x l)71(Graph(F))
= (lxtf)(Graph(G)) (
Graph(tfDG) = (G x l)+1(Graph(tf))
as well as formulas which state that the inverse of a product is the
product of the inverses (in reverse Order):
' i) {HoG)-\z) = G-\H~\z)) = (G-1 о Я"1) (ζ)
ii) (tfDG)"1^) = G+Htf-1^))

38 1- Continuity of Set-Valued Maps
We also observe that
' i) χ € (tfDG)"1^) <=*> G(s) С Я"1^)
<
и) х е (С-^пЯ"1)^) <=► Я"1^) с G(s)
and thus, that
G{x) = H'\z) «=► χ e (G^Dff-^WniffDG)-1^)
Let us also point out the following relations: When Μ is a subset of
Z, then
' i) {HoG)~\M) = G-^H-^M))
<
^ и) (ЯоС)+1(М) = С+1(Я+1(М))
1.4 Continuity of Set-Valued maps
The concepts of semi-continuous maps have been introduced in 1932
by G. Bouligand6 and K. Kuratowski7. We begin with the upper
semicontinuous set-valued maps:
1.4.1 Definitions
In this section, Χ, Υ, Ζ denote metric spaces. We describe the
concepts of semicontinuous set-valued maps introduced by Bouligand,
Kuratowski and Wilson in the early thirties.
Definition 1.4.1 A set-valued map F : Χ ~> Υ is called upper
semicontinuous at χ £ Dom(F) if and only if for any neighborhood
U of F{x),
3 η > 0 such that V x' € Βχ(χ,η), F(x') С U.
6who wrote: "Peut-on rendre plus profond hommage a la memoire de Rene
Baire qu'en poursuivant les consequences d'une idee degagee par lui et dont
l'importance se revele chaque jour accrue, la semi-continuite? Elle echappa tout
le XIXе siecle aux adeptes de la theorie des fonctions, et a plus forte raison, aux
purs geometres, qui s'adonnaient a des occupations moins subtiles."
7 who also wrote:
"D'apres Monsieur Baire, une fontion est dite semi-continue superieurement
La notion de semi-continuite dont nous allons nous servir ici est tout a fait
analogue a celle-ci, mais concerne le cas ou la fonction F(x) admet comme valeurs
des sous-ensembles fermes...."

1.4. Continuity of Set-Valued maps
39
It is said to be upper semicontinuous if and only if it is upper semi-
continuous at any point of Dom(F).
When F(x) is compact, F is upper semicontinuous at χ if and only
if
Ve > 0, 3 η > 0 such that Vs' G Βχ(χ,η), F(x') с BY(F(x),e)
We observe that this definition is a natural adaptation of the
definition of a continuous single-valued map. Why then do we use
the adjective upper semicontinuous instead of continuous? One of the
reasons is that the celebrated characterization of continuous maps —
a map f is continuous at χ if and only if it maps sequences converging
to χ to sequences converging to f(x) — does not hold true any longer
in the set-valued case.
Indeed, the set-valued version of this characterization leads to the
following definition.
Definition 1.4.2 A set-valued map F : X ~> У is called lower semi-
continuous at χ £ Dom(.F) if and only if for any у G F(x) and for
any sequence of elements xn G Dom(F) converging to x, there exists
a sequence of elements yn G F(xn) converging to y.
It is said to be lower semicontinuous if it is lower semicontinuous
at every point χ G Dom(F).
Actually, as in the single-valued case, this definition is equivalent to:
For any open subset U <ZY such that U П F(x) φ 0,
3 η > 0 such that V χ' G Βχ(χ, η), F(x') Π U φ 0
Unfortunately, there exist set-valued maps which enjoy one
property without satisfying the other.
Examples — The set-valued map i<\ : R -^> R defined by
*Л*) - \ {0} if χ = 0 ·
is lower semicontinuous at zero but not upper semicontinuous at zero.
The set-valued map F2 : R ~> R defined by
Fo(x) - I {0} if X ф °
Ы*)- | μ1)+1] if x = 0

40 1- Continuity of Set-Valued Maps
Figure 1.3: Semicontinuous and Noncontinuous Maps
.
0
,
lower s.c. h not upper s.c.
.
0
upper s.c. & not lower s.c.
is upper semicontinuous at zero but not lower semicontinuous at zero.' □
We are therefore led to introduce still another
Definition 1.4.3 We shall say that set-valued map F is continuous
at χ if it is both upper semicontinuous and lower semicontinuous at
x, and that it is continuous if and only if it is continuous at every
point o/Dom(-F).
Remark — We can use the concepts of inverse images ала
cores to characterize upper and lower semicontinuous maps:
Proposition 1.4.4 A set-valued map F :X ^Y is upper
semicontinuous at χ if the core of any neighborhood ofF(x) is a neighborhood
of χ and a set-valued map is lower semicontinuous at χ if the inverse
image of any open subset intersecting F(x) is a neighborhood of x.
Hence, F is upper semicontinuous if and only if the core of any
open subset is open and it is lower semicontinuous if and only if the
inverse image of any open subset is open.
IfOom(F) is closed, then F is lower semicontinuous if and only if
the core of any closed subset is closed and F is upper semicontinuous
if and only if the inverse image of any closed subset is closed.
We shall also need to adapt to the set-valued case the concept of
Lipschitz applications.

1.4. Continuity of Set-Valued maps
41
Definition 1.4.5 When X and Υ are normed spaces, we shall say
that F : Χ ~> Υ is Lipschitz around χ £ X if there exist a positive
constant I and a neighborhood U С Dom(F) of χ such that
\/X!,X2&U, F(xi) С F(x2) + l\\xi - Χ2\\Βγ
In this case F is also called Lipschitz (or I—Lipschitz) on U.
If у is given in F(x), F is said pseudo-Lipschitz around (x,y) €
Graph(F) if there exist a positive constant I and neighborhoods U С
Dom(-F) of χ andV of у such that
Vii,i2eW, F(xi)nV С F{x2) + l\\xi - x2\\BY
We can also adapt the definitions of upper and lower limits to the
case of a set-valued map F : X ~»· Y. For that purpose, the notation
x1 -+p χ means that x' remains in Dom(.F) and converges to x.
Definition 1.4.6 When F : Χ ~> Υ is a set-valued map, we say
that
IAmsu-ox,^xF{x') := \y £ Υ \ liminf d(y,F(x')) = θ)
is the upper limit of F(x') when x' —> χ and that
Um.mix,^xF{x') := \y <Ξ Υ | lim d(y,F(x')) = θ]
is the lower limit of F(x') when x' —> x.
They are obviously closed. We also see at once that
Liminix/^xF(x') С F(x) С Limsnpx,^xF(x')
Remark — The use of the concept of filter would avoid duplicating
these definitions in the discrete and continuous cases. We have preferred
this longer, hopefully more pedagogical, presentation. D
Remark — As in the case of sequences of sets, we can easily obtain
the following characterization of upper and lower limits. Set К := Dom(F).
Then
Limsupa^I^a/) = Γ\η>0{]χ>ζΒκ(χ,η)Ρ(χ')
= Πε>ο ί\>ο ϋχ>ζΒκ(χ,η) B(F(x'), ε)

42
1 - Continuity of Set-Valued Maps
and
Lmnnix^xF(x') = Π U Π B{F{x'),e) □
ε>0 7?>0χ'ςβκ(χ,7?)
The connections between semi-continuity of set-valued maps and
semi limits are given by
Proposition 1.4.7 A point (x, y) belongs to the closure of the graph
of α set-valued map F : Χ ~> Υ if and only if
у € Limsup3.,_t3.F(a;')
and F is lower semicontinuous at χ £ Dom(-F) if and only if
F(x) С Liminf-c'-^-F^')
Then we can measure the lack of closedness (of the graph) or the
lack of lower semicontinuity by the discrepancy between the sets
F(x), L]mmix>^>xF(x') and Limsup3./_t3.ir(a;/)
Remark — This proposition led several authors to call upper semi-
continuous maps the ones which are closed in our terminology. Naturally,
these two concepts coincide for compact-valued maps, since Theorem 1.1.4
can be easily adapted to the case of set-valued maps:
Proposition 1.4.8 The graph of an upper semicontinuous set-valued
map F : X ~> У with closed domain and closed values is closed.
The converse is true if we assume that Υ is compact.
More generally, the "continuous version" of Proposition 1.1.5
reads:
Proposition 1.4.9 Let F and G be two set-valued maps from X to
Y. Assume that F is closed, that G{x) is compact and that G is
upper semicontinuous at χ G Dom(F Π G). Then F Π G is upper
semicontinuous at χ.
The above proposition provides an easy way to construct upper semi-
continuous set-valued maps, by "cutting" set-valued maps with closed graph
by set-valued maps with ball values, the radius of which are upper semi-
continuous (real-valued) functions:

1.4. Continuity of Set-Valued maps
43
Corollary 1.4.10 Let F : X ^У be α closed set-valued map and
r:X U R+
he an upper semicontinuous function. If the dimension of Υ is finite , then
the cut set-valued map Fr : Χ ~> Υ defined by
Fr(x):=F(x)Dr(x)B
is upper semicontinuous.
This is due to the fact that χ ~> r{x)B is upper semicontinuous
(respectively lower semicontinuous, Lipschitz) whenever r is upper semicontinuous
(respectively lower semicontinuous, Lipschitz.)
Remark — We shall prove later (Theorem 2.2.5) that the set-valued
analogues of continuous linear operators, called closed convex processes,
from a Banach space to another are also upper semicontinuous (and even,
Lipschitz): this is an extension of Banach's Closed Graph Theorem. □
We can extend the concept of proper maps to set-valued maps in the
following way:
Definition 1.4.11 (Proper Set-Valued Map) Let us consider a closed
set-valued map F : X ~> Y. The two following properties are obviously
equivalent:
1. — The projection πγ : Graph(F) иУ is proper
2. — If a sequence yn £ F{xn) converges in Υ, then the sequence
(xn)neN has a cluster point.
If they are satisfied, we shall say that the set-valued map F : X -^У is
proper.
When F is proper, the images F{K) of closed subsets К С X are closed
and the inverse images F~l(M) of compact subsets Μ С Υ are compact.
Indeed, we can write
i) F{K) = πγ (Graph(F) n(Kx Y))
ii) F-X{M) = έχ (Graph^nTiy^M))
In particular, the image of a proper set-valued map is closed.

44
1 - Continuity of Set-Valued Maps
Proposition 1.4.12 Let us assume that X is locally compact and that for
every compact subset К С X, the graph of the restriction F\k ofF : X ^>Y
to К is compact. Then
1. — F is upper semicontinuous
2. — F~l is proper (and thus, its domain is closed)
Proof— Since for all a; ε Dom(F), there exists a compact neighbor-
-hood К of x, the restriction F\k is upper semicontinuous on K, having a
compact graph. Then F is upper semicontinuous at x.
Let a sequence
xn € F~l{yn) с Dom(F)
converge to some x. We have to check that the sequence yn has a cluster
point. Since the convergent sequence (£n)neN remains in a compact
subset K, and since the pair (хп,Уп) belongs to the graph of F\k, which is
compact, the sequence (xn, yn) has a cluster point of the form [x, y), which
belongs to the graph of F. Hence у £ F~1(x) is a cluster point of (j/n)^eN-
D
1.4.2 Generic Continuity
Although the notions of upper semicontinuity and lower semicontinuity are
distinct, they generically coincide, i.e., a semicontinuous map is continuous
on a residual.
Recall that a residual of a metric space X is a countable intersection
of dense open subsets An с X. Countable intersections of residuals are
residuals. Baire's Theorem9, states that a residual of a complete metric
space is dense. A property which is true for every element of a residual is
called generic.
Theorem 1.4.13 (Generic Continuity) Let F be a set-valued map from
a complete metric space X to a complete separable metric space Υ.
1. — If F is upper semicontinuous, it is continuous on a residual
ofX.
2. — If F is lower semicontinuous with compact values, it is
continuous on a residual of X.
3. — If F is lower semicontinuous with closed values, then there
exists a residual R of X such that
Viefi, Limsupa.^iV) = F(x)
proved in 1899...

1.4. Continuity of Set-Valued maps
45
Proof— Since У is a separable metric space, there exists a countable
family of open subsets Vn С Υ, stable by finite union satisfying the following
property:
V open subset V С Υ, V у ε V, 3 V„ such that у ε V£ с V
1. — Assume first that F is upper semicontinuous. We associate
with any open subset Vn the subset
Ln := F-X(K) = {x£X \ F(x)n%^4>}
which is closed by Proposition 1.4.4 since the Vn are closed.
We then note that if χ ε X is such that
V ra e N, x&Ln => xe Int(Ln)
then F is lower semicontinuous at x. Indeed, let V С Υ be any open subset
such that F{x) Π V φ 0. Then there exists η ε Ν and
ν ε f(i)ni; с F(a;)nv
Thus χ € Ln and therefore it belongs to Int(jLn.) Since
Ln = F-ЦЩ С F-lQ>)
we deduce that F is lower semicontinuous at x.
Therefore the subset Db of points where F is not lower semicontinuous
is contained in the countable union of the subsets dLn, that are closed
subsets with empty interior.
By taking the complements, we infer that the complement of Db, which
is the set of elements at which F is continuous, contains a residual.
2. — If F is lower semicontinuous, then the subsets
Kn := F+1(K) = {xeX \ F{x) cK}
are closed by Proposition 1.4.4. If χ ε Χ is such that
VneN, x<=Kn => χ ε Ы(Кп)
F is upper semicontinuous at x. Indeed, let an open set V С У be such
that F(x) с V. For every у ε F(x), let V„u be such that у ε Vny С V.
Since F has compact images, F{x) can be covered by a finite number of
sets Vnj ··= Vny., j = 1, ···, m. Thus
τη - - τη
F{x) С (J Щ := (J Vnj С V

46
1 - Continuity of Set-Valued Maps
The family of Vn being stable by finite union, we deduce that for somen,
χ € F+l (pVnjj = Kn С F+X(V)
Thus, a; £ Int(F+1(V) and we have proved that F is upper semicontinuous
at x.
Therefore the subset D$ of points where F is not upper semicontinuous
is contained in the countable union of the subsets dKn := Kn\ Int(.fiTn).
The interiors of these subsets are empty because they are closed.
By taking the complements, we deduce that the complement of D",
which is the set of elements at which F is continuous, contains a residual.
3. — We follow Choquet's proof to deduce the third statement. It
is based on the fact that any complete separable metric space Υ is homeo-
morphic to a subset Zq of a compact metric space Z.
Denote by φ the homeomorphism from Υ onto Zq, and set
G0 := φ ο F : X ~» Z0 k G(x) := G0(x)
the closure of Gq{x) in Z.
Since F is lower semicontinuous, so is Go and G. By the preceding
statement, there exists a residual R of X such that G is upper
semicontinuous from R to Z. Therefore, for any ε > 0, there exists η such that, for
any χ' ε Β(χ,η),
G0(x') С G(x') С B(G(x),e) = B(G0(x),e)
On the other hand, Go(a;) = G(x) Π Zq since the images Go(a;) are closed
in Zq for the induced topology. Therefore
MxGR, Limsupx,_xGo(a/) = Go(a;)
Since ψ is an homeomorphism, F enjoys the same property. □
Remark — It is easy to construct a lower semicontinuous set-valued
map F (with closed but noncompact values) which is nowhere upper semi-
continuous: Take X = R, Υ = R2 and F(t) := {{x, y)\y = tx}. Π
1.4.3 Example: Parametrized Set-Valued Maps
Let us consider three metric spaces Χ, Υ and Z, a set-valued map
U:X ~> Ζ

1.4. Continuity of Set-Valued maps
47
and a single-valued map
/ : Graph (17) ■-» Υ
We associate with these data the set-valued map F : Χ ~> Υ defined
by
Vx€X, F{x) := (f(x, u))ueu{x)
Proposition 1.4.14 Assume that f is continuous from Graph(!7)
toY.
— If U is lower semicontinuous, so is F.
— If U is upper semicontinuous with compact values, so is F.
Proof — Let us consider a sequence xn £ Oom(F) converging
to χ € Dom(-F) and take у := f(x,u) belonging to F(x), where
и €U(x). Since U is lower semicontinuous, there exists a sequence
un G U(xn) converging to u. Then the sequence yn = f(xn,un),
where yn belongs to F(xn), converges to у because / is continuous.
Hence F is lower semicontinuous.
Let us fix χ € Dom(l7), e > 0 and consider the neighborhood
B(F(x), e) of F(x). Since it is a neighborhood of each f(x, u) when
и € U(χ), the continuity of / implies the existence of щ > 0 and
6U > 0 such that
V (x1, v) £ Graph([/) П (B(x, η») χ B{u, 6U)), f{x', v) € B{F(x), e)
The subset U(x) being compact, it can be covered by ρ balls В(щ, 6щ).
Since U is upper semicontinuous, there exists 770 > 0 such that
ρ
Vx' € Β(χ,η0), U(x') С \jB(Ui,6Ui)
i=l
We take η := min(77o,mini=i)...)P щ{) > 0. Then we infer that
Vs'__e Β(χ,ή), F{x') с B(F(x),e)
i.e., that F is upper semicontinuous at x. □

48
1- Continuity of Set-Valued Maps
1.4.4 Marginal Maps
Definition 1.4.15 (Marginal Functions) Let us consider α set-
valued map F : Χ ~> Υ and a function f : Graph(F) i-> R. We
associate with them the marginal function jiIhRu {+°o} defined
by
g{x) := sup f(x,y)
yeF(x)
Theorem 1.4.16 (Maximum Theorem) Let metric spaces X, Y,
a set-valued map F : Χ ~> Υ and a function f : Graph(F) i-> R be
given.
— If f and F are lower semicontinuous, so is the marginal
function.
— // / and F are upper semicontinuous and if the values of
F are compact, so is the marginal function.
Proof — Let us" consider a sequence xn converging to x, fix
λ < g(x) and choose у € F(x) such that λ < f(x,y). Then there
exist elements yn G F(xn) converging to у (because F is lower semi-
continuous) and we know that f{xn,yn) < g{%n)· Since / is lower
semicontinuous, we infer that
λ < f(x,y) < liminf/(a;n,yn) < liminf^(a;n)
η—юо τι—юо
By letting λ converge to g(x), the claim ensues.
For proving the second statement, pick χ € X and fix e > 0.
Since / is upper semicontinuous, we can associate with any у G F(x)
open neighborhoods V(y) of у and Uy{x) of χ such that
Vue Uy{x) and ν £V(y), f{u,v) < f{x,y) + e (1.4)
Since F(x) is compact, it can be covered by η neighborhoods V(yi),
i = 1, · · · ,p, the union of which makes up a neighborhood of F(x).
Then there exists a neighborhood Uq{x) such that
Vx'€ U0(x), F(x') С \JV(yi)
i=l

1.5. Lower Semi-Continuity Criteria
49
because F is upper semicontinuous. By taking и in the neighborhood
ρ
U{x) := UQ{x) П f]UVi{x)
we observe that
\fueU(x), Vv£F(u), f(u,v)< sup f{x,yi)+e < g{x)+e
i=l,---,p
(thanks to (1.4)) and we deduce that
\fu£U(x), g{u) < g{x) + e □
We will use quite often the following corollary·.
Corollary 1.4.17 // α set-valued map F is lower semicontinuous
(resp. upper semicontinuous with compact values), then the function
(x,y) ι—» d(y,F(x)) is upper semicontinuous (resp. lower
semicontinuous) on Dom(-F).
1.5 Lower Semi-Continuity Criteria
We provide in this section lower semicontinuity criteria, some of them
being adaptations of results on lower limits of sequences to the
continuous case.
Proposition 1.5.1 Consider a metric space X, two normed spaces
Υ and Z, two set-valued maps G and F from X to Υ and Ζ
respectively, and a (single-valued) map f from Χ χ Ζ to Υ satisfying the
following assumptions:
. i) G and F are lower semicontinuous with convex values
< ii) f is continuous
Hi) V χ £ X, и η-► f(x, и) is affine
We posit the following condition:
Vx £X, 37>0,£>0, c>0, r>0 such that V x' £ B(x, δ)
we have
ΊΒΥ С f{x',F{x')nrBz)-G{x')

50
1 — Continuity of Set-Valued Maps
Then the set-valued map R:X~^Z defined by
R(x) :=-{u€F(x) | f(x,u)eG(x)} (1.5)
is lower semicontinuous with nonempty convex values.
The proof is a trivial adaptation of the one of Proposition 1.2.6.
We state now another condition which is less symmetric.
Proposition 1.5.2 Consider a metric space X, two normed spaces
Υ and Z, two set-valued maps G and F from X to Υ and Ζ
respectively and a (single-valued) map f from X x Ζ to Υ such that
' i) F is lower semicontmuous with convex values
ii) f is continuous
< Hi) V χ G X, u i-> f(x, u) is affine
iv) V χ G X, G{x) is convex and its interior is nonempty'
v) the graph of the map X э χ ~» Int(G(a;)) is open
We posit the following condition:
Vx eX, 3u€ F(x) such that f(x, и) е Iht(G(s)) (1.6)
Then the set-valued map R defined by (1.5) is lower semicontinuous
with convex values.
Proof
1. — We introduce the set-valued map S : Χ ~> Ζ defined by
S(x) ={u£ F(x) | f(x,u) e Int(G(s))} С R{x)
Assumption (1.6) implies that S(x) is not empty. We claim that
S is lower semicontinuous. Indeed, if xn —> χ and if и belongs to
S(x) С F(x), there exists a sequence of elements un G F(xn) which
converges to и because F is lower semicontinuous. Since
{xn-,f{xn-,un)) converges to (x, f(x, u)) € Graph(Int(G(·)))
by continuity of / and since the graph of Int(G(·)) is open, the
elements f(xn-,un) belong to Int(G(xn)) for η large enough and thus,
the elements un belong to S(xn) and converge to u.

1.5. Lower Semi-Continuity Criteria
51
2. — Convexity of F(x) and G(x) implies that S(x) = R(x).
Indeed, let us fix и € R{x) and ад € 5(ж). Then г/g := Quo + (l — 6)u
belongs to S(x) when θ e]0,1[, because C?(:e) is convex and f(x,uo)
belongs to the interior of G(x), so that for every θ e]0, l[,
f{x, u) + 0(/(s, u0) - f{x, u)) = f(x, у + вуо- ву) = f(x, υθ)
belongs to the interior of G(x). Then и is the limit of vq when θ > 0
converges to 0.
3. — The theorem follows because the closure of any lower
semicontinuous set-valued map is still lower semicontinuous. Π
We extend now the above lower semicontinuity criterion to
infinite intersection of set-valued maps.
Theorem 1.5.3 Let us consider α metric space X, normed vector-
spaces Υ and Ζ and set-valued maps F : XxY ~> Ζ and Η : X ~> Y.
We assume that
i) F is lower semicontinuous with convex values
ii) Η is upper semicontinuous with compact values
and that there exist positive constants 7, δ, с such that for every single-
valued map е:Ун jB we have
Vx'e Β(χ,δ), сВП П (F(x',y)-e(y)) φ 0 (1.7)
вея(х')
Then the set-valued map G : Χ ~> Ζ defined by
Vx€l, G{x) := f) F(x,y)
y£H(x)
is lower semicontinuous (with nonempty convex images.)
Remark — When the set-valued map F is locally bounded (in
the sense that it maps some neighborhood of each point to a bounded
subset), we do not need the constant с and we can replace (1.7) by
\/χ'€Β(χ,δ), Π (F(x',y)-e(y)) φ 0 D
yetf(x')

52
1 - Continuity of Set-Valued Maps
Proof— Let us choose any sequence of elements xn € Dom(-F)
converging to χ and ζ € G(x). We have to approximate ζ by elements
zn eG(xn).
We introduce the following numbers:
en := sup d(z,F(xn,y))/2 (1.8)
yeH(xn)
Now, let us choose for each у £ H(xn) an element un(y) €
F(xn,y) satisfying
ll«-«n(y)ll < Щг,Р(рп,у)) < en
and set θη := 7/(7 + en). Consequently,
6n(«-«n(y)) € 6„enS = {1-θη)ΊΒ
So that there exists ап(у) € 7В such that
θη(ζ - un(y)) = (1 - вп)оп(у)
Therefore, assumption (1.7) implies the existence for all η large
enough of elements wn G cB and elements vn(y) € -Ρ(2;π, у) such that
an{y) = vn{y) ~ Wn for all у € Я(яп).
Hence we can write
0n(z - Un(y)) = (1 - 6n)(vn(y) - wn)
So that the common value:
zn := θηζ + (1 - ^η)ωπ = 9nun(y) + (1 - 6n)vn{y)
does not depend on y, belongs to all F(xn,y) (by convexity),
converges to ζ because
\\z-zn\\ = (l-en)\\z-wn\\ < (1-θη)(\\ζ\\+α)
and, because 1 — θη = en/(-y + en) converges to 0 for en, converges
to 0 thanks to the following lemma. □
Lemma 1.5.4 Let us assume that F is lower semicontinuous and
that Η is upper semicontinuous with compact images. Then the
numbers en defined by (1.8) converge to 0.

1.5. Lower Semi-Continuity Criteria
53
Proof— Since F is lower semicontinuous, Corollary 1.4.17 to
the Maximum Theorem implies that the function
(x,y,z) н-> d(z,F{x,y))
is upper semicontinuous. Therefore, for any e > 0 and any у £ Η {χ),
there exist an integer Ny and a neighborhood Vy of у such that
Vy'eVj,, Vn>Ny, d{z,F(xn,y')) < ε (1.9)
because d(z,F(x,y)) = 0. Hence the compact set H{x) can be
covered by ρ neighborhoods Vyi. Furthermore, Η being upper
semicontinuous, there exists an integer iVo such that,
Vn>iV0, H(xn) с (J Vy.
i=l, ...,p
Set N := тахг-=о,...,р NVi. Then, for all η > N and у G Я(жТ1), у
belongs to some Vyi, so that, by (1.9), d(z,F(xn,y)) < ε. Thus,
Vn>iV, en := sup d(z,F(xn,y))/2 < e/2
y£H{xn)
i.e., our lemma is proved. □
For set-valued maps with non convex images, we deduce from
Theorem 1.2.9 its continuous version:
Theorem 1.5.5 Let G : Χ ~> Ζ be α closed lower semicontinuous
set-valued map from a metric space X to a Banach space Ζ and
f : X xY н-> Ζ a continuous (single-valued) map, where Υ is another
Banach space. Let us assume that f is differentiable with respect to
у and that there exist constants с > 0 and η > 0 such that
' Vi£B(i0,i)), je%,i?), 2£B(/(a:o,5o),i))nG(i)
<
Bz С cfy(x,y)(BY)-TG[x)(z)
(1.10)
Then the set-valued map R defined by
R(x) := {y€ Υ | f(x,y)€G(x)}
is lower semicontinuous at xq-

Chapter 2
Closed Convex Processes
Introduction
Naturally, the first question which arises is: What are the set-
valued analogues of continuous linear operators!
Since the graph of a continuous linear operator A £ £(X, Y) is
a (closed) vector subspace of Χ χ Y, it is quite natural to regard
set-valued maps, with closed convex cones as their graphs, as these
set-valued analogues. Such set-valued maps are called closed convex
processes1 and the maps the graph of which are vector subspaces are
called linear processes.
The main class of examples of closed convex processes is provided
by derivatives of set-valued maps which are introduced and studied
exhaustively in Chapter 5.
We shall prove that closed convex processes enjoy (almost) all
properties of continuous linear operators, including Banach's Open
Mapping and Closed Graph Theorems (Section 2) and the Uniform
Boundedness Theorem (Section 3.)
As continuous linear operators, closed convex processes can be
transposed and the Bipolar Theorem can be adapted to closed convex
processes. They thus enjoy the benefits of a duality theory exposed
in Section 5. For instance, a duality criterion of invariance by a
'The term "process" has been coined by R.T. Rockafellar for denoting maps
the graph of which are cones in a study of economic "processes" (with constant
return to scale.)
55

56
2- Closed Convex Processes
closed convex process is given in this section for linear processes and
in Chapter 4 (Section 2) in the general case.
We shall also provide in Chapter 3 a theorem on existence of
eigenvectors of closed convex processes (Theorem 3.6.2.)
For proving these results, we recall in Section 4 the Bipolar
Theorem for continuous linear operators, the Closed Range Theorem and
the properties of support functions. The latter allow characterization
of closed convex subsets by an (infinite) family of linear inequalities
thanks to a version of the Hahn-Banach Separation Theorem. It also
enables one to deal with the class of upper semicontinuous convex
positively homogeneous functions instead of handling closed convex
subsets.
We conclude the chapter with a short section on upper hemicon-
tinuous set-valued maps, characterized by the upper semicontinuity
of the support functions of their values, a more familiar property.
This characterization is mainly useful for set-valued maps with closed
convex images.
2.1 Definitions
Let us introduce the set-valued analogues of continuous linear
operators, which are the closed convex processes.
Definition 2.1.1 (Closed Convex Process) Let F : Χ ~> Υ be
a set-valued map from a normed space X to a normed space Y. We
shall say that F is
— convex if its graph is convex
— closed if its graph is closed
— a process (or positively homogeneous,) if its graph is a cone
— a linear process if its graph is a vector subspace.
Hence a closed convex process is a set-valued map whose graph
is a closed convex cone.
We shall see that most of the properties of continuous linear
operators are enjoyed by closed convex processes.
Let us begin with the following obvious statements.

2.2. Open Mapping and Closed Graph
57
Lemma 2.1.2 A set-valued map F is convex if and only if
' Vsi, X2 € Dom(F), VA e [0,1],
<
AF(xi) + (1 - AjFCsa) С F(Asi + (1 - A)s2)
It is a process if and only if
\/x e X, V λ > 0, AF(s) = F(As) and 0 e -F(O)
and a convex process if and only if it is a process satisfying
Vsi, ж2 € X, FCsiJ + F^) С F(x1+x2)
We observe that the domain and the image of a closed convex
process are convex cones (not necessarily closed.)
The main examples of closed processes are provided by contingent
derivatives of set-valued maps that we shall introduce in Chapter 5.
We associate with a closed convex process its norm defined in the
following way.
Definition 2.1.3 (Norm of a Closed Convex Process) Let F :
X~> Υ be a closed convex process. Its norm ||.F|| is equal to
' \\F\\ ··= supieDom(F)d(0,F(a;))/||a;||
= suPxeDom(F)mW(x)IM|/IM| (2.1)
= suPxeDom(F)nB 'miv eF{x) 11ν11
2.2 Open Mapping and Closed Graph
Theorems
The Banach Open Mapping Theorem can be extended to closed
convex processes:
Theorem 2.2.1 (Open Mapping) Let Χ, Υ be Banach spaces.
Assume that a closed convex process F : Χ ~> Υ is surjective (in the
sense that Jm(F) =Y). Then -F-1 is Lipschitz:

58
2- Closed Convex Processes
There exists a constant I > 0 such that, for all x\ G F 1(yi) and
for any ?/2 € У, we can find a solution x^ G F~1(y2J. satisfying:
\\xi -ж2|| < l\\yi — 2/21|
Actually, this theorem holds true for closed convex maps:
Theorem 2.2.2 (Robinson-Ursescu) LetX, Υ be Banach spaces,
F : Χ ~> Υ be a closed convex set-valued map. Suppose that yo
belongs to the interior of the image of F and let xq G F~1(yo).
Then there exist positive constants I and 7 such that for any у G
г/о + 7-В, there exists a solution χ to the inclusion F(x) Э у satisfying
\\x-x0\\ < l\\y-yo\\
Proof — For simplicity, we prove this theorem only when the
Banach space X is reflexive2.
Let us introduce the function ρ defined by
p(y) := inf ||s-soil = dix^F-^y)) (2.2)
xeF-Цу)
(It takes the value +00 when у does not belong to Im(F).)
Since the set-valued map F is convex, this function is obviously
convex. Assume for a while it is lower semicontinuous.
Then, Baire's Theorem implies that ρ is continuous3 on the
interior of Im(F), which is not empty by assumption. Since continuous
convex functions are locally Lipschitz, there exists a ball of radius
7 > 0 centered at yo and a constant I' > 0 such that for all у in this
ball,
llp(y)ll = llp(y)-p(yo)|| < %-уо||
2See for instance [35, Theorem 3.3.1] for the nonreflexive case and the original
papers of Robinson and Ursescu.
3Let us recall this result: The domain of ρ is the union of the sections
Sn := {y I p(y) < n)
which are closed because ρ is lower semicontinuous. Since the interior of the
domain of ρ is not empty, the interior of one of these sections is not empty. So,
the convex function ρ being bounded on an open subset Int(5n) for some n, it is
continuous (and even, locally Lipschitz) on the interior of its domain.

2.2. Open Mapping and Closed Graph
59
because p(yo) = 0. Therefore
d^F-'iy)) < i'||y-yo||
By setting I = 21' we end the proof. It remains to check that the
function ρ is lower semicontinuous. Π
Lemma 2.2.3 Let us consider α reflexive Banach space X, a Ba-
nach space Y, a closed convex process F : X ~> У'.
Then the function ρ defined by (2.2) is lower semicontinuous.
Proof — It is enough to prove that nonempty sections
{y I p(y) < λ}
are closed. Let us consider a sequence of elements yn of such a
section converging to some y. Then, by reflexivity of X, there exists
Xn € F 1(yn) satisfying || Xn -CO II = р(Уп) < λ. Hence the elements
xn remain in the ball xq + \B. Since X is reflexive, there exists a
(weak) cluster point χ of the sequence xn. Then (x, y) is a weak
cluster point of the sequence (хп,Уп), which thus belongs to the
graph of F because, being closed and convex, it is closed in Χ χ Υ
when X is supplied with the weak topology and Υ with the norm
topology. Since the elements xn belong to the ball xq + \B, which is
weakly closed, the cluster point χ belongs also to this ball, so that
p{y) < \\x — xo\\ < λ Π
Proof of Theorem 2.2.1 — We take xq = 0, yo = 0 in
Theorem 2.2.2. To say that 0 belongs to the interior of the cone
Im(F) amounts to saying that Im(F) is equal to Y, i.e., that F is
surjective. By Theorem 2.2.2 for a constant I > 0 we have:
Vy € У, 3x € F~1(y) such that ||x|| < 2||y||
Fix ?/i, г/2 € Υ and let us choose any solution x\ € F~1{y{) and
e e F~1(y2 - 2/1) satisfying ||e|| < l\\yi— 2/2II· Then X2 := xi + e
belongs to i?_1(2/2), since F is a convex process and satisfies the
estimate
||zi-S2|| = ||e|| < i||yi-y2|| °

60
2- Closed Convex Processes
Corollary 2.2.4 Let us consider Banach spaces X, Y, a continuous
linear operator A G C(X, Y) and a_ closed convex subset К С X. Let
us assume that there exists xq G К such that Axq G Int(A(K)).
Then there exist positive constants I and j such that for any у G
A(xq) + 7-B, there exists a- solution χ £ К to the equation Ax = у
satisfying \\x — xq\\ < l\\y — A(xq)\\.
- —*--
Remark — Actually, Corollary 2.2.4 is equivalent to
Theorem 2.2.2; we apply it with К := Graph(F) and A := πγ. □
Since the restriction F := Α\κ of a continuous linear operator
A G C(X, Y) to a closed convex cone is a closed convex process, we
obtain the following consequence:
Corollary 2.2.5 Let us consider Banach spaces X, Y, a
continuous linear operator A G C(X, Y) and a closed convex cone К С X
such that A(K) = Y. Then the set-valued map у ~> A~1{y) Π Κ is
Lipschitz: there exists a positive constant I such that,
Vyi, 2/2 € Y, А-1{У1)ПК С А-1{у2)ПК + i||yi-y2||£
As in the case of continuous linear operators, the Open Mapping
Theorem is equivalent to the Closed Graph Theorem, which can be
stated as follows.
Theorem 2.2.6 (Closed Graph Theorem) A closed convex
process F from a Banach space X to another Υ whose domain is the
whole space is Lipschitz: there exists a (Lipschitz) constant I > 0
such that
Vxu x2 GX, F(xi) С F(x2) + i||si-s2||-B (2.3)
Thus, the norm of F is finite whenever Dom(F) = X.
Proof — It is sufficient to apply the Open Mapping
Theorem 2.2.1 to the closed convex process -F-1. □

2.3. Uniform Boundedness
61
2.3 Uniform Boundedness Theorem
We can now adapt the Uniform Boundedness Theorem to the case
of closed convex processes.
Theorem 2.3.1 (Uniform Boundedness) Let X and Υ be Ba-
nach spaces and Fh be a family of closed convex processes from X to
Y, "pointwise bounded" in the sense that
V χ € X, 3yh € Fh(x) such that вирЦу^Ц < +oo (2.4)
h
Then this family is "uniformly bounded" in the sense that
sup 11iX11 < +oo
h
Hence we can speak of bounded families of closed convex
processes, without specifying whether it is pointwise or uniform.
Proof — Let us consider the positively homogeneous convex
lower semicontinuous functions ph defined by
ph{x) := inf \\y\\ = d(0,Fh(x))
j/eFh(x)
(which are lower semicontinuous because the closed convex processes
Fh are Lipschitz) and the function ρ defined by
У χ £ Χ, ρ(χ) := supphfa)
h
Assumption (2.4) implies that this function ρ is finite. Since it is also
positively homogeneous, convex and lower semicontinuous (being the
supremum of such functions), it is continuous at 0. Hence there
exists a constant I such that saphd(0,Fh(x)) = p(x) < l\\x\\, i.e.,
\\Fh\\ < l< oo. □
The following consequence of Theorem 2.3.1 extends to closed
convex processes the following useful convergence result.
Theorem 2.3.2 (Crossed Convergence) Consider a metric space
U, Banach spaces Χ, Υ and a set-valued map associating to each

62
2- Closed Convex Processes
и £ U a closed convex process F(u) : X ~> Y. Let us assume that
the family of closed convex processes F(u) is pointwise bounded.
Then the following conditions are equivalent:'
i) the map и ~> Graph(.F(ii)) is lower semicontinuous
<
ii) the map (u,x) ~> F{u)(x) is lower semicontinuous
Proof — For proving that г) implies ii), let us consider a
sequence of elements (un, xn) converging to (u, x) and an element
у € F(u)(x). We have to approximate it by elements yn £F(un)(xn).
Since и ~> Gr&ph(F(u)) is lower semicontinuous, we can
approximate (x, y) by elements (хп,Уп) £ Graph(-F(un)). By the pointwise
boundedness assumption and Theorem 2.3.1, there exist I > 0 and
solutions fn € F{un){xn — xn) satisfying
||/π|| Ъ. ΙΊ\Χη 2·π||
The right hand side of the above inequality converges to zero when
η goes to infinity. Because F(un) is a convex process, the element
Уп '■= Уп + fn does belong to F(un)(xn). Consequently, yn converging
to y, we deduce that the set-valued map (u, x) ~> F(u)(x) is lower
semicontinuous at (u, x).
The converse is obviously true even when the family (F(u))ueu
is unbounded. Π
2.4 The Bipolar Theorem
Definition 2.4.1 Let К be a nonempty subset of a Banach space
X. We associate with any continuous linear form ρ £ X*
σκ{ρ) '■= σ(Κ,ρ) := sup < ρ, χ> € RU{+oo}
хек
The function σκ ■ X* nRU {+oo} is called the support function of
K. Its domain is a convex cone called the barrier cone denoted by
b{K) := Όοτη(σκ) := {реГ| σκ{ρ) < oo} (2.5)

2.4. The Bipolar Theorem
63
We say that the subsets of X* defined by
' i) K°
ii) Κ-
ίη) K+
iv) K1
= {p & X*\ σκ(ρ) < 1}
= {ρ & X*\ σκ{ρ) < 0}
= -κ-
= {ρ € Χ* I V χ € К, < ρ, χ > = 0}
are the polar set, (negative) polar cone, positive polar cone and
orthogonal of К respectively.
When L С I*, we define the polar set L° С X as the subset of
elements χ £ X (and not X**) satisfying < ρ, χ >< 1 for all ρ G L.
The polar cone L~ С X and the orthogonal L1 С X of L are defined
in the same way. The subsets
K°° := {K°)° С X к К— := {К')' С X
are called respectively the bipolar set and bipolar cone of a subset
К С X and the subspace K±J- := {К1)1- С X the biorthogonal of
jr.
It is clear that K° is a closed convex subset containing 0, that
K~ is a closed convex cone, that AT is a closed subspace of X* and
that
K1 = К~ПК+ С K~ С K° С b(K)
Examples
о When К = {χ}, then σκ(ρ) — <p,x>
© When К = Дх, then σΒχ (ρ) = ||ρ||*
® If AT is a cone, then
, . / 0 if ?€Г
σ*ω = ( +oo if pi К' D
When AT = 0, we set σ%(ρ) = — oo for every ρ G X*.
The Separation Theorem can be stated in the following way:

64
2- Closed Convex-Processes
Theorem 2.4.2 (Separation theorem) Let К be a nonempty
subset of a Banach space X. Its closed convex hull is characterized by
linear constraint inequalities in the following way:
co(K) ={ieI|Vj)6l*, <ji,i>< σκ{ρ)}
Furthermore, there is a bijective correspondance between nonempty
closed convex subsets of X and nontrivial lower semicontinuous
positively homogeneous convex functions on X*.
Remark — The Separation Theorem holds true not only in Banach
spaces, but in any Hausdorff locally convex topological vector-space. In
particular, we can use it when X is supplied with the weakened topology.
The geometrical interpretation can be stated as follows: the closed
convex hull of a nonempty subset is the intersection of all closed half-spaces
containing it. □
We observe that a subset К is bounded if and only if its support
function is finite.
We mention the following consequence, known as the Bipolar
theorem.
Theorem 2.4.3 (Bipolar Theorem) Let X,Y be Banach spaces
and К С X. The bipolar cone К is the closed convex cone
spanned by K.
If A G C(X, Y) is a continuous linear operator from X to Υ and
К is a subset of X, then
(A(k))- = a*-\k-)
where A* denotes the transpose of A.
Thus the closed cone spanned by A(K) is equal to (A* {K~~)j .
We provide now a simple criterion which implies that the image
of a closed subset is closed.
Theorem 2.4.4 (Closed Range Theorem) Let X be a Banach
space, Υ be a reflexive space, К С X be a weakly closed subset and
A G C(X, Y) a continuous linear operator satisfying
lm(A*) + b{K) = X*
(2.6)

2.4. The Bipolar Theorem
65
Then the image A(K) is closed. In particular, if К is a closed convex
cone and if
bn(A*) + K- = X*
then
A(K) = (a*-\k-))~
Proof — Let us consider a sequence of elements xn G К such
that A(xn) converges to some у in Y. We shall check that this
sequence is weakly bounded, and thus, weakly relatively compact.
Let us take for that purpose any ρ € X*, which can, by assumption
(2.6), be written ρ := A*q + r, where q<=Y*, r G b(K). Therefore,
supn < p,xn > = supn(< q, Axn > + < r,xn >)
<
< supn(||g||||.Aa:n|| + aK{r)) < +oo
since the converging sequence (Axn)nej^ is bounded.
Therefore the sequence (£n)neN has a weak cluster point χ which
belongs to К since it is weakly closed. D
Remark — Actually, arguments of the above proof imply that,
under assumptions of the Closed Range Theorem, every sequence
xn € К such that A(xn) is weakly converging, has a weak cluster
point □
For the convenience of the reader, we list below some useful
calculus of support functions and barrier cones4.
4 See [35, Chapter 3] for instance.

66 2- Closed Convex Processes
Table 2.1: Properties of Support Functions.
> If К С L С X, then 6(L) С b(K) and σ^ < aL
> If AT; С X, i £ I, then
№^))сае/Р«)
o"(co((Jie/ Ki)iP) = suPie/ °X-Cp)
> Ιΐ KiCXi, (i = 1,..., n), then
ΚΠ?=ι^) = Π?=ιΚ^)
*(Π?=ι *i> (Pi. ■ ■ ■. Pn)) = Σ?=ι ^Х-Ы
> If A e ДА", У), then
6(Α(ϋΤ)) = Α*_16(ϋΤ)
°аЩ)^ = σκ{Α*Ρ)
> If Κι and Й2 are contained in X, then
6(ΛΓι + K2) = b(Ki) П 6(ϋΤ2)
σκ1+κ2 (ρ) = σΚι (ρ) + σκ2 {ρ)
In particular, if К С X and P is a cone, then
b(K + P) = b(K) П Ρ- and
σκ+ρ(ρ) = σχ(ρ) if ρ £ P~ and +oo if not
(5) > If L С X and Μ С У are closed convex subsets and
A G C{X, Y) is a continuous linear operator such that
the following constraint qualification condition
0 € Int(M - A(L)) holds true, then
6(L П A-1 (M)) = b(L) + A*b(M) and
VpeifLnA-^M)). 3 5 e У* such that
οί,οα-ΜλοΟρ) =<tl(p- A*q) + σΜ{Ά)
= miqeY*(aL(p - A*q) + σΜ{θ))
(5)a) > If Μ С У is a closed convex subset and if
A € C{X, У) is a continuous linear operator such that
0 € Int(Im(A) - M), then ^^(M)) = A*b{M)
and, for every ρ £ 6(Л_1(М)), there exists g € 6(M)
such that σΛ-ι(Μ)(ρ) = σΜ(?) = ΐηίΑ*9=ρ(σΜ(?))
(5)b) > If K\ and ΛΓ2 are closed convex subsets of X such that
0 € Int(UTi - ЯГ2), then 6(JTi η K2) = b{K{) + b{K2)
and V ρ € b(Ki Π ΑΓ2), 3pP,(i= 1,2) such that
с^пАГгСр) = σΚι{θΊ) + VK2{qi)
= να.ΐρ=ρι+ρ2{σΚλ{Ρι) + σκ2{Ρ2))
(1)
(2)
(3)
(4)a)
(4)b)

2.5. Transposition
67
2.5 Transposition of Closed Convex Process
Definition 2.5.1 (Transpose of a Process) Let X,Y be Banach
spaces, F : Χ ~> Υ be a process. Its left-transpose (in short, its
transpose,) F* is the closed convex process from Y* to X* defined by
ρ € F*(q) <==> У χ £ X, У у £ F(x), <p,x>~< < q,y >
(2.7)
In particular, the transpose F* of a linear process F is defined
by
p £ F*(q) <{=> Ух £ X, У у £ F(x), <p,x> = < q,y >
The graph of the transpose F* of F is related to the polar cone
of the graph of F in the following way:
Lemma 2.5.2 (Graph of the Transpose) Let us consider Banach
spaces Χ, Υ and let F : X ~> У be a process. Then
(q,p) € Graph(F*) «=!►(?,-?) € (Graph(F))"
In the case of linear processes, we observe that ρ G F*(q) if and
only if (p,—q) belongs to Graph(F)-1- and we see at once that the
bitranspose of a closed linear process F coincides with F.
The definition of a bitranspose of a convex process is not
symmetric: If G : Y* ~> X* is a convex process, we define its transpose
G* : Χ ~> Υ by the formula
(-y,x) € (Graph(G))-
(instead of the formula (y, —x) € (Graph(G))" obtained by
exchanging the roles of X and Υ*, Υ and X* respectively.)
With this definition, the bitranspose of a closed convex process F
coincides with F.
We provide now a formula for transposing the product of closed
convex processes.

68
2- Closed Convex Processes
Theorem 2.5.3 (Transpose of a Product) Let W, Χ, Υ, Ζ be Ba-
nach spaces, F be a closed convex process from X to Υ', A G C{W, X)
and В G C(Y, Z) be continuous linear operators. Assume that
lm(A)-Oom(F) = X (2.8)
Then the transpose of BFA is equal to:
(BFA)* = A*F*B*
Proof — First, we prove that the formula
(BF)* = F*B*
always holds true, since the graph oiBF is equal to (1 xi?)Graph(.F).
Consequently, thanks to the Bipolar Theorem 2.4.3
((1 χ B)Graph(F))~ = (1 x B)*'1 (Graph(F)-)~
so that (p, —q) belongs to Graph(i?.F)"~ if and only if (p, —B*q)
belongs to Graph(.F)"~, i.e., if and only if ρ belongs to F*{B*q).
The graph of FA being equal to (A x l)_1Graph(.F), we need to
assume the constraint qualification property
Im(A x 1) - Graph(F) = Χ χ Υ (2.9)
for deducing from the properties of polar cones that
(Graph(FA))- = (A* x 1) (Graph(F)-)
If this is the case, we infer that r G (FA)*(q) if and only if there
exists ρ € F*(q) such that r = A*p.
It remains now to check that assumption (2.8) implies the
constraint qualification property (2.9.)
Indeed, let (x, y) belong to Χ χ Υ. Since χ can be expressed as
χ = Axq — χι where xq belongs to W and x\ to the domain of F, we
can write
(x,y) = (AxQ,yQ)- (xi,yi)
where y\ G F(x\) and yo = у + y\. Hence (χ,у) belongs to Im(A χ
l)-Graph(F).
We then deduce from the above proof the conclusion of the
theorem. □

2.5. Transposition
69
Corollary 2.5.4 Let X,Y and Ζ be Banach spaces, F a closed
convex process from X to Z, G a closed convex process from Υ to Ζ and
A£C(X,Y). If
A(Oom{F))-Oom(G) = Υ (2.10)
then {F + GA)* = F* + A*G*.
Proof— We set
H(x) := F(x) + G{Ax), B(y,z) := у + ζ
so that the set-valued map Η can be written Η = B(F χ G)(l χ A).
Since Dom(.F χ G) is equal to Dom(.F) χ Dom(G), assumption (2.10)
implies that
Im(l χ A) - Oom(F xG) = XxY
Therefore from Theorem 2.5.3 follows that H* = (1 χ A)*(F χ G)*B*.
Since
{F χ G)* = F* xG* к (1 χ A)* = 1 + A*
we infer that H* = F* + A*G*. Π
Corollary 2.5.5 (Transpose of the Restriction) LetX,Y be
Banach spaces, F : Χ ~» Υ be a closed convex process and К С X be a
closed convex cone. Assume that
K-Oom{F) = X
Then the transpose of the restriction F\k of F to К is given by
(F\k) (q) - | 0 otherwise
Proof— We apply Corollary 2.5.4 with A = 1 and G defined
by
G(x) = [ {0} if X € K
10 otherwise
whose domain is К and whose transpose is the constant set-valued
map denned by G*{q) =K~. Π
We shall adapt to the case of closed convex processes the Bipolar
Theorem 2.4.3. To this end, we begin by stating the following simple
result.

70
2- Closed Convex Processes
Proposition 2.5.6 Let X,Y be Banach spaces and F : X ~> У be
a process. Then
(Im(F))- = -F*~\0) к F(0) = (Dom(F*))+
Therefore, if F is a convex process, the image of F is dense if and
only if the kernel F* (0) of its transpose is equal to 0.
Furthermore a convex process F is surjective if F* (0) = {0}
and either the dimension of Υ is finite or the image of F is closed.
Proof — To say that q belongs to the polar cone of the image
of F amounts to saying that the pair (0, q) belongs to the polar
cone of the graph of F, i.e., that (—q, 0) belongs to the graph of the
transpose F*, in other words, that 0 belongs to F*(—q). The proof
of the second statement is naturally analogous. □
The extension of the Bipolar Theorem 2.4.3 to closed convex
processes is then a consequence of Corollary 2.5.5 and Proposition 2.5.6.
Theorem 2.5.7 (Bipolar Theorem) Consider Banach spaces Χ, Υ
and let F : Χ ~> Υ be a closed convex process, and К С X be a cone
satisfying Dom(.F) — К = X. Then
(F{K)y = -F*'1 (K+)
Proof — We apply Proposition 2.5.6 to the restriction F\k
whose image is F(K) and whose transpose is F*(·) + K~ thanks to
Corollary 2.5.5. D
The above condition is obviously satisfied when the domain of F
is the whole space. In this case, we obtain
Corollary 2.5.8 Let F : Χ ~> Υ be a strict closed convex process.
Then
Dom(F*) = F(0)+
and F* is upper hemicontinuous (see Definition 2.6.2 below) with
bounded closed convex images, mapping the unit ball to the ball of
radius \\F\\. In particular, F*(0) = {0}.
The restriction of F* to the vector space
Oom(F*) П (Dom(F*))
is single-valued and linear.

2.5. Transposition
71
Proof — We observe that
V? € Dom(F*), sup ||p|| < ||F||||g||
p£F*(q)
because for all χ € Dom(.F) = X and for all ρ G F*(q), we have, by-
definitions of the transpose and the norm of a closed convex process
\\p\\ := suPxeB <P,X> < supieB miyeF{x) <q,y>
<
< supxejB infj,ejF(x) ||e||||y|| = ||ЛНЫ1
Then F* maps bounded sets to bounded sets. Therefore, the cone
F*(0) being bounded, is equal to {0}. Since the domain of F is the
whole space, the assumptions of Proposition 2.6.4 below are met, so
that for all χ £ X, the function q i-> a(F*(q),x) is upper semicon-
tinuous.
The domain of F* is closed, thanks to the Closed Range
Theorem 2.4.4 applied to the projection πγ* from Χ* χ Υ* and the cone
Graph(F*). Then
Dom(F*) = πγ* (Graph(F*))
is closed because, the domain of F being equal to X,
1т((тгк*)*) - (Graph(F*))~ = X xY
This and Proposition 2.5.6 imply that Dom(F*) = F(0)+.
If q belongs to both Dom(.F*) and -Dom(.F*), then
F*(q) + F*{-q) С F*(0) = {0}
Hence F*(q) contains only one element and F*(q) = —F*(—q). D
Example: Case of linear processes
When F : Χ ~> Υ is a linear process from a Banach space X to
another Y, then, from the very definition of the adjoint, it follows
that F* is also a linear process from Y* into X*. Hence its domain
is a subspace of Y* and, by Proposition 2.5.6, Dom(.F*) = F^)1.
In general the space Dom(.F*) is not closed. However this is
always the case when Υ has a finite dimension.

72
2- Closed Convex Processes
Furthermore Corollary 2.5.8 implies that when F is strict, then
Dom(P*) = F(0)±
and F* is a linear operator from the subspace Ρ(Ο)1· into X*.
When X = Y, a closed subspace Ρ С Dom(P) is called invariant
under F if F{P) С Р. We have the following
Proposition 2.5.9 Assume that Dom(P*) is closed. If a closed sub-
space Ρ is invariant under F, then its orthogonal space P± is
invariant under F*.
Consequently, if both Dom(P) and Dom(P*) are closed, then a
closed subspace Ρ is invariant under F, if and only if its orthogonal
space P^ is invariant under F*.
Proof— Let Ρ С Dom(P) be an invariant closed subspace. Then
P(0) С Р and therefore
P± С P(0)± = Dom(P*)±± = Dom(P*)
Fixing q € P1 and ρ € F*(q), for every χ € Ρ, у & F(x) С Р we
have
<p,x> = < q,y > = 0
Thus ρ € Ρ1- and we have proved that if a subspace Ρ is
invariant under F, then its orthogonal P-1 is invariant under the adjoint
process F*. Since F = F** the last statement follows. Π
For every χ £ X define recursively
F1{x) = F{x) and for every integer η > 1, Fn+1{x) = F(Fn(x))
Then for every n, Fn(0) is a subspace of X and
Pn(0) С Pn+1(0)
Consider the subspace
Q = U ^п(°)
π>1

2.5. Transposition
73
Clearly F(Q) С Q. If F is strict, then it is Lipschitz on X. Thus
F(Q) С Q and therefore in this case Q is the smallest closed subspace
of X invariant under F. We also have the following implication
Fk(0) = Fk+1(0) ==» VreeN, Fk(0) = Fk+n(0) (2.11)
In particular this yields that when X = Rn, then Q = Fn(0). Indeed
in this case {Fk(0)}k>i is a nondecreasing sequence of subspaces of
Rn satisfying (2.11.) The dimension of Rn being equal to re, the last
claim follows.
Thus .Fn(0) is the smallest subspace of Rn invariant under F.
О
The sum of two closed convex processes or the product BF is not
necessarily closed. We have to provide sufficient conditions implying
that they are still closed.
Proposition 2.5.10 (Closed Graph Criterion) Let Χ, Υ and Ζ
be reflexive Banach spaces, F : X ~> Υ and G : Χ ~> Ζ be closed
convex processes and В G C(Y, Z) be a continuous linear operator.
If
B*(Dom(G*)) - Oom(F*) = Y*
then the convex process BF + G is closed.
Proof — This is a consequence of the Closed Range
Theorem 2.4.4 with К := Graph(.F χ G) and the continuous linear
operator A defined by A(x, y, z) := (x, By + ζ). Π
Proposition 2.5.11 Let X,Y be Banach spaces, G : Χ ~> Υ be a
closed convex process and Ρ С X and Q С Υ be closed convex cones.
Let us consider the convex process F defined by
- .pM ._ ί G{x) + Q ifxeP
[ > - \ 0 if χ i Ρ
It is closed when we suppose that
Dom(G*) + Q- = У*
If we assume that Dom(G) — Ρ = X, then its transpose is given by
F*(n) ._ ί G*(q)+P- ifq € Q+
{q) " 10 ifqi Q+

74
2- Closed Convex Processes
Proof — We observe that F is the sum of the closed convex
process G and the closed convex process Η defined by H{x) := Q if
χ € Ρ and 0 if χ £ Ρ, whose domain is Ρ and whose transpose is
defined by H*(q) := P~ if q € Q+ and 0 if not. Corollary 2.5.4 ends
the proof. □
2.6 Upper Hemicontimious Maps
We associate with a set-valued map F from a metric space X to a
normed space Υ the family of functions
χ >-^ a(F(x),p) := sup <p,y>
yeF{x)
indexed by the continuous linear functionals ρ £Y*.
We observe that
Corollary 2.6.1 // a set-valued map F from a metric space X to
a normed space Υ is weakly upper semicontinuous and has compact
values (resp. lower semicontinuous), then the function
(x,q) € XxY* ^ a{F{x),q)
is upper semicontinuous (resp. lower semicontinuous).
Therefore, it is quite convenient to introduce the following
definition.
Definition 2.6.2 (Upper Hemicontinuous Map) We shall say
that a set-valued map F : Χ ~> Υ is upper hemicontinuous at xq G
Dom(.F) if and only if for any ρ G Y*, the function χ н-> a(F(x),p)
is upper semicontinuous at xq.
It is said to be upper hemicontinuous if and only if it is upper
hemicontinuous at every point ofOom(F).
Proposition 2.6.3 The graph of an upper hemicontinuous set-valued
map with closed convex values is closed.
Proof— Let us consider elements (хп,Уп) € Graph(-F)
converging to a pair (x,y). Then, for every ρ €Υ*,
<p,y>= lim <p,yn>< limsupa(F(xn),p) < a(F(x),p)
η~>0° 71-ЮО

2.6. Upper Hemicontinuous Maps
75
by the upper semicontinuity ofiM a(F(x),p). This inequality
implies that у € F(x) since these subsets are closed and convex, thanks
to the Separation Theorem 2.4.2. We have shown that (x, y) belongs
to Graph(.F), which ends the proof. D
It is useful to compare the support functions of F(x) and F*(q).
Proposition 2.6.4 (Support Function of the Transpose) Let
Χ, Υ be Banach spaces and F : Χ ~> Υ be a closed convex
process. Then for every χ G X and q £ Y*
a(F*(q),x) + a(F(x),-q) < 0
Furthermore for every xQ in the interior of the domain of F there
exists po € F*(q0) such that < pQ, x0 > is equal to the common value
a{F*(q0),x0) = -а{Р{хо),-до) (2.12)
In the same way, if qQ belongs to the interior of the domain of F*,
there exists y0 € F(x0) such that < q0,y0 > is equal to the common
value (2.12.)
Therefore for every χ € Dom{F), the function q н-> a(F*(q),x)
is upper semicontinuous on the interior of the domain of F* and for
every q £ Dom{F*) the function χ н-> a(F(x), -q) is upper
semicontinuous on the interior of the domain of F.
Proof— The first claim follows from the very definition of the
support function.
Denoting by π χ the projector from Χ χ Υ to X, we can write
a(F(x0),qo) = a(GT&ph(F)mrx1(x0),(0>qo))
We apply the formula (5) of Table 2.1 on the support functions of
an intersection and inverse image with L := Graph(.F), Μ := {xq}
and A := πχ. The constraint qualification assumption is satisfied
because 0 belongs to the interior of
KX(GT&ph(F)) - x0 = Dom(F) - x0
by assumption. Then
a(F(x0),qo) = M{<p,xQ> +a(Graph(F),(0,g0)-7Γ^(ρ)))

76
2- Closed Convex Processes
We observe that πχ(ρ) = (ρ,Ο) and that
a(Graph(F),(-Pi?o)) = 0_
if and only if ρ € i?*(^o)) so that (2.12) ensues. Since the function
—a(F(xo), ■) is upper semicontinuous, the proof of the last claim
ensues. □

Chapter 3
Existence and Stability of
quilihrium
Introduction
This chapter is devoted to the major questions of nonlinear
functional analysis: provide criteria allowing to solve1 a (nonlinear)
equation or an inclusion (with or without constraints) and, also,
approximate their solutions and study their stability.
We tried to present the shortest introduction to this so important
field by choosing only the theorems we judge the most powerful, i.e.,
the theorems from which we obtain the most results with the least
efforts.
Historically, nonlinear analysis is based on two fixed point
theorems:
» Banach-Picard Successive Approximation Theorem for
problems set on complete metric spaces
β Brouwer Fixed Point Theorem in a convex compact
environment
Using these tools amounts to transforming the problem under
investigation to a fixed point problem, for which one can apply one of
1 Apparently Cauchy was the first to prove an existence theorem, instead of
assuming a priori that the solution exists and compute it, risking in this way to
obtain paradoxes.
77

78
3- Existence ала Stability of ал Equilibrium
these fixed point theorems or their numerous variations or extensions.
The "fixed point" format being quite rigid, one encounters the risk of
more or less considerable loss of information by doing so. It also often
happens that this transformation may require additional assumptions
and useless technical difficulties.
This is the reason why so many statements logically equivalent
to the Brouwer Fixed Point Theorem — constituting the corpus of
nonlinear analysis — have been designed to be readily adapted to
classes of specific problems.
Still, among these equivalent results, one can look for those
statements which, in some sense, incorporate more labor-value2, and
therefore, might be more useful.
This is the case of the Ky Fan and Ekeland Theorems, which
provide less elegant but more efficient statements than the corresponding
Fixed Point Theorems.
These two more flexible and powerful theorems, equivalent to
each of the above Fixed Point Theorems, appeared at the beginning
of the seventies:
β The Ky Fan Inequality, in 1972
β Ekeland's Variational Principle, in 1974.
We thus begin by deducing Ky Fan's Inequality from the Brouwer
Fixed Point Theorem in the first section, and use it to derive in
Section 2 not only the traditional fixed point theorems (including
the Kakutani-Fan Fixed Point Theorem for set-valued maps), but
also a Constrained Equilibrium Theorem, providing an equilibrium
2Although two statements V and Q may well be equivalent, it is common
experience that the proof of one of the implications, say V => Q, is more difficult
or involves deeper results than the proof of the converse. In this case, one can
say that Q incorporates more labor value than V and thus, expect as a general
rule that the statement Q may be more useful than V.
One could still follow Aristotle's observation that most theorems of geometry
have valid converses, so that one can go leisurely from V => Q (called analysis)
to Q =>■ V. However, more serious greek mathematicians, conscious of the
pitfalls, used to add supplementary conditions (called diorismoi.) The reverse
inference was called synthesis.
By the way, the most pleasant or intuitive statements are quite often the ones
with the least labor value.

3.0. Introduction
79
ж of a set-valued map F (i.e., a solution to the inclusion 0 € F(x)
satisfying the constraints described by χ € К.) Besides standard
mild conditions on F, we shall assume that К is a convex compact
viability domain of F; this means that we can find at every point χ
of the compact convex subset К an element ν € F (x) tangent to К
at x.
We then prove Ekeland's Variational Principle in Section 3, a
result we shall use extensively in this book, above all for establishing
the calculus of tangent cones, of derivatives of set-valued maps and
epiderivatives of extended functions in the forthcoming chapters.
But we start to apply it in Section 4 to extend Graves' Inverse
Function Theorem to constrained problems of the form:
К being a closed subset describing the constraints,
look for a solution xq G К to the equation f(xo) = yo-
In the unconstrained case, Graves' Theorem states that if / is C1
at xq and if f'(xo) is surjective, then the equation f(x) = у still has
a solution for any right-hand side у close enough to f(xo), and the
set-valued map /-1 behaves in a Lipschitz way.
When constraints requiring that χ belongs to К are present, we
"differentiate" them by using the contingent cone Тк{х) to К at x,
which is a most reasonable way to translate the concept of tangency
for an arbitrary subset. Roughly speaking, if /'(·) is continuous at
xq, if Τκ{·) is lower semicontinuous at xq and if /'{хо)(Тк(хо)) is
the whole space, then the equation f(x) = у has a solution χ € К
for any у in a neighborhood of f(xo).
This stability property, often regarded as an existence theorem of
a solution to nonlinear problems, is actually a particular case of an
approximation procedure, and conceals the famous Lax Principle:
{Consistency and Stability imply Convergence
for approximating solutions to given problems,
which underlies numerical functional analysis. The constrained
problem is approximated by the following constrained problems:
find xn € Kn satisfying fn{xn) = Уп

80 3- Existence and Stability of an Equilibrium
Consistency means that in some sense, fn converges to / and Kn
to K. Stability means that linear equations
find un € TKn{xn) satisfying f^(xn)un = vn
have solutions satisfying the stability condition ||ιιπ|| < c||wn|| and
Convergence means that the convergence of the right-hand sides yn
to ?/o implies the convergence of approximate solutions xn to xq.
These theorems will be extended to inclusions as soon as we have
defined the derivatives of set-valued maps in Chapter 5.
We describe in Section 5 the basic results on monotone and
maximal monotone maps, which share many properties with continuous
linear operators. Maximal monotone maps can be approximated by
single-valued Lipschitz maps (Yosida approximations), a feature that
is very useful3.
We provide in Section 6 a result on existence of eigenvalues and
eigenvectors of a closed convex process.
3.1 Ky Fan's Inequality
After what we said in the introduction, we state and prove the Ky
Fan4 inequality.
Theorem 3.1.1 (Ky Ean Inequality) Let К be a compact convex
subset of a Banach space5 X and φ : Χ χ X \—t R ie α function
satisfying
i) У у € К, х 1-^><р(х,у) is lower semicontinuous
< ii) Ух £ К, г/(-► (^(ж,?/) is concave (3.1)
^ in) У у € К, ip{y,y) < 0
Then, there exists χ G K, a solution to
У у € Κ, φ{χ,ν) < 0 (3.2)
3We shall use it in Chapter 9 to construct regular selections of set-valued maps.
4A student of Prechet. He visited Paris for one year in 1939, with a small grant
and the map of Paris' metro. He then had to survive there during the whole war.
5Actually, the proof we give shows that this theorem remains true for any
Hausdorff locally convex topological vector space and in particular, when the
Banach space X is endowed with the weak topology.

3.1. Ky Fan's Inequality-
Si
Proof — We shall prove it first in the finite-dimensional case,
from which we shall derive the proof in Banach spaces.
— The finite-dimensional case. We shall derive a
contradiction from the negation of the conclusion:
V χ £ K, 3y € К such that φ(χ, у) > О
so that К can be covered by the subsets
Vy := {x € K\ φ(χ, у) > 0}
which are open by assumption (3.1) г). Since К is compact, it can
be covered by η such open subsets Vyi. Let us consider a continuous
partition of unity6 (скг)г=1,...,п associated with this open covering of
К and define the map / : К н-► X by
η
Ух £ К, f(x) := Y^ai(x)yi
г=1
It maps К to itself because К is convex and the elements yi belong
to K. It is also continuous, so that Brouwer's Fixed Point Theorem
implies the existence of a fixed point у = f(y) G К of /. Assumption
(3.1) ii) imply that
π π
г=1 г=1
Let us introduce
I(V) ■= {» = l,...,n| a{(y) > 0}
It is not empty because Σά-ι щ{у) = 1. Furthermore
η
Y^ai(y)<p(y,yi) = Σ ац(у)<р(у,у{) > 0
г=1 iel(y)
6 A continuous partition of unity associated with a covering of К by η open
Subsets Vi is a sequence of η continuous maps са : К н-> R such that,
η
Vie Jf, ^аг(ж) = 1, Vi = l,...,n, m(x) > 0 & support(ai) С Vi
Such continuous partitions of unity do exist when К is a compact metric space.

82 3- Existence and Stability of an Equilibrium
because, whenever г belongs to I(y), щ{у) > 0, so that у belongs
to Vyv and thus, by the very definition of this subset, if{y,yi) > 0.
Hence, we have proved that ip(y, y) is strictly positive, a contradiction
of assumption (3.1) Hi).
— The infinite-dimensional case. Let us introduce the family
S of finite subsets Μ := {у ι,... ,ym} С К of К and the number
υ := sup inf max (p{x,yi)
This number is nonpositive. Indeed, the function φΜ (λ, μ) defined on
Sm χ Sm (where the simplex Sm is a subset of elements (Ab ..., Xm) ε Rm
such that Xi > 0 and Υ^χ λί = 1) by
m η
ψΜ{μ,Χ) ■■= Σ)μί¥Έλί2/ί>2/.ί)
ί=1 ί=1
satisfies the assumption of Ky Fan's Theorem in finite dimension (it is
obviously lower semicontinuous with respect to λ and linear with respect
to μ, and ψΜ(μ,μ) < 0 because φ is concave with respect to у and by
assumption (3.1) iii). Hence there exists λ £ Sm such that
m
У μ £ 5m, J2^M^Vj) = ΨΜβ,μ) < 0
where we set χ := ]Γ^ι λΐί/ΐ e co(M) С К. We then deduce that
τη
vM ■■= inf max <p(x,yj) < max ^(ϊ,^·) < sup У^МФ&Уз) ^ °
xeKyjeM yjEM ues™rr
J = l
and thus, that ν = supMeS vm is nonpositive.
It remains now to prove the existence of ж £ if such that
sup φ(χ, у) < ν
уек
This results from the following
Lemma 3.1.2 Let К С X be α compact subset of a topological space X, L
be any subset and φ : Κ χ L н-> R a function such that χ н-> φ(χ, у) is lower
semicontinuous for every у £ L. Let S be the family of finite subsets of L.
Then there exists χ £ К satisfying
sup</j(5, y) < υ := sup inf max φ(χ, yj)
yeL MesxeKvoEM

3.2. Equilibrium and Fixed Point Theorems
83
Proof — We introduce the subsets
Sy := {χ ε Κ Ι ψ{χ,ν) < ν)
They are nonempty closed subsets of the compact К because </?(·, у) is lower
semicontinuous. The very definition of υ implies that these subsets enjoy
the finite intersection property: indeed, consider η such subsets Sy. and set
Μ := {г/ι,. · · ,yn}· The compactness of К and the lower semicontinuity of
</?(·, y) implies the existence of an element xM ε Κ minimizing the function
тах^емУ^Уг)) which then satisfies
max φίχΜ,νί) = inf max φ(χ,уЛ < ν
ViEM xeKyiEM
so that it belongs to Γ)Γ=ι ■%/; ■
Hence there exists some χ in their nonempty intersection f]yeK Sy,
which is a solution to the desired inequality. D
3.2 Equilibrium and Fixed Point Theorems
3.2.1 The Equilibrium Theorem
We are ready to derive from the Ky Fan inequality the Constrained
Equilibrium Theorem. It provides sufficient conditions for an upper
semicontinuous set-valued map F from a Banach space X to closed
convex subsets of X to have an equilibrium satisfying constraints,
i.e., a solution Τ to the inclusion 0 € F(x) required to belong to a
given closed subset K.
Such a set К defining the constraints will be assumed to be at
least convex and compact, properties which are usually demanded
for any existence theorem derived from the Brouwer Fixed Point
Theorem7.
But we have to provide a further assumption which links the set
К with the set-valued map F: we shall require that К be a viability
domain of the set-valued map F. We recall that the tangent cone
Tk(x) to a convex subset К at χ € К is the closed cone spanned by
Κ — χ, which is convex:
im*)= UK~X
h>o h
7up to the reduction to this case, using coerciveness to trade-off with
compactness, as we shall see at the end of this section, for instance.

84
3- Existence and Stability of art Equilibrium
(see Section 2 of Chapter 4 for a detailed exposition of tangent cones
to convex subsets.)
A subset К С Dom(F) is said to be a viability domain of F if
and only if
Vx€K, F{x)nTK(x) φ 0
This means that for any point χ € AT, there exists at least a
direction υ € F(x) which is tangent to К at x. We shall see in
Chapter 10 that this means also that from any initial state xq € K,
there exists a solution x(-) to the differential inclusion
x' € F(x), x(0) = x0
which is viable in К in the sense that x(t) € К for any t > 0.
Theorem 3.2.1 (Equilibrium Theorem) Assume thatX is a'Ba-
nach space8 and that F : X ~> X is an upper hemicontinuous set-
valued map with closed convex images.
If К С X is a convex compact viability domain of F, then it
contains an equilibrium of F.
Proof — We proceed by contradiction, assuming that the
conclusion is false. Hence, for any χ € К, 0 does not belong to F(x).
Since the images of F are closed and convex, the Hahn-Banach
Separation Theorem implies that there exists px € X* such that
a{F{x),px)<0.
By setting
Vp := {x €K\ a{F{x),p) < 0}
the negation of the existence of an equilibrium of F in К implies that
К can be covered by the subsets Vp when ρ ranges over the dual of X.
These subsets are open by the very definition of upper hemicontinuity
of F. So К can be covered by η such open subsets VPi. Let us
consider a continuous partition of unity (αί)ί=ι,...,π associated with
8 Actually, the proof we give shows that this fundamental theorem remains
true for any HausdorfF locally convex topological vector space and in particular
for a Banach space endowed with the weak topology.

3.2. Equilibrium and Fixed Point Theorems
85
this finite open covering and introduce the function φ : Κ χ Κ —* R
defined by
η
Ч>{х,У) ■= Y^Oii{x)<Pi,x-y>
i=l
Being continuous with respect to χ and affine with respect to y, the
assumptions of Ky Fan's Inequality (Theorem 3.1.1) are satisfied.
Hence there exists χ € К such that for ρ ·.= Σ?=1 oii(x)pi we have
Vy€K, φ{χ,ν) = <p,x-y > < 0
The above inequality means that —p belongs to the polar cone Τκ{χ)~
of the convex subset К at x.
Since К is a viability domain of F, there exists υ € F(~x) Γ\Τκ(χ),
and thus
a(F{x),p) > <p,v > > 0
We set
I(x) := {i = l,...,n\ oti(x) > 0}
which is not empty. Hence
a(F(x),p)< Σ <*i@MF@),Pi) < 0
because, for any i € I(x), щ{х) > 0, and thus, χ belongs to the
subset VPi, which means precisely that a(F(x),pi) < 0. The latter
inequality is then a contradiction of the previous one. □
By modifying slightly the proof of the Equilibrium Theorem, we can
prove the existence of zeros of a set-valued map from a Banach space X to
another Banach space Y.
Theorem 3.2.2 (Solvability Theorem) Let К be α convex compact
subset of a Banach space X and F be an upper hemicontinuous set-valued map
with closed convex values from К to another Banach space Y.
Let us consider also a continuous map В : К н-> £(X, Υ). If К, F and
В are related by the condition
Vx(=K, F(x) П B(x)TK(x) Φ 0
then
(i) 3 χ ε Κ such that 0 ε F{x)
ii) My ε Κ, 3χ ε Κ such that B(x)y ε B(x)x + F(x)

86 3- Existence ала Stability of ал Equilibrium
Proof — The proof of the existence of ал equilibrium χ £ К of F
is the same as the one of the Equilibrium Theorem, where we define the
function φ by
η
Ч>(х,У) ·"= Σ<*ί{χ) <Pi,B(x)(x-y) >
Ky Fan's Inequality thus implies the existence of χ € К such that
-B{x)*p € TK(x)~
so that, taking a sequence un £ Τχ{χ) such that B(x)un converges to some
ν £ F(x)nB(x)TK{x)
(which exists by the tangential condition), we infer that
f a(F(x),p)
\ > <p,v> = limn-^oo < P, B{x)un > = limn-^oo < B(x)*p,un > > 0
This inequality is contradicted as in the proof of Theorem 3.2.1.
Take now у € if and introduce the set-valued map G : X-^Y defined
by
G{x) := F(x) + B{x)(x - y)
which also satisfies the assumptions of our theorem. Then there exists a
zero χ £ К of G, which is a solution to the inclusion B(x)y £ B(x)x+F(x).
Π
3.2.2 Fixed Point Theorems
These theorems are equivalent to the Brouwer Fixed Point Theorem;
we shall prove here only this equivalence9.
We begin by showing that Theorem 3.2.1 implies the Kakutani
Fixed Point Theorem10, which is the set-valued version of the Brouwer
9See [45, Appendix B] for a proof of the Brouwer Fixed Point Theorem
based on Sperner's Lemma and [35, Chapter II] for a proof based on
differential geometry.
10called Ky Fan's Fixed Point Theorem in infinite dimensional spaces.
The story began in 1910 with the Brouwer Fixed Point Theorem, which was
proved later in 1926 via the Three Polish Lemma, the three Poles being Knaster,
Kuratowski and Mazurkiewicz. Knaster saw the connection between Sperner's
Lemma and the fixed point theorem, Mazurkiewicz provided a proof corrected
by Kuratowski. The extension to Banach spaces was proved in 1930 by their
colleague Schauder.
Von Neumann did need the set-valued version of this Fixed Point Theorem in
game theory, which was proved by Kakutani in 1941.

3.2. Equilibrium and Fixed Point Theorems
87
Fixed Point Theorem.
Theorem 3.2.3 (Kakutani Fixed Point Theorem) Let К be а
convex compact subset of a Banach space X and G : Χ ~> Κ be an
upper hemicontinuous set-valued map with nonempty closed convex
values. Then G has a fixed point11 χ € if. Π G(x).
Proof— We set F(x) := G(x)—x, which is also upper
hemicontinuous with convex values. Since К is convex, then К — χ С Тк{х),
and since G(K) С К, we deduce that К is a viability domain of F
because
F{x) С K-x С Тк{х)
Hence there exists a viable equilibrium χ € К of F, which is a fixed
point of G. □
Actually, we do not need to assume that G maps К to itself. It is
enough to assume that if is a viability domain of F := G — 1, which can
be written in the following form
\/x<=K, G{x)n(x + TK{x)) φ 0 (3.3)
This leads to the following
Definition 3.2.4 (Inward L· Outward Maps) A map G : Κ'~> Χ
satisfying property (3.3) is said to be inward. It is called outward if
Vx&K, G(x)n(x-TK{x)) φ 0
Since if is a viability domain of F := G — 1 when G is inward and of
Л := 1 — G when G is outward, and since the equilibria of F and F_ are
fixed points of G, we obtain the useful
о
Theorem 3.2.5 Let К be a convex compact subset of a Banach space X
and G : Κ ~> Χ be an upper hemicontinuous map with nonempty closed
convex values. If G is either inward or outward, it has a fixed point
~x £ Kr\G{x)
nwhich can be regarded as an equilibrium for the discrete set-valued dynamical
system xn+i e G{xn).

88 3- Existence and Stability of an Equilibrium
Since the Kakutani Fixed Point Theorem implies obviously the Вгошгег
Fixed Point Theorem, we have proved that all these statements are
equivalent. □
Remark — We can deduce from the Viability Theorem and other
properties of differential inclusions two other results on the existence of an
equilibrium of a set-valued map F.
We shall mention them in Chapter 10. Π
We now provide a convenient sufficient condition implying that a set-
valued map G is outward. We say that
δσκ(ρ) := {x e К | <p,x>= σκ{ρ)}
is the extremal face of К at p. Observe that χ ε δσκ(ρ) if and only if
peTK(x)~.
Proposition 3.2.6 Let К be a nonempty closed convex subset of X and
let G be a set-valued map from К to К satisfying
\/p£X*,Vx£daK(p), G(x)ndaK(p) φ 0
Then G is outward. In particular, this is the case of a map G satisfying
Vpel*, G(daK(p)) с δσκ(ρ)
Proof — Let χ ε Θσκ(ρ) and
Уо е G{x)f\daK{p)
be fixed. Hence, ν = ya — χ belongs to G(x) — χ and — Тк(х) because,
Vpe TK(x)~, <p,v> = <p,y0 > - <p,x> = σκ{ρ)-σκ{ρ) =0 D
We can relax the assumption that К is compact, replacing it by a
coerciveness assumption.
Theorem 3.2.7 Let К be a closed convex subset of a finite dimensional
space and let F be an upper hemicontinuous map with nonempty compact
convex images satisfying the coerciveness assumption
limsupCT(F(a;),a;) < 0 (3.4)
ll«ll-.oo
хек
We posit the tangential condition
\fx £ К, F{x)r\TK{x) φ 0

3.2. Equilibrium and Fixed Point Theorems
89
Then there exists an equilibrium χ £ К of F. If we posit the stronger
coerciveness assumption,
a(F(x),x)
hmsup v I (' ; = -oo (3,5)
IMI—oo \\X\\
хек
Then for all у £ К, there exists a solution χ ς. Κ to the inclusion у £
x-F{x).
Proof— Coerciveness assumption (3.4) implies that there exist e > 0
and a > 0 such that
sup a(F(x),x) < -e < 0
INI>«
хек
This implies that for all a; £ if with ||ж|| < a, F(x) с Тав{х). By taking
о
the number a large enough so that Κ Π α Β^ 0, we know that
Ткпав{х) = TK(x)nTaB{x)
(see Chapter 4, Section 2.) So the tangential condition implies that
УхеКПаВ, F{x)nTKnaB{x) φ 0
It is sufficient to apply Theorem 3.2.1. To prove the second part of the
theorem, we fix у £ if and replace F by the map G defined by
G{x) = F{x) + y-x
which satisfies the coerciveness assumption (3.4) whenever F satisfies the
stronger coerciveness assumption (3.5.) Π
3.2.3 The Leray-Schauder Theorem
We can deduce the set-valued-version of the Leray-Schauder Theorem12
on the existence of stationary points from Theorem 3.2.1 by Poincare's
continuation method. We take X = Rn and К to be a compact
convex subset with a nonempty interior, so that the boundary
8K = K\ Int(UT)
of К is distinct from K.
proved in 1934,

90
3- Existence and Stability of an Equilibrium
Theorem 3.2.8 (Leray-Schauder) Let us consider α compact
convex subset К С Rn with nonempty interior and an upper hemicontin-
uous set-valued map F from [0,1] x К to Rn, with nonempty closed
convex values. Suppose the set-valued map χ —» F(0, x) satisfies the
tangential condition
\/x € dK, F(0,x)nTK(x) φ 0 (3.6)
and
Vx € dK, VA € [0,1[, 0 i F(X,x) (3.7)
Then there exists χ € К such that 0 €'F(l,x).
Proof — We suppose that the conclusion is false and derive a
contradiction. We set N := dK, a closed subset of К and introduce
the subset
Μ := {χ € K\ 3 λ € [0,1] satisfying 0 € F(\,x)}
The subset Μ is nonempty because it contains a solution χ € К to
the inclusion 0 € F(0,x), which exists by Theorem 3.2.1 (thanks to
assumption (3.6).) It is closed, since the graph of F is closed. The
intersection Μ Π Ν is empty : If χ € N and if λ € [0,1[, assumption
(3.7) implies that χ φ Μ.
Consider a continuous function ψ from К to [0,1]
( л = ^>AQ
Ψ{Χ) ' d{x,M) + d{x,N)
which is equal to zero on N and to one on Μ and define the set-valued
map G on К by
G{x) := F(<p(x),x)
Then G is clearly upper hemicontinuous with nonempty closed
convex values. It coincides with F(0, ■) on N = dK and, consequently,
satisfies the assumptions of Theorem 3.2.1. Hence, there exists a
solution χ € К to the inclusion
0 € G{x) = Ρ{ψ{χ),χ)
But this implies that χ € Μ and, therefore, ψ(χ) = 1, so that 0 €
F(l, x) which is a contradiction. □

3.3. Ekeland's Variational Principle
91
3.3 Ekeland's Variational Principle
Ekeland's Theorem, we are about to prove, provides an approximate
minimizer of a bounded from below lower semicontinuous function in
a given neighborhood of a point. This localization property is very-
useful and explains the importance of this result which has been used
extensively since its discovery in 1974.
Theorem 3.3.1 (Ekeland's Variational Principle) Let
y:lHRu{+oo}
be α lower semicontinuous nontrivial extended bounded from below
function defined on a complete metric space X. Let xq € Dom(V)
and ε > 0 be fixed. Then there exists χ € X, a solution to
(i) V{x) + ed{x0,x) < V{x0)
(3.8)
ii) Vi^i, V{x) < V(x) + ed(x,x)
Proof— We can take e := 1 and assume that V is non negative
by replacing it by V(-) — inf V if needed. We associate with V the
set-valued map F : X ~> X defined by
F(x) := {y € X | V(y) + d(x,y) < V(x)}
The subsets F(x) are closed and F is a preorder (given by у У- х Ч=>
у € F{x)) in the sense that
г) Vi e Dom(V), χ € F(x) (reflexivity)
ii) у € F{x) =^ F{y) С F(x) (transitivity)
Indeed, the second condition is obvious when χ fi Dom(V), since in
this case F(x) = X.
Assume now that V{x) is finite. By taking у € F(x), ζ € F(y),
adding inequalities
V(z) + d(y,z) < V{y) L· V(y) + d(x,y) < V(x)
and using the triangular inequality, we obtain V(z) + d(z, x) < V(x),
i.e., ζ € F(x).

92
3- Existence and Stability of an-Equilibrium
Let us define the function g on Dom(V) by
-~Vi/"€ Dom(V), 3(y) :=" inf V(z)
It is clear that
VyeF(x), d(x',y) < V{x)-g{x)
so that the diameter supy ζς.ρ{χ) Ну — z\\ °f F(x) is no^ greater than
2(V(x)-g(x)).
Define now the following sequence starting at xq: we take xn+i €
F(xn) such that
V(xn+1) < g(xn) + 2"n
Since F(xn+i) С F{xn) by transitivity, we deduce that
g(xn) < gfan+i)
On the other hand, inequality g{y) < V(y) implies that
g{xn+i) < V{xn+i) < g{xn) + 2~n < g{xn+i) + 2~~n
and thus, that
0 < V(xn+1) - g(xn+1) < 2-n
Therefore, the diameters of the closed subsets F(xn) converge to 0.
Since the sequence of these closed subsets is nonincreasing and since
the metric space is complete, we infer that
Π F(xn) = {x}
7l>0
This implies that χ € F(xo), i.e., inequality (3.8) г). We also observe
that F(x) = {x}. Indeed, χ belonging to all the subsets F(xn), then,
by transitivity, F(x) is contained in all the ir(xTl)'s, and thus, in {x}.
Consequently, when χ φ χ, then χ does not belong to F(x), i.e.,
V{x) + d{x,x) > V(x)
This is inequality (3.8) и). ■ □

3.4. Constrained Inverse Function Theorem 93
3.4 Constrained Inverse Function Theorem
3.4.1 Derivatives of Single-Valued Maps
We recall first some classical definitions and notations.
Definition 3.4.1 Let Ω be an open subset of a normed space X and
f-.П ι-+ Υ
be a single-valued map from Ω to a normed space13 Y. We denote by
Vhf(x):v - {f{x + hv)-f{x))/h
13The concepts of functional and of derivatives of functionals are due to
Volterra in 1887. Then Gateaux, in a note written in 1913 and published in
1919 after his death during the First World War, introduced the concept of first
variation
Sf{x):=\-^-f(x + X6x)
L d\ Ι λ=0
As we can see, Gateaux's definition did not involve linearity with respect to the
increment — long before our time when we are at last ready not to require linearity
anymore in order to go forward — and for that, was criticized by Hadamard
and his former student, Frechet. Mathematicians of this period still asked many
properties for the derivatives of functionals and were not ready to give them away.
But Gateaux was forgiven, and even rewarded posthumously since linearity was
introduced in the definition of the Gateaux derivative!
Frechet proposed his concept of derivative (with the mandatory linearity) for
a function as early as in 1912 in the case of functions and in 1925 for maps from
a normed space to another one.
However, he did not choose the stronger concept (called strong or strict
differentiability) derived from a definition of the derivative of a function that Peano
suggested in 1892
f{x) := Hm /fri)-/fo)
on the ground of convincing physical motivations. This track was not used until
1961 (by S. Leach to give an improved statement of the Inverse Function
Theorem.) However, Peano's point of view on how to take limits is very much the
one which underlies the concepts of Clarke tangent cones and paratingent cones
which will be studied in Chapter 4.
Since then, the history of derivatives in Banach spaces is a kind of
mathematical striptease, where one by one, and very shyly, the required properties of the
derivative of a functional are taken away. We shall go quite far in Chapter 6 to
leave the derivatives with the bare minimum.
This led to a menagerie of concepts: strong or weak Frechet and Gateaux
derivatives, Hadamard, bounded (Suchomlinov), locally uniform (Vainberg)
derivatives, directional semiderivatives or derivatives from the right, Sindalowsky

94
3-—Existence and Stability of an Equilibrium
the differential quotient of / at χ in the direction v.
If the differential quotients V/i,/(:e)(i/) converge to a limit denoted
by df(x)(v)~, then df(x)(v) is called the Dini directional derivative of
f at χ in the direction v.
If the map υ ι—» df(x)(v) is linear and continuous, we say that f
is Gateaux differentiable at χ and the continuous linear operator
f'{x) : υ н-» df{x){v) =: f'{x)v
is called the derivative of f at x. When f is real-valued, we say that
f'(x) € X* is the gradient of f at x.
If furthermore
Шп /Ы-/(я)-/'0с')О/-и) = 0
y->x \\y — x\\
then f is said to be Prechet differentiable at x. When the derivative
у ι—» f'(y) is continuous at x, f is called continuously differentiable
(or of class C1) at x.
Recall that if / is continuously differentiable, it is Prechet
differentiable and when / is Prechet differentiable, it is continuous.
Naturally, there are many other ways to take limits14 of the
differential quotients, using the weak topologies of either X or Υ or
both, taking limsup's or liminf's, etc.
3.4.2 Constrained Inverse Function Theorems
Let us consider now a Banach space X, a normed space Y, a closed
subset К С X and a continuous (single-valued) map f : К i-^Y.
or Muravev variations, and the ones we shall study in Chapter 6 and quote in the
bibliographical comments of this chapter.
We shall face a similar situation in the set-valued case. This is quite natural,
though, because each problem demands its own amount of properties that the
derivative should enjoy, i.e., its own degree of regularity. Without going too far
by always requiring minimal assumptions, some problems could not be solved
by sticking to the richest structure. The right balance between generality and
readability is naturally a subjective choice.
14too many of them having names of French mathematicians..., according at
least to some American ones.

3.4. Constrained Inverse Function Theorem
95
We shall deduce from Ekeland's Theorem a "nonlinear
extension" of Graves' Theorem 3.4.2, which allows one to solve (locally) a
"constrained problem" of the form
find χ € К such that f(x) = у
for any right-hand side у in a neighborhood of some
У0 := f(xo) € /(IT)
When К is equal to the whole space X, a solution to this
unconstrained problem is given by Graves' version of the Inverse-Function
Theorem15:
Theorem 3.4.2 (Graves' Theorem) Let Χ, Υ be Banach spaces
and f : Χ ι—> Υ a continuous (single-valued) map. We assume that
f is continuously differentiable on a neighborhood of xq and that
f'{xo) is surjective
Then the set-valued map у ~»· /-1(y) is pseudo-Lipschitz around
{f{xo),xo).
Proof — It is an obvious consequence of Theorem 3.4.10 below
with К := X. □
For constrained problems, i.e., when К is no longer the whole
space, we need to introduce right now the concept of contingent
cone, the exhaustive study of which is the purpose of Chapter 4.
The contingent cone Тк{х) to К С X at χ is the upper limit of
the subsets {K — x)/h when h —> 0+:
ν € Tk(x) if and only if liminf — , V) = 0
/i-»o+ h
Graves' Inverse-Function Theorem can be extended to our
constrained problem provided that an adequate transversality condition
holds true:
15This result is sometimes called in the literature Ljusternik's theorem.
However Graves was the first to state it in 1950,, while Ljusternik investigated in 1934
tangent spaces to level sets of a function. Both Graves and Ljusternik used in their
proofs an approximation procedure which is a standard technique of functional
analysis.

96
3- Existence and Stability of an Equilibrium
Theorem 3.4.3 (Constrained Inverse Function Theorem) Lei
X be α Banach space and Υ be a normed space. We consider a
(single-valued) continuous map f : X *-*Y, a closed subset К С X
and an element xq of K.
We assume that f is Frechet differentiable on a neighborhood of
xo and we posit the following transversality assumption:
there exist constants с > 0, a € [0, l[ and η > 0 such that
Vx € ΚΠΒ(χ0,η), Βγ С f'(x)(TK(x)ncBx) + aBY
Then f(xo) belongs to the interior of f(K) and the set-valued map
у ~» /~1(y) Π Κ is pseudo-Lipschitz around (f(xo),xo).
This theorem is a consequence of the still more general
Theorem 3.4.5 below:
Indeed, not only are we interested in knowing whether a solution
to the constrained problem does exist, but we wish to approximate
it by solutions to the approximate problems
find xn € Kn such that fn{xn) = Уп
where yn converges to yo, fn converges to / and Kn to К in the
following sense:
Definition 3.4.4 We say that the maps fn are consistent with f on
Kn at xq Ε Κ if there exist elements XQn € Kn converging to xq such
that fn{%0n) converges to f(xo).
We have to add to this consistency assumption a stability
assumption to be able to approximate the solution of the constrained
problem:
Theorem 3.4.5 Let X be a Banach space and Υ a normed space.
Consider a sequence of continuous single-valued maps fn from X to
Y, a sequence of closed subsets Kn С X and an element xq in the
lower limit of the subsets Kn such that the maps fn are consistent
with f on Kn at xq.
We assume that fn are Frechet differentiable on a neighborhood
of xq and verify the following stability assumption:

3.4. Constrained Inverse Function Theorem
97
there exist constants с > Ο, a E [0, l[ and η > 0 suc/i that
\/x € ЛГпПБ(хо,?7), -Ву С /;(х)(Г/сп(х)ПсВл:) + аВу (3.9)
JTien ί/iere exist I > 0 and 7 > 0 such that for any xqu € Вкп(хо, η)
satisfying fn{xQn) € B(f(xQ),j), we have
Vyn € J3(/(so),7)i ^(^./„"'(ynjnlfn) < i||l/n-/n(a;on)||
When the functions fn = f and the subsets ΛΓπ ξ Κ are const ant,
the conclusion of the theorem means simply that /-1 is pseudo-
Lipschitz at (f(xo), xo), so that Theorem 3.4.3 follows obviously from
it.
Before proving the above theorem we show that it implies
Theorem 1.2.9 from Chapter 1.
Proof of Theorem 1.2.9 — Let uq € /-1 ( Liminf^ooMn):
There exists a sequence of elements vqu € Mn converging to f(uo).
We shall construct a sequence of elements un € f~1(Mn) converging
to ?xo thanks to Theorem 3.4.5.
Indeed, we apply it with Χ χ Υ playing the role of X, the map
fQl:{u,v) i-> f(u) - υ
playing the role of / and fn, the sets Kn := Χ χ Mn, XQn = (uq, von)
and yn = 0. The transversality assumption of Theorem 1.2.9 implies
that for every χ € U
Βγ С f(x)(dBx)-TM71(y)ndBY
(with d = csupieW \\f'(x)\\ + 1) which can be written
By С (fei)'(x)(TKn(x)n2dBx)
Hence the transversality condition is satisfied with a = 0.
Therefore, there exist a constant I > 0 and solutions
Xn = (un,vn) € X x"Mn to%(fei){xn) = f{un)-vn = 0
satisfying
{IIEn - xon\\ = ||"o —ип|| + ||υοη — vn\\
< J||l/n-(/ei)(son)|| = l\\f(uo)-von\\

98
3- Existence and Stability of ал Equilibrium
Hence un Ε f г(Мп) and converges to щ, so that uq belongs to the
lower limit of the inverse images /~1(Μπ). □
Proof of Theorem 3.4.5 — We choose ρ > 0, ε > 0 such that
So 1-a
— < ε <
η с
and consider elements xon satisfying
||EOn-Eo|| < η/3 & ||/πΟθπ) - 2/0|| < 7?/3
and elements yn G B(fn(x0n),p).
By Ekeland's Variational Principle 3.3.1 applied to the function
V{x) := \\Vn-Ux)\\
on the complete metric space Kn, we know that there exists a solution
xn £ Kn to
0 hn - fn(£n)\\ + e\\xn - xon\\ < \\Уп - fn(xon)\\
<
ii) Vi„ e /Q, ||yn -/n(a?n)|| < ||yn-/n(a:n)|| + e||a;n-in||
(3.10)
We deduce from inequality (3.10) г) that
\\2n-xon\\ < -Ьп - fn(xon)\\ < - < ?
ε ε ο
so that
\\xn - xo\\ < η/З + \\xon - жо|| < 2η/3
Stability assumption (3.9) implies that there exist un Ε Ткп(хп) and
wn Ε Υ satisfying
г) Уп - fn{xn) = fn{xn)un + wn
H) \\un\\ < c\\yn- fn(xn)\\ & ||wn|| < a\\yn - fn(xn)\\
By definition of the contingent cone, there exist elements hp > 0
and ep Ε Χ converging to 0+ and 0 respectively such that, fn being
Prechet differentiable,
Xn "= Xn τ fipUn -f- π,ρβρ Ε li-n
fn(xn) = fn{xn) + hvUn(xn){un + ep) + e(hp))

3.4. Constrained Inverse Function Theorem
99
where e{hv) converges to 0 with hp.
By taking in inequality (3.10) it) such an xn, by observing that
yn-fn(xn) = (1 ~ hp)(yn - fn(xn)) + hpwn - hp{f'Jxn)ev + e{hv))
we deduce that
hp\\yn -/n(2n)|| < ^pll^nll +hp\\fL(xn)ep + e{hp)\\ + ehp\\un + ep\\
Dividing by hv > 0 and letting ρ —* +oo, we get:
\\yn - fn{xn)\\ < ||юп|| + е||ип|| < (" + ec)||y„ -/n(in)||
Since we have chosen e such that а + ее < 1, we infer that xn is a
solution to
xn e Kn & fn{xn) = Уп
satisfying
-X0n\\ < ~\\Уп— fn(x0n)\\
from which the error estimate follows.
Observe also that by letting e converge to c/(l — a), we obtain
the estimate
dixonJn^y^nKn) < c||y„-/n(a:on)||/(l-aO Π
Remark — This theorem extends to nonlinear constrained
problems Lax's principle that consistency and stability imply
convergence. Due to the importance of this principle in numerical analysis,
we reformulate it in this framework:
Theorem 3.4.6 Let us consider a sequence of continuous single-
valued maps fn from a Banach space X to a normed space Y, a
sequence of closed subsets Kn С X and an element
x0 £ Liminfn-H.00 ifn
such that the maps fn are consistent with f on Kn at xq.
Assume that fn are Frechet differentiable on a neighborhood of
xo and that stability assumption (3.9) holds true.

100
3- Existence and Stability of an Equilibriwx
Then there exists α constant I > 0 such that:
if yn converges to f(xo), then xq can be approximated by solutions
Xn € Kn to the equation fn{xn) — Уп satisfying the following erroi
estimate:
Given any "reference " sequence of elements xon € Kn converging
to xq such that fn{xon) converges to f(xo)i for all η large enough
\\xn - xq\\ < K\\xq - xon\\ + ||/(zo) - /η(^οη|| + \\yn - yo\\)
Remark — Case of Linear Operators
When we consider the case of continuous linear operators /n := An Ε
£{X, Y), we can restrict yn to belong to the subset An(Kn) in the proof
of Theorem 3.4.5 and thus, to relax somewhat the stability assumption.
Indeed, in this case,
2/n — Anxn £ An[Kn — xn)
When xn £ Kn, we denote by
SKn{xn) ■
the cone spanned by Kn — xn:
Theorem 3.4.7 (Inverse Stability Theorem) LetX be a Banach space
and Υ a normed space. Consider a sequence of continuous linear operators
An ε £(X, Y) and a sequence of closed subsets Kn С X such that An are
consistent with Α ε £{X, Y) on Kn at xq ε Liminfn_00i;sTn.
We posit the following stability assumption:
there exist constants с > 0, α ζ. [0,1[ and η > 0 such that
I ^ Xn
ε ΚηΠΒ{χ0,η),
{ (з.п)
( AnSKn(xn)r\BY С Ап(ТкЛхп)ПсВх) + аВу
Then there exist I > 0 and 7 > 0 such that for any x0n £ Вкп(ха,гу)
satisfying Anxon ε Β(Αχο,ί) we have
ν Уп 6 An {Κη)ΐΛΒ{Αχ0,Ί), d(xQri,A-1(yn)r\Kn) < l\\
In particular, for fixed Kn and An, we obtain:
= U
Λη Χγ,
h>0
h

3.4. Constrained Inverse Function Theorem
101
Corollary 3.4.8 (Criterion of Pseudo-Lipschitzeanity) Consider any
closed subset К of a Banach space X and let A £ C{X,Y) be a
continuous linear operator from X to a normed space Y. Assume that for some
x0 £ К, there exist constants с > 0, a £ [0,1[ and η > 0 such that
\fx £ ΚηΒ(χ0,η), ASK{x)C\BY с А(Тк(х)ПсВх) + аВу (3.12)
Then the set-valued map
A{K) Э у ~> A-1(y)nK
is pSeudo-Lipschitz around (Axq,xq).
3.4.3 Pointwise Stability Conditions
The presence of the constant a G [0, l[ in the transversality
conditions of the Inverse Function Theorem 3.4.3 is useful because it
enables its deduction from a weaker pointwise condition and a kind
of "continuity." Namely, we introduce the following definition:
Definition 3.4.9 (Sleek and Uniformly Sleek Sets) A closed
subset К is called sleek at xq G К if the cone-valued map
К Э χ -^ TK{x)
is lower semicontinuous-at xq. It is called uniformly sleek at xq € К
lim f sup d(u,TK{x))\ = 0
ϊ->ϊ° \ιι6Τ*(ζ0)ηΒ* /
We shall prove in Theorem 4.1.8 that the contingent cone Τκ{χο) ίο
a subset К sleek at xq is convex.
Theorem 3.4.10 (Pointwise Inverse Function Theorem) Let
X and Υ be Banach spaces, К be a closed subset of X and f : X i-+Y
be a continuously differentiable at xq G- К map. Assume that
f'(xo)TK(xo) = Υ
1. — If К is uniformly sleek atxQ, then f(xo) belongs to the
interior of f(K) and the set-valued map
у ~> Г1{у)пк

102 3- Existence and Stability of an Equilibrium
is pseudo- Lipschitz around (f(xo),xo)·
2. — If the dimension of Υ is finite, it is sufficient to assume"
that К is sleek at xq to obtain the same conclusion.
Proof — Replacing К by a closed neigEborhood of xq in K,
we may assume that / is continuous on K. We have to prove that
in both cases, the stability assumption is satisfied.
1. — The proof of the first case is easy. There exists a
constant с > 0 such that, for all υ in the unit sphere Sy-, there exists
a solution uq to the equation f'(xo)u = υ such that ||uo|| < c\\v\\,
thanks to Robinson-Ursescu's Theorem (because Tk(xq) is a closed
convex cone.)
Since К is uniformly sleek at xq and /'(·) is continuous at xq, we
can associate with any e > 0 an η > 0 such that, for all uq G X and
all χ G Bk(xo,v)> there exists и G Тк{х) satisfying
II"-«oil < e||u0|| & H/'(aO-/'(so)|| ^ ε
Hence any ν G Sy can be written ν = f'(x)u + w where и G
Tk{x), \\u\\ < (1 + e)c||u|| and w = f'(xo)uo — f'(x)u. Thus
IN < e||u0||(||/'(so)||+l + e) < εφ||(||/'(χο)||+ l + e)
Taking e > 0 small enough and using Theorem 3.4.3 we complete the
proof of the first statement.
2. — When the dimension of Υ is finite, the unit sphere
Sy is compact. We know from the Robinson-Ursescu Theorem that
for any Vi € Sy, there exits a solution uoi € Tk{xq) to the equation
f'(xo)uoi = iH such that ||здг|| < CIK|| = c. Fix e > 0.
Since К is sleek at xq, for any Vi, there exists щ > 0 such that,
for all χ € Βκ{χο,η), we can find щ G Тк(х) satisfying
||uj - uoj|| < е||здг||
The map /'(·) being continuous at xq, there exists щ > 0 such that
||/'(х)-/'Ы11<е.
We can cover Sy by ρ balls В(щ,е) so that, by taking
m щ
°,<p

3.4. Constrained Inverse Function Theorem
103
we obtain that for any ν € Sy and any χ € Βκ(χο,η), there exist
щ £Тк{х) and Wi eY satisfying
ν = f'(x)ui+Wi, \\щ\\ < c(l+e), Wi = v-Vi+f'(xo)u^-f'(x)ui
As above, we can show that
Nil < e(l + c||/'(so)|| + (l+e)c)
and use Theorem 3.4.3 with e small enough. □
3.4.4 Local Uniqueness
We conclude this section with a local uniqueness criterion of a
solution to a constrained nonlinear problem:
Proposition 3.4.11 Let К be α subset of α finite dimensional vector-
space X and f : Χ ι—» Υ a single-valued map to a normed space Y.
Let xq G К and assume that f is Frechet differentiable at xq .
If
Тк(х0)П ber(/'(s0)) = {0}
then xq is the only local solution χ € К to the equation f(x) = f(xo),
in the sense that there exists a neighborhood Μ{xq) ofxQ such that
\/x € λί{χ0)ηΚ, χ^χο, f{x) φ /(so)
Proof — Assume the contrary: for all η > 0, there exists
xn € К П B(xo,l/n), xn φ xo
such that J(in) = /(^o)· Let us set
hn '■= \\χη — xo\\ > 0
which converges to 0 and
Xn xq
Vn ■■= 7 "
hn
Since vn belongs to the unit sphere, which is compact, a subsequence
(again denoted) vn converges to some element ν of the unit sphere.

104 3- Existence and Stability of an Equilibrium
Figure 3.1: Example of Monotone ала Maximal Monotone Maps
,
0
/
)
Monotone Map
,
0
/
I
Maximal Monotone Map
This limit ν belongs also to the contingent cone Tk{xq) because,
for all η > 0, xo + hnvn = xn belongs to K. Furthermore, / being
Frechet differentiable at xo, we obtain the equation f'{xo)vn = e{hn)
where e(hn) converges to 0 with hn. We thus infer that ν belongs
also to the kernel of f'(xo)· Since υ φ 0, we derive a contradiction
of the assumption. □
3.5 Monotone and Maximal Monotone Maps
In this section, we assume once and for all that X is a Eilbert space
identified with its dual.
3.5.1 Monotone Maps
Definition 3.5.1 We shall say that a set-valued map A from X to
X is monotone if its graph is monotone in the sense that
V (χ,ρ) Ε Graph(A), V (y, q) G Graph(A), <p-q,x-y>> 0
Example The map
{{x - 1} if χ < 0
{-1,+1} if χ = 0
{x + 1} if χ > 0

3.5. Monotone and Maximal Monotone Maps
105
is a monotone map as well as the map
A(x) = <
{x - 1} if χ < 0
[-1,1] if χ = 0
{x + 1} if χ > 0
The terminology comes from the monotone maps from R to R.
For instance, a nondecreasing map F from R to R is monotone.
More generally, if / is a nondecreasing map from R to R, the
map χ н-> A(x) := \f(x—), f(x+)] Π R is monotone.
Let us give another example of monotone map. Let / be a non-
decreasing function from an interval Dom(/) С R to R. Let Ω С Rn
be open and X := L2(fi). Define the set-valued map A from X to
*by
for almost allu€fi, Α(χ(·))(ω) : = /(χ(ω))
It is clear that A is monotone.
We shall see in Chapter 6 that the subdifferential map χ ι—> dV(x)
of a nontrivial convex function V : X —> RU {+00} is monotone. □
Since monotonicity is a property bearing on the graph of A,
A is monotone if and only if A~l is monotone
If A and В are monotone and λ > 0, μ > 0, then \A + μΒ
is monotone. If we supply Χ χ Χ with the product of the strong
topology and the weak topology, we can see that the closure of a
monotone graph is still the graph of a monotone map.
-" We begin by mentioning the following characterization:
Proposition-.3.5.2 A set-valued map A from X to X is monotone
if and only if fbr every λ > 0
V(z,p), (2/, ?) € Graph(A), ||s-y|| < ||s -y + X(p - q)\\ (3.13)
Proof — We compute
\\x - У + \(p - q)\\2 = \\x-y\\2 + \2\\p-q\\2 + 2\<p-q,x-y>

106
3- Existence ала Stability of an Equilibrium
Hence, if A is monotone, inequality (3.13) ensues; conversely, if
(3.13) holds true, we deduce that
λ2||ρ - ill2 + 2X<p-q,x-y>> 0
and obtain monotonicity by dividing by λ > 0 and letting λ converge
to zero. □
Remark — We observe that condition (3.13) does not involve
the scalar product; therefore, it can be-used on Banach spaces. In
this case, maps A satisfying (3.13) are called accretive maps. □
We associate with the map A its resolvents
VAX), Jx := (1 + XA)-1
An important consequence of property (3.13) is given in
Proposition 3.5.3 The resolvent J\ of a monotone map A is a
single-valued nonexpansive map from Im(l + XA) to X.
Proof— Let χ G J\{u) and у € J\{v). We can write
и € x + XA(x), ν € y + \A(y)
Property (3.13) implies that
(x-y)+X^—x —j
к - 2/|| <
= \\u — v\\
This shows that J\ is nonexpansive and, by taking и = v, that
J\ contains a unique point. □
3.5.2 Maximal Monotone Maps
We shall characterize monotone maps A such that
VA > 0, lm(I + XA) = X
They are the maximal monotone maps.
Definition 3.5.4 A monotone set-valued map A is maximal if there
is no other monotone set-valued map whose graph strictly contains
the graph of A.

3.5. Monotone and Maximal Monotone Maps
107
We begin by pointing out the following:
A set-valued map is maximal monotone if and only if its inverse
A"1 is maximal monotone.
Also, the graph of any monotone set-valued map is contained in
the graph of a maximal monotone set-valued map by Zorn's lemma,
because the union of an increasing family of graphs of monotone
set-valued maps is the graph of a set-valued monotone map.
The following characterization of maximal monotone maps
provides a useful and manageable way for recognizing that an element
и belongs to A{x).
Proposition 3.5.5 A necessary and sufficient condition for a set-
valued map A to be maximal monotone is that the property
V (y, v) € Graph(A), <u-v,x-y > > 0
is equivalent to
и £ A(x)
Proposition 3.5.6 Let A be maximal monotone. Then
1. — Its images A(x) are closed and convex
2. — Its graph is strongly-weakly closed in the sense that if
xn converges to χ and if un G A(xn) converges weakly to u, then
и € A{x).
Proof —
1. — By the preceding proposition, A(x) is the intersection
of the closed half-spaces
{u£ X \ <u-v,x-y>>0}
when (y, v) ranges over the graph of A. Hence, A(x) is closed and
convex.
2. — Let xn converge to χ and let un G A(xn) converge
weakly to u. Let us choose (y, v) in the graph of A. Inequalities
< un - v, xn - у > > 0

108 3- Existence and Stability of an Equilibrium
imply, by going to the limit, inequalities
< u — ν, χ — у > > 0
Hence, и £ A(x) by Proposition 3.5.5. □
Proposition 3.5.7 // the restriction of α monotone single-valued
map A : X ι—> X to finite dimensional vector subspaces is continuous,
then A is maximal monotone.
Proof — Let χ £ X and и £ X such that
(u - A(y), x-y) > 0 for all у £ X
To show that A is maximal monotone, we have to check that и =
A(x) (Proposition 3.5.5.) For that purpose, we take16 у = x — \(z—x)
where λ G]0, l[ and ζ £ X. The above inequality becomes
(u - A(x - \(z - x)), z-x) > 0 for all ζ € X
By letting λ converge to zero and by the continuity of A on finite
dimensional vector subspaces, we deduce that
\f ζ £ X, {u-A(x),z-x) > 0
that is и = A(x). D
The Minty Theorem we are about to prove provides a very
important characterization of maximal monotone maps.
Theorem 3.5.8 (Minty) A monotone map A is maximal if and
only if the map 1 + A is surjective.
Proof—
1. — Assume that 1 + A is surjective. Let χ £ X and и £ X
satisfy
V (y,v) £ Graph(A), <u-v,x-y>>0 (3.14)
16This method is known under the name of Minty's trick.

3.5. Monotone and Maximal Monotone Maps
109
By Proposition 3.5.5, we have to prove that и G A(x). Since
1 + A is surjective, we can choose у in (3.14) to be a solution г/о of
the inclusion
u + x € yo +A(yo)
Let vq € A(yo) such that гх + χ = г/о + зд. Then, by (3.14),
||я-2/о||2 = <х-уо,х-уо> = - <u-v0,x -yo> < 0
Hence, x = yo, and thus
и = v0 € Л(г/0) = ^(ж)
2. — Assume now that A is maximal monotone. Let у € X.
We have to prove that there exists χ such that у ζ. χ + A(x). It
is sufficient to choose у = 0, since this amounts to replacing A by
x —> — г/ + Л (ж), which is also maximal monotone. So we have to
show that there exists χ such that —x G A(x). By Proposition 3.5.5,
we must prove that there exists χ satisfying
V (г/, ν) € Graph(A), < -χ - ν, χ - у > > 0
For convenience, set φ(χ; (г/, υ)) :=< χ + υ, χ — у >, or, equivalently,
ip(x;(y,v)) := \\x\\2+ < χ, ν - у > - < ν, у >
We have to check that there exists χ such that
V(y,u) € Graph(A), φ(χ;(ν,υ)) < 0 (3.15)
Fix (г/о, г;о) to be any point in the graph of A. If a solution to (3.15)
exists, it certainly belongs to
L := '{x £ Dom(A) | φ(χ· (г/0, υ0)) < 0}
The map χ —> (^(ж; (г/о, vq)) is quadratic. Consequently, the set
{x | <^(ж;(г/0,г;о)) < 0}
is convex, closed, and bounded, hence, weakly compact. Let L
denote its intersection with Dom(A). Then co(L) is weakly compact.

110
3- Existence and Stability of an Equilibrium
Also, the maps χ ι-> φ(χ; (у, υ)) are convex and (strongly)
continuous. _ Therefore, their lower sections are convex and closed, hence
weakly closed. It follows that the above maps are weakly lower semi-
continuous.
We use Lemma 3.1.2: Set <S to be the family of all finite subsets
Μ := {{yi,vi),...,(yn,vn)} of Graph(A). Then there exists χ £
W{L) such that
suP(»,«)6Graph(A)V(^ (»'"))
< supMe5 infz6gB(x,) тах(=1,...)П φ{χ; (у», «())
< supMe5 Мхесо[уи^Уп) maxi=if...,n ψ{χ; (у», ъ))
< SUpMe5 infz6co(»i,...,»n) SUPm£5" Σ"=1 /ijVfa (Vj,Vj))
< supMe5 infAe5n sup^n ψμ{\ μ)
where for all λ, μ in the simplex Sn
η η
ψΜ{\μ) ■= Σμιφ{β{\); (yj,Vj)), and β(\) := J^XiVi
3=1 г=1
Every function ψΜ is continuous with respect to λ and
ψΜ{μ, μ) = E£fc=i A*Vfc < β(μ) + vh Ук-Уг>
= Σα=ι ^Vfc < β(μ), у к -уг>+ Σ£*=ι μιμΗ <ъ,ук-уг>
The first term is zero for reasons of symmetry, while the second
can be written
E%k=iμίμ'ΐ < miУк-Уг> + Σ£ί=ι/*V < -щ,Уз -yi>
= E%k=iμίμ]ζ <Уг-ук,Ук-Уг> < о
since A is monotone. Hence, assumptions of the Ky Fan inequality
(Theorem 3.1.1) are satisfied. Therefore, for all M,
inf sup ψΜ{Κμ) < 0
so that inequality (3.15) holds true. We have shown that ж is a
solution to 0 G χ + A(x). □

3.5. Monotone and Maximal Monotone Maps
111
3.5.3 Yosida Approximations
Suppose that A is maximal monotone. We show that A can be
approximated in some sense by single-valued maps Αχ that are also
maximal monotone. These maps, called Yosida approximations, play
an important role, thanks to:
Theorem 3.5.9 Let A be maximal monotone. Then for every λ >
0, the resolvent J\ = (l + \A)~~l is a nonexpansive single-valued map
from X to X and the map A\ := (1 — J\)/\ satisfies
Г i) Ух € X, Ax(x) e A{Jx{x))
I ii) Αχ is Lipschitz with constant l/λ and maximal monotone
Let m(A(x)) denote the element of A{x) with the smallest norm. We
also have
VxGDom(A), \\Ax{x)-m{A{x))\\2 < \\m(A(x))f - \\Ax(x)\\2
and for all χ G Dom(A),
(i) J\{x) converges to χ when λ —> 0+
ii) A\(x) converges to m{A{x)) when λ —> 0+
Definition 3.5.10 The maps A\ are called the Yosida
Approximations of A.
Proof
1. — Let Xi (i = 1,2) be solutions to the inclusions
Уг&Хг + Щхг) (« = 1, 2)
So Уг = χι + Xvi where Vi G A(xi). We obtain
II2/I-2/2II2 = \\xi ~ X2 + λ(νι - V2)\\2
< = \\xi-x2\\2 + \2\\т-щ\\2+ 2X <υι-υ2,χι-Χ2>
. > \Ы - X2W2 + >?\\vi - v2\\2

112
3- Existence and Stability of an Equilibrium
Hence,
Hzi-ajall < II2/1-2/2II & ||«1-«з|| < llyi-yall/λ (3.16)
By taking ?/i = ?/2) (3.16) proves the uniqueness of the solution. We
note that
Xi = J\Vi and Vi = A\{yi)
Hence, inequalities (3.16) prove that J\ and Ax are Lipschitz with
constants 1 and l/λ, respectively.
2. — By the very definitions of J\ and Αχ, we have
Му) = \(У ~ Ыу)) € A(J\(V)) for all у е X
Therefore, since yi = J\{yi) + XA\{y{), we obtain
< Ax(yi) - А\{У2),У1 ~ 2/2 >
- = < Ax{y1)-Ax{y2),Jx{y1) - Jx{y2) > +X\\Ax(yi) - Ax(y2)f
k > Х\МУ1)-МУ2)\\2 > 0
Hence, Αχ is monotone (and by Proposition 3.5.7, maximal
monotone.)
3. — Let χ € Dom (A.) Then
\\Αχ(χ)-τη(Α(χ)ψ
= \\Αχ(χψ + \\m(A(x))\\2 - 2 < Αχ(χ),τη(Α(χ)) >
= ||m(A(s))||2 - \\Ax(x)\\2 - 2 < AA(s),m(A(s)) - Ax(x) >
Using that A is monotone, that т(Л(ж)) € Л(ж) and A\(:c) £
Л(7л(ж)), we obtain
< AA(s),m(A(s))-AA(s) >= γ < m(A(x))-Ax(x),x-Jx(x) > > 0

3.5. Monotone and Maxima,! Monotone Maps
113
Therefore, we have proved inequality
\\Ax(x) - m(A(x))\\2 < \\m(A(x))f - \\Ax(x)\\2 (3.17)
4. — Then when χ G Dom(A), we have
\\х-Ых)\\ = λ||Αλ(χ)|| < A||m(A(x))||
therefore, J\(x) converges to χ when λ converges to zero.
5. — We deduce that у = A\(x) is a solution to the equation
у € A(x — \y). Indeed, setting ζ = χ — Xy, this equation becomes
χ £ ζ + \A(z). Hence,
ζ = Jx{x) к у = {x-Jx{x))/\ = Ax{x)
This remark implies that
Αμ+χ{χ) = (Αμ)χ{χ)
Indeed, у = Αμ+χ{χ) is a solution to the equation у G A(x — \y — μy)■,
then у € Αμ(χ — Ay). Applying again the preceding remark to the
Yosida approximation Αμ, which is maximal monotone, we deduce
that у = (Αμ)χ(χ).
6. — Now we use inequality (3.17), replacing A by Αμ. Since
πι(Αμ(χ)) = Αμ(χ), we obtain
||Αμ+λ(χ)-Αμ(χ)||2 < ||Αμ(χ)||2-||Αλ+μ(χ)||2
Then the sequence ||Αμ(α;)||2 is monotone and bounded from above
by ||т(Л(ж))||2, so that it converges to some real number α when
A —> 0+. This implies that
lim ||Αμ+λ(χ)-Αμ(χ)||2 < а-а = О
Hence, Αχ(χ) satisfies the Cauchy criterion and converges to some
element ν in X. Since A\(x) G A(Jx(x)) and the graph of Л is closed,
we deduce that ν £ A(x). Also
IMI = lim ||Αλ(χ)|| < ||m(A(x))||
A—*U4~

114
3- Existence and Stability of an Equilibrium
Since A(x) is closed and convex, the projection of zero onto A(x) is
unique and consequently, ν = m{A{x)). Therefore, A\(x) converges
to m(A{x)) for all χ € Dom(A). D
We shall provide an existence and uniqueness theorem for
differential inclusions the right-hand side of which is minus a maximal
monotone in Chapter 10.
Remark — We shall study in Chapter 5, Section 2 the
derivatives of monotone maps and in Chapter 7, Section 1, the graphical
convergence of maximal monotone maps. □
3.6 Eigenvectors of Closed Convex Processes
We shall prove now that a closed convex process F does have an
eigenvector in cones with compact soles.
Let К be a closed convex cone in Rn. We recall that the following
conditions are equivalent:
' г) К is spanned by a convex compact set disjoint from 0
< ii) the interior of the polar cone K+ is not empty
Hi) S := {x € K\ < po,x > = 1} where po € Int(UT+) spans К
The compact convex subset S is called the sole, and such closed
convex cone are called cones with compact sole.
Definition 3.6.1 We say that a solution χ to the problem
χ £ Κ, χ ^ 0, AGR & Xx £ F(x) (3.18)
is an eigenvector χ G К of the closed convex process F associated
with an eigenvalue λ.
Theorem 3.6.2 Let X be a finite dimensional vector-space and F :
X ~»· X be a closed convex process. Assume that a closed convex
cone К С X enjoys the following properties:
г) К has a compact sole
ii) К is a viability domain of F
Hi) the norm \\R\\ of the map R defined by :
\/x<EK, R(x) := F(x)nTK(x) φ 0 is finite

3.6- Eigenvectors
115
Then there exists a nonzero eigenvector χ G К of the closed convex
process F associated with an eigenvalue λ.
Proof — We associate with the closed convex cone К and an
element po € Int(.ftT+) the "compact sole"
S := {x € K\ <po,x > = 1}
By Lemma 4.2.5 below, its tangent cone is equal to
Ts(x) = {v€TK(x)\ <po,v > = 0}
For an element у G S the projection w(y) onto the orthogonal hy-
perplane to po is defined by
VzG-X", w{y)z := z- <p0,z> у
We then remark that:
V2/G5, ш(у)Тк(у) С Ts{y) (3.19)
Indeed, by Lemma 4.2.5 below, if и € Тк(у), then
w(y)u :=u-<p0,u>y € {K + Ry) + Ry С К+ Ry = TK(y)
because К is a closed convex cone. Furthermore
< po,и > = < po, и > - < po, и >< po,y>= 0
because < po,y >= 1. We deduce that w{y)u belongs to Ts{y).
Let us associate with the closed convex process -F the set-valued
map G defined on the compact sole S by
G{y) := w{y){F{y) П ||Д||В)
It is obviously a set-valued map with closed convex images contained
in the ball ||-R||-B, which is compact.
Since its graph is closed, we deduce that G is upper semicontin-
jaous from S to X.
By (3.19), S is a viability domain of the set-valued map G since
the cone К is a viability domain of the closed convex process F.

116 3- Existence and Stability of an Equilibrium
Since the sole S is a convex compact subset, we can apply
Equilibrium Theorem 3.2.1 to the set-valued map G. There exists an
equilibrium χ € S of G, i.e., a solution to 0 € G(x), in other words,
a solution to
χ € S, 0 = y— < po, у > χ where у € F(x)
By setting λ :=< po) У >i we see that the pair (λ,χ) is a solution to
inclusion (3.18.) □
Remark — Other theorems dealing with positive eigenvalues
and an extension .of Perron-Probenius's Theorem to closed convex
processes can be found in [35, Chapter 3]. □

Chapter 4
Tangent Cones
Introduction
We already had to use tangent cones in the preceding chapter,
both for solving inclusions under constraints and studying stability
and convergence of approximations of their solutions.
They also play an important role in viability theory and in
optimization. It is time now to conduct a thorough investigation of their
properties in order to apply theorems involving tangent cones.
There are many ways to describe technically the concept of tan-
gency to a given subset К at a point χ € К. The idea can well
be expressed in terms of constraints: We regard a (closed) subset
К as the subset of elements of the whole space X obeying given
constraints.
Pick any direction ν G X and start from χ in the direction v,
ranging over the line χ + hv when h > 0.
We would like to distinguish those directions which, for small h,
do not lead us too far away from K. Such directions encapsulate the
idea of tangency.
But how can we translate rigorously this idea in mathematical
terms? It certainly involves the distance άκ(χ + hv) from the points
x + hv to K, which should tend to 0 faster than h, in the sense that
άκ(χ + hv)/h should converge to 0 with h.
Naturally, but unfortunately, there are many ways of taking such
limits; we can take either the liminf or the limsup, we can require
117

118
4- Tangent Cones
a little more uniformity in both cases. We feel like opening the
door of a menagerie of tangents, and facing the choice of a favorite
pet! We are in the situation of the particle physicists when so many
unsuspected particles popped up, destroying the harmonious pre-
particle description of matter. We also face a whole collection of
more or less colored "quark cones", each one having a charm of its
own.
Even though the subjective choice of a given category of tangents
can be justified by accumulated experience, the reasons to
emphasize one concept at a given time can very well disappear later on.
This is why we shall use the concept of "tangent cone" as a generic
term to designate any of the particular implementations inherited
from history (as the contingent and paratingent cones since Bouli-
gand.) Fortunately, most of these different implementations coincide
for familiar classes of subsets (differential manifolds, convex subsets.)
An analogous situation happened with the concept of
derivative of a function, rigorously defined after Cauchy and Weierstrass,
and severely taught to generations of young, properly brainwashed
mathematicians. The acquired dogma were difficult to overrun when
Sobolev (very timidly) and then L. Schwartz (much more boldly)
introduced the weak derivatives leading on to derivatives in the sense of
distributions, when Dini involved liminf and limsup to let differential
quotients converge (more on this in next chapters) and when Bouli-
gand introduced the concepts of contingent and paratingent sets.
Furthermore, even for the usual derivatives, different ways of taking
the limits and various topologies were used, giving rise to a
manifold of concepts (Gateaux, Hadamard, Frechet, C1, etc., derivatives)
which were adapted more or less reluctantly, each problem requiring
its particular brand of derivative.
Geometers became eagerly "differential geometers" one century
ago, and would hardly accept leaving the luxurious environment
where the sets of tangents are comfortable vector spaces, with
smoothness all around them.
But non smooth sets do exist, unfortunately, as soon as objects
of investigation are no longer freely given; they are encountered in
numerous problems as the result of some operations. This may
destroy smoothness, so that we lose the possibility of imposing a priori

4.0. Introduction
119
regularity conditions .
Many of us did actually encounter them. The pressure coming
from many problems2 in control theory and differential games, in
economy theory, in biological regulation, in systems theory, etc., led
mathematicians from many countries and schools, motivated by
different problems,.to investigate various ways of overcoming regularity
requirements.
The first step away from differential geometry was to adapt many
of the results to the case of convex subsets, after Fenchel, J.-J. Moreau
and Rockafellar. It was a surprise to see that tangent cones and
normal cones to convex subsets shared many properties of tangent
and normal spaces to differentiable manifolds.
After being born with Bouligand in the thirties, and then revived
during the early days of control theory, the problem of defining and
using tangent cones to arbitrary subsets got a new start in 1975,
when Clarke introduced a tangent cone Ск{х) to a set К at χ G К
which he proved to be always a closed convex cone. The price of
this nice property, however, was quite high since this tangent cone
may often be too small or even reduced to the singleton 0. At that
time, notwithstanding this cost, it was a good candidate to be "the"
tangent cone that was waited for as a messiah!
Further investigations both in viability theory and control theory
suggested however that, for the time being at least, the tangent cone
which was the most often adequate was the contingent cone Тк{%)
introduced by Bouligand: it is the upper limit of the differential
quotients3 {K — x)/h when h —> 0+.
It happens however that if the set-valued map у ~> Тк{у) is
lower semicontinuous at x, then the contingent cone Тк{х) coincides
1However Differential Geometers can find solace in the 2,600 year old Xeno-
phanes of Colophon's statement that God was spherical, unique, motionless and
"shaking" all things by the power of thought. He made fun of the
anthropomorphic character of the gods of the Olympian theology, arguing that cows would
make their gods in their own image.
Of greater philosophic interest, he stated that men can have no certain
knowledge, only opinions.
2 Just remember to begin with that the intersection of two smooth sets may
no longer be smooth.
3whereas the tangent cone С к (х) is the lower limit of the differential quotients
{K — x')/h, when h —> 0+ and χ' —>κ χ.

120
4- Tangent Cones
with Ck(x) and is thus convex. This fortunate situation deserves
a name: we say that K_ is sleek at _x. Both smooth manifolds and
closed convex subsets are sleek:
This, and elementary properties of the contingent cones, are
presented in Section 1. Section 2 is devoted to the tangent cones to
convex subsets, where we recall (most often, without proof) the
properties of the tangent cones from convex analysis.
The main motivation for introducing sleek subsets (and/or Clarke
tangent cones) is that they play a crucial role in the transversality
assumptions that we need for stating that a tangent cone to an inverse
image (respectively an intersection) is equal to the inverse image
(respectively the intersection) of the tangent cones. These crucial
formulas are indeed a consequence of the Constrained Inverse Function
Theorem from Chapter 3, and are proved in Section 3.
We then devote Section 4 to dual concepts: the normal space to a
differentiable manifold being defined as the orthogonal space to the
tangent space, the normal cone to a convex subset as the polar cone
to the tangent cone, we consider the polar cones to the tangent cones
and translate by duality the properties of the corresponding tangent
cones to normal cones. We shall not dwell on this, since many other
monographs emphasize this view point, couched in the traditional
Lagrange formulation of necessary conditions for optimality. In this
sense, one can say that this book is "orthogonal" to most of the texts
devoted to optimization and nonsmooth analysis!
Section 5, which can be omitted in a first reading, deals with
other tangent cones.
When convexity is required, we can use what we call the convex
kernel of the contingent cone, which is a convex cone lying between
the Clarke tangent cone and the contingent cone, and which is large
enough: it is the intersection of the maximal closed convex cones
contained in the contingent cone.
We then examine the paratingent cones introduced also by Bouli-
gand, which contain the contingent cones. The graph of the
paratingent map is closed (an important property indeed.) We prove Cho-
quet's Theorem which states that the contingent and paratingent
cones are generically equal.

4.1. Tangent Cones to a Subset
121
We study in Section 6 the limits of tangent cones Ткп{хп) to a
sequence of subsets Kn and relate them to the upper and lower limits
of differential quotients (Kn — xn)/hn.
Since we can regard tangent cones to a subset К as first-order
approximations of K, we conclude this chapter with a short
introduction to higher order approximations, which are various limits when
h i-> 0+ of "differential quotients"
К — χ — hu\ — ■■ ■ — hrn~1vm-i
_
They enjoy many of the properties of first-order approximations.
4.1 Tangent Cones to a Subset
4.1.1 Contingent Cones
We begin with a presentation of the contingent cones:
Definition 4.1.1 (Contingent Cones) Let К с X be α subset of
α normed vector space X and χ € К belong to the closure of K. The
contingent4 cone Тк{х) is defined by
TK{x) := {v\limw.fdK(x + hv)/h = 0}
We see at once that the contingent cone Тк{х) is the upper limit
of the subsets (K — x)/h, so that Τχ(χ) is a closed cone.
4from the Latin contingere, to touch on all sides, introduced by G. Bouligand
in the 30's. This term was already used by R. Descartes, in a 1638 letter to
Mersenne criticizing P. de Fermat's method on tangents:
"Puis, outre cela, sa regie pretendue n'est pas universelle comme il lui semble,
et elle ne peut s'etendre a aucune des questions qui sont un peu difficiles, mais
seulement aux plus aisees, ainsi qu'il pourra eprouver si, apres 1'avoir mieux
digeree, il tache de s'en servir pour trouver les contingentes, par exemple, de la
ligne courbe BDN, que je suppose etre telle qu'en quelque lieu de sa circonference
qu'on prenne le point B, ayant tire la perpendiculaire ВС, les deux cubes des
deux lignes ВС et CD soient ensemble egaux au parallelepipede des deux memes
lignes ВС, CD et de la ligne donnee P."
This is quite a long and nasty sentence indeed.

122
4 - Tangent Cones
Figure 4.1: Contingent Cone at a Boundary Point may be the Whole
Space
_
/^
\ °
.
~~2^)
~^)
Subset К such that TK{0) = X
Denote by
SK(x) := U
K-x
h>0
h
the cone spanned by К — χ. Hence the contingent cone Тк(х) is
contained in Sk(x)-
It is very convenient to have the following characterization of this
cone in terms of sequences:
{υ € Тк{х) if and only if 3 hn —»· 0 + and 3 υη —»· υ
such that V n, ж + hnvn € AT
We also observe that
if χ € Int(ur), then Гя-(х) = X
This situation may also happen when χ does not belong to the
interior of К (see Figure 4.1.)
Unfortunately, the graph of the set-valued map
К Э χ ~> Тк{х) С X
is not necessarily closed. This important property is lacking
whenever inequality constraints are involved in the definition of K, as the

4.1. Tangent Cones to a Subset 123
Figure 4.2: The Graph of T[a>b](·)
ΤΓ
following simple example, where К := [а,Ь], shows (see Figure 4.2.)
We shall see in a while that this set-valued map is closed when К is
described by equality constraints and that it is lower semicontinuous
when К is convex.
Example Let us consider a normed space X, a continuous map
д = (gi,---,9P) -X >-> R-p
and the subset К of X denned by inequality constraints
К := {χ € X I gi(x) > 0, i= l,...,p}
Fix χ in K. We denote by
I(x) ■- {i=i,...,p| 9i(x) = 0}
the subset of active constraints. We observe that Тк{х) = X whenever
I(x) = 0 and that, otherwise, inclusion
TK(x) С {и £ X I Vie I(x), < д-(х), и > > 0}
holds true when g is Frechet differentiable at ж. If we posit moreover the
constraint qualification assumption:
3 v0 £ X such that V г € I(x), < g'i(x), v0 > > 0

124 ■ —4 - Tangent Cones
Table 4.1: Properties of Contingent Cones.
> If К С L and χ € K, then Тк(х) С TL(x)
> If Kid X, (i = 1,. .., n) and ж € Ц ^i> then
where /(ж) := {г | χ € ^}
> If ΛΓί С Xj, (г = 1, ... j n) and χι € if,;, then
2nr=ilf«^1,"-,a:n^ C ПГ=1^(Жг)
> IfgeVpf.y), ififCl,xeIaBdMc7, then
д'{х){Тк{х)) С Гв(к?(5(х))
Tff-i(i0(s) С ^(хНад^а:))
о If Κι С X, (г = 1,..., η) and x € fli ^'> tnen
then the contingent cone
Гк:(а;) = {u ε Χ | Vie I(x), <д'{(х),и>> 0}
Indeed, let и satisfy < д[{х), и >> 0 for any г ε I(x). For г φ, I(x),
strict inequalities 5i(a;) > 0 imply that for some а > 0, we have
Vfte [Ο,α], Vi ^ /(ж), gi(x + hu) > 0
Consider first the case when (^(a;),u) > 0 for any г ε I{x)- Then
V г ε Ι(χ), gi{x + hu) = gi(x + hu) - gi{x) = < д{(х), u> + hei(h)
where Si(h) converges to 0 with h. This implies that gi{x + hu) > 0 for h
small enough and all г ε I(x), and thus, for all г = 1,... ,p. Then such an
element и belongs to the contingent cone Тк(х)·
Consider now the general case. By assumption, we deduce that for any
β ε]0,1[, υ,β := (1 — P)u + βυο satisfies strict inequalities (д[(х), up) > 0
for any г ε 1{χ) and, by what precedes, belongs also to the contingent cone
Tk{x)· Letting β converge to 0, we infer that the limit и of the up's belongs
also to the contingent cone Τχ(χ). Π
(1)
(2)
(3)
(4)
(5)

4.1. Tangent Cones to a Subset
125
4.1.2 Elementary Properties of Contingent Cones
Proposition 4.1.2 Assume that the Banach space X is smooth,
i.e., that the norm of X is Gateaux differentiable off the origin, and
denote by J(x) its gradient at x. Let К be a closed subset of X.
We set
Пк-(у) := {ζ € K\ \\z - ?/|| = dK(y)}
Let у £ К be such that Лк(у) is n°t empty. Then
Vz€UK(y), \/v£cd(TK(z)), (J(y-z),v) < 0
Proof — Fix υ e Tk{z): There exist hn > 0 and vn e X
converging to 0 and ν respectively such that ζ + hnVn belongs to
К for all η > 0. Since the function || · || is convex and Gateaux
differentiable off the origin, the following inequality holds true: for
every ?/i ^ 0 and 2/2,
< J(yi),V2 - 2/1 > < II2/2II - H2/1II
We thus deduce that
< J(y - ζ - hnvn),vn >
< = < J(y - ζ - hnvn), y-z-(y-z- hnVn) > /hn
< (\\y-z\\-\\y-'z-hnvn\\)/hn < 0
Since J is continuous from X to X* supplied with the weak-* topol- "
ogy, we infer that J{y — ζ — hnvn) converges weakly to J{y — z).
Hence < J{y — ζ — hnvn),vn > converges to < J(y — ζ),υ > because
vn converges strongly to v, so that
\/v€TK(z), <J{y-z),v><0
and thus the same inequality holds true for any ν in the closed convex
hull of TK(z). □
We introduce the Dubovitskij-Miljutin cone defined by
5which is a map from X to the unit sphere of X*, called the duality map,
satisfying in particular < J(x),x >= \\x\\. The duality map is continuous from
X to X* supplied with the weak-* topology.

126
4 .-- Tangent Cones
Definition 4.1.3 The "Dubovitskij-Miljutin tangent cone" Dk(x)
to К at χ € К is defined by:
υ € Dk{x) if and only if 3 e > 0 such that χ + ]0,ε](υ+εΒ) С К
This cone is the complement to the contingent cone to the
complement:
Lemma 4.1.4 Let χ belong to the boundary of К and К denote
the complement of К'. Then the complement of the contingent cone
Tg(x) to К at χ is the "Dubovitskij-Miljutin cone" Ώχ(χ) to К at
x.
Remark — The last example shows that when
К := {χ I gj(x) > 0, j = 1,... ,p}
then the open subset
{v\ <sj(i),0 > 0, j £l(x)}
is contained in Dk{x)· α
4.1.3 Adjacent and Clarke Tangent Cones
To go further by having equalities in some of the formulas of
Table 4.1, we need more assumptions in order to use the Constrained
Inverse Function Theorems of the preceding chapter.
These further assumptions are regularity assumptions, which
require that the contingent cone Тк(х), which is the upper limit of the
subsets (K — x)/h when h —>■ 0+, coincides with the lower limit of
these subsets when /i-+0+ and even, of the subsets (K — y)/h when
both h —> 0+ and у £ К converge to x.
This is the reason why we have to introduce two more tangent
cones:
Definition 4.1.5 Let К С X be a subset of a normed vector space
X and χ £ К belong to the closure of K.

4.1. Tangent Cones to a Subset
127
1. — the intermediate or adjacent6 cone Т\{х) is defined by
Ί*κ(χ) := {v\ lim dK{x + hv)/h = 0}
2. — the Clarke7 tangent cone or circatangent cone С к {я) is
defined by
CK{x) ■■= {v | lim dK(x' + hv)/h = 0}
h—>0+,i'—+Д-1
where —+K denotes the convergence in K.
We shall say that a subset К С X is derivable at χ ς. Κ if
and only ifl\(x) = Тк(х) and tangentially regular at χ г/Тк(х) =
Ск(х)·
We see at once that these tangent cones are lower limits
Ί%(χ) = Ummih^Q+—τ—
h
and
К — χ'
Ск(х) = Liminf/l_t0+, кэх'^х—τ—
so that they are closed cones, that
Ск(х) С Ί*κ(χ) С Тк(х) с 5^М
that these tangent cones to К and the closure К of К do coincide
and that
if χ e Int(UT), then CK{x) = X
We shall see later that the converse is true when the dimension
of X is finite.
These tangent cones may all be different: see Figure 4.5.2 of
Section 4.5.2 for an example when these tangent cones as well as
other ones are distinct. In the example of the set of Figure 4.1, the
6from the Latin adjacere, to lie near, also used under the name of derivable
cone by R. T. Rockafellar.
7from the Canadian Frank H. Clarke; we shall use the adjective circatangent
to mention properties derived from this tangent cone, for instance, circatangent
derivatives and epiderivatives.

128
4 - Tangent Cones
adjacent and the contingent cones at zero coincide (with the whole
space) and the cone Ск{0) is equal to the horizontal axis R χ {0}.
It is very convenient to use the following characterization of these
cones in terms of sequences.
f ν € Тьк{х) if and only if V hn -* 0+,
1 3vn^*v such that V η, χ + hnvn € К
and
ν € Ck{x) if and only if V hn —»· 0+, V xn —*к x,
3vn —* ν such that V η, χη + /ιπυπ € AT
They are not necessarily equal (see Figure 4.1) for instance.
Let us single out an astonishing fact: the tangent cone Ск{х) is
always a closed convex cone.
Proposition 4.1.6 The tangent cone Ск{х) is a closed convex cone
satisfying the following properties
CK(x) + TK(x) С Tk(x) L· Ск{х)+Тьк{х) С Тьк(х)
Proof— Let vi and V2 belong to Ск(х)· To prove that vi +щ
belongs to this cone, let us choose any sequence hn > 0 converging to
0 and any sequence of elements xn € К converging to x. There exists
a sequence of elements v\n converging to щ such that the elements
xin := xn + hnvin do belong to К for all n. But since the sequence
x\n does also converge to χ in K, there exists a sequence of elements
V2n converging to V2 such that
V П, Xin + hnV2n = Xn + Κ(υίη + V2n) & К
This implies that v\ +V2 belongs to Ск(х) because the sequence of
elements v\n + V2n converges to υχ + V2.
The proof of the two other inclusions is analogous and left as an
exercise. □
Unfortunately, the price to pay for enjoying this convexity
property of the Clarke tangent cones is that they may often be reduced
to the trivial cone {0}.

4.1. Tangent Cones to a Subset 129
Table 4.2: Properties of Adjacent Tangent Cones.
> If К С L and χ £ К, then Ί+Κ(χ) С lj{x)
> If K{ С X, (г = 1,..., η) and χ G (Ji ^i, then
where I(x) := {i|a; G iQ}
> li Ki С Xi, (i = 1,. .., n) and Xi G ifj, then
> IfggC^y), if К С Χ, χ € F and M С У, then
tf'COCUOO) с 2j(i0GKaO)
> If Ki С X and lefli^i (i = l,...,n), then
As an example, consider the subset
ЛГ := {(x,y)eR2||x| = |y|}
We see that
Tk(0) = 2^(0) = ЯГ and C^(0) = {0}
But we shall show in just a moment that the Clarke tangent cone
and the contingent cone do coincide at those points χ where К is
sleek, i.e., where the set-valued map χ ~»· Тк{х) is lower semicon-
tinuous. Hence the cone Ск{х) can be seen as a "regularization" of
the contingent cone Тк{х).
The elementary properties of the adjacent cones are the same
than the ones of contingent cones, except for the formula of the
intermediate tangent cone to the product, which is the product of
the intermediate tangent cones, and the formula for the union. They
are summarized in Table 4.2.
(1)
(2)
(3)
(4)
(5)

130
4 - Tangent Cones
4.1.4 Sleek Subsets
We recall the definition of sleek subsets:
Definition 4.1.7 We shall say that a closed subset К is sleek at
xq € К if the cone-valued map
К Э χ ~> TK{x)
is lower semicontinuous at xq and that it is sleek if it is sleek at every
point of K.
Theorem 4.1.8 (Tangent Cones to Sleek Subsets) Let K~be a
closed subset of a Banach space. If К is sleek at χ G K, then the
contingent and Clarke tangent cones to К at χ do coincide, and
consequently, are convex.
It follows from the "lower semicontinuous version" of the next
result:
Theorem 4.1.9 Let К be a nonempty closed subset of a Banach
space X and xq belong to К. Then
LiminfI_tic3;or^(a:) С Cr{xo)
because if К is sleek at xq, then
Tk{xq) С Liming кХ0Тк {х) С Ck{xq) С Тк(х0)
Before proving Theorem 4.1.9 in full generality, we shall prove
stronger versions first in the case of finite dimensional vector-spaces,
and then, in the case of Hilbert spaces and more generally, in the
case of uniformly smooth Banach spaces X such that the norm of its
dual X* is Prechet differentiable away from zero.
4.1.5 Limits of Contingent Cones; Finite Dimensional
Case
Theorem 4.1.10 Let X be a finite dimensional vector-space and К
be a closed subset of X. Then for every χ £ К
lAnnxiiy^KXTK{y) = Limmiy^KXcd{TK{y)) = CK{x)
Consequently, К is sleek at χ if and only if it is tangentially regular
at x.

4.1. Tangent Cones to a Subset
131
Proof
1. — Let us take ν φ 0 in the lower limit of the closed convex
hull of the contingent cones Τκ{ζ) when ζ converges to χ G K: This
means that for any e > 0, there exists η > 0 such that for
V ζ£Β{χ,η)Γ]Κ, Β(ν,ε)ΠΈοΤκ{ζ) φ 0
For proving that ν G Ск(х), we take any
y£B{x,V/4)nK, t < η/4\\υ\\, zEUK{y + tv)
and we introduce the function д defined by g(t) := ак{у + tv),
which, being Lipschitz, is almost everywhere differentiable. Let
ί€[0,τ,/4|Η]
be any point where g is differentiable.
We claim that g'(t) < ε whenever g(t) > 0, i.e., whenever у + tv
does not belong to K.
Let h > 0 be small enough. Since
dK(y + (t + h)v)-dK(y + tv) < ||y-z + nt,||-||y-z||
and since the euclidian norm is convex and differentiable off 0, we
infer that
' 9(t + h)-g(t) < \\y-z + hv\\-\\y-z\\ < h(^j±j^,v)
<
{ Ь fi \\\y-z+hv\\ \\v-z\\>v/ + n \W^>O/
We check next that ζ G B(x, η) Π Κ because
\\z — x\\ < \\z — у — tv\\ + ||?/ — χ +iw|| < 2||y — a;+ίϋ|| < η
whenever у G Β(χ,η/4) and t < 77/4||w||. Therefore, there exists
w G co(Tk{z)) such that \\v — w\\ < e, so that, by Proposition 4.1.2,
' (9(t + h)-g(t))/h
У - e + \\\y-z+hv\\ "Й'7

132
4 - Tangent Cones
Taking the limit when h —»· 0+ we deduce that g'{t) < ε whenever
g{t) > 0. Hence the claim is proved.
We prove next that g(t) < et. This being obviously true when
g{t) = 0, assume that g(t) > 0. Let
ίο := sup{T€[O,i]|0(T) = 0}
Since g is continuous, g(to) is equal to 0, so that we can write
g(t) = f g'{r)dr < e(t - i0) < et
because g'{r) < ε for any r €]ίο>ί]· This shows that υ G Ск{х)·
2. — Let us prove now that Ск{%) is contained in the lower
limit of the contingent cones Тк(у) when у converges to χ in K. '
Indeed, let υ belong to Ск(х) and be a sequence of elements
xn € К converging to x. Then, for all e > 0, there exist N and
β > 0 such that, for all 0 < h < β, η > Ν
ак(хп + hv) < Ηε
Let us associate with such xn elements y% € К satisfying
\\Уп~хп -Ml ^ ^ε
We set υ^:= (y^ — xn)/h. Since ||ϋ£ —ϋ|| < ε and since the dimension
of the space X is finite, there exists a cluster point vn G υ+εΒ of the
sequence (υ^)/ι>ο· Such a vn belongs to the contingent cone Τκ{%η)·
Hence υ is the limit of the elements vn. □
4.1.6 Limits of Contingent Cones; Infinite
Dimensional Case
The proof of the first part of the above theorem can readily be
extended to the case of Hilbert spaces when К is weakly closed. This is
too strong an assumption, since we need to use such a result for closed
subsets. We can overcome this difficulty and extend Theorem 4.1.Ю

4.1. Tangent Cones to a Subset
133
to uniformly smooth8 Banach spaces by using the following Edel-
stein Theorem9 (see [35, Theorem 5.7.13. p.294] for instance), which
states that in Hilbert spaces and some Banach spaces, we can
approximate any point by another which has a unique projection of
best approximation on a closed subset K.
Theorem 4.1.11 Assume that X is reflexive and that the norms of
X and X* are Frechet differentiable off the origin. Let К be a closed
subset of X. Then there exists a dense subset D of X of points у
with a unique projection on K.
In infinite dimensional spaces, we have to replace the contingent
cones Tk{x) by the weak contingent cones Τχ(χ) defined by
Vh — X
υ € Tpc{x) <£=Φ- υ is a weak cluster point of —-— when h —> 0+
h
where ун belongs to K.
We observe that Proposition 4.1.2 can be extended to the case
when the contingent cones Τκ {ζ) are replaced by the weak contingent
cones T%(z).
Proposition 4.1.12 Assume that the norm of the Banach space X
is Frechet differentiable off the origin and denote by J(x) its gradient
at x. Let К be a closed subset of X and у ф К be such that Цк-(у)
is not empty. Then
Vz€nK(y), \/v€co(T£(z)), <J(y-z),v><0
Proof — Since the norm is Frechet differentiable at у — ζ φ 0,
we know that for any e > 0, there exists η > 0 such that whenever
IIHI ^ Vi
\\\y — ζ + w\\ — \\y — z\\— < J(y — z),w > I < e||w||
8A Banach space is uniformly smooth if and only if its norm is uniformly
Frechet differentiable in the sense that there exists e(h) converging to 0 with
h > 0 such that
V χ φ 0, ж 6 Б, υ SB, \\\χ + HI " N1 -h< J(x),v >| < he(h)
In this case, J is uniformly continuous from X to X* supplied with the norm
topology. Uniformly smooth Banach spaces are reflexive. Lebesgue spaces Lp(0)
and Sobolev spaces Wm,p(n) are uniformly smooth for 1 < ρ < +oo.
9which can be derived from Ekeland's Variational Principle.

134
4 - Tangent Cones
Let ν belong to the contingent cone T£(z). Then there exist
sequences hn > 0 converging to 0 and vn g X converging weakly to ν
such that ζ + hnvn belongs to К for any η > 0, so that
\\y ~z|| < \\y-z- hnvn\\
For η large enough, the norm of w := hnvn is smaller than or equal
to r\ since vn is bounded by a constant с and hn converges to 0. The
above inequalities imply
< J(y - z),vn > --— , .
< (||y -z -hnvn\\ - \\y -z\\ + hn < J(y - z),vn >)/hn
< e\\vn\\ < ec
and thus, by letting vn converge weakly to v, that < J(y — z),v >< ec
for any e > 0. Hence < J(y — z),v >< 0 for every υ € Τχ(ζ), and
thus, for any ν in its closed convex hull. □
Theorem 4:1.13 Assume that X is uniformly smooth10 and that
the norm of X* is Frechet differentiable off the origin. Let К be a
closed subset of X. Then
Liminfy-^ Тк{у) = 1лшш1у^кхсо{Т^{у)) = Ск(х)
Proof — The proof is similar to the one of Theorem 4.1.10
1. — To prove the first inclusion, let us take ν φ 0 in the
lower limit of the closed convex hull of the weak contingent cones
Τχ(ζ) when ζ € К converges to χ € К: This means that for any
e > 0, there exists η > 0 such that
V ζ€Β(χ,η)ηΚ, Β{υ,ε)Γ)ΈδΤκ{ζ) φ 0
We have to show that it belongs to Ск(х)· For that purpose, we
take у € B(x, 77/6) П K, t < 77/6||t^|| and we introduce the function
g(t) := dK(y + tv)
10Actually, we only need that the norm of X is Frechet differentiable off the
origin and that J is uniformly continuous from the unit sphere of X to X* supplied
with the weak-* topology.

4.1. Tangent Cones to a Subset
135
which, being Lipschitz, is almost everywhere differentiable. Let
t € [0,η/6\\υ\\]
be any point where д is differentiable. We claim that g'{t) < ε
whenever g(t) > 0.
Let ft > 0 be small enough. By Edelstein's Theorem 4.1.11, we
can associate with any ft > 0 an element ун € В (у + tv, ft2) which
has a unique projection Zh G K. Therefore,
f dK(y + {t + h)v)-dK{y + tv) < dK(yh + hv)-dK(yh) + 2h2
\ < \\Vh - Zh + hv\\ - \\yh - zh\\ + 2h2
Let J : Χ η-> J5* be the gradient of the norm of X. Since X
is uniformly smooth, we infer that there exists e{h) converging to 0
with h such that
\\yh -zh + hv\\ - \\yh - zh\\ < h< J(yh - zh),v > +he{h)
Finally, we check that Zh € Β(χ,η) because
\W -z|| < \\zh -Vh\\ + \\yh -y -tv\\ + ||y-a;+iu||
< dK{y+tv) + 2h2 + ||y-a;+iu|| < 2(\\y - χ + tv\\ + h?) < η
whenever h < -у/77/6, у € Β(χ,η/6) and t < rj/6||u||. Therefore,
there exists Vh € W(T^(zh)) such that \\v — υ^\\ < ε, so that, by
Proposition 4.1.12,
' g(t + ft) - g(t) = dK(y+tv + hv) -dK(y + tv)
< ft < J(yh-zh),vh > +Λ(||ϋ-ϋ/»||+ε(Λ)+2Λ)
< ft(e + e(ft) + 2ft)
so that g'(t) < ε whenever g(t) > 0.
This yields that g(t) < et in the same way as in the proof of
Theorem 4.1.10 and shows that ν £ Ск{х)·

136
4 - Tangent Cones
2. — The second inclusion is a consequence of
Lemma 4.1.14 Let X be α reflexive Banach space, К С X a closed
subset and xq belong to K. Then
Ck{xq) С итт£х->кХ0Тк(х)
Proof — Let υ belong to Ck(xq) and be a sequence of elements
xn € К converging to xq. Fix e > 0 and let N and β > 0 be such
that, for all 0 < h < β, η > Ν
ак(хп + hv) < he
Let us associate with such xn elements y% £ К satisfying
\\уЬ-хп-кь\\ < 2Ле
We set v% ■= {Уп — xn)/h. Since \\v^ - v\\ < 2e and since the space
is reflexive, there exists a weak cluster point vn € ν + 2εΒ of the
sequence (w^)/l>0. Such a vn belongs to the contingent cone Τχ(χη).
Hence ν is the (strong) limit of the elements vn. □
Proof of Theorem 4.1.9 — Since the lower limit of the
contingent cones Tk(x) when χ £ К converges to xq in К is a cone, it is
enough to show that any υ G Limmix-yкХ0Тк(х) with norm equal to
1 belongs to Ck{xq)· Fix such a v. Then for any e > 0, there exists
iV > 1 such that
Vn>iV, VzeBjf(i0ll/n), Β{υ,ε/2)ΓλΤκ{ζ)φ% (4.1)
Assume for a while that υ does not belong to Ск(%о)· This
means that there exist e G]0, l/4[ and sequences xn € Βχ(χο, 1/2η),
К е]0, l/4n[ with
{xn + hnB{v,e))DK = 0 (4.2)
Fix iV such that (4.1) holds true, n> N and set δ := e/(l + ε).
We use Ekeland's Variational Principle (Theorem 3.3.1) with the
closed subset
Q := Kn{xn + \0,hn]B{v,e))3xn

4.1. Tangent Cones to a Subset
137
and the continuous function V defined on Q by V(x) := — \\x — xn\\,
which is bounded from below on Q: There exists then a solution
ζ G Q to the «^-minimization problem
Vi£Q, Цж-ЖпЦ < \\z —xn\\ +δ\\ζ —x\\ (4.3)
We shall exhibit an element χ € Q which, plugged in the above
inequality, yields a contradiction.
For that purpose, we observe that ζ can be written
ζ = xn + (1 — a)hnwn
with a € [0, l] and ||wn —ϋ|| < ε. We first infer from (4.2) that α φ 0.
We check next that
z + [0,ahn]B(v,e) С xn + [0,hn]B(v,e) (4.4)
Indeed, the subset xn + [0, hn]B(v, e) being convex,
ζ + [0, ahn]B(v, e) = (1 - a)(xn + hnwn) + a(xn + [0, hn]B(v, e))
Cxn+ [0,hn]B(v,e)
Finally, ζ belongs to the ball Вк(хо, Vn)- Indeed,
ll^-^nll < (1-α)/ιη(||ϋ||+ε)
an,d thus, the norm of ν being equal to one,
II2 — xq\\ < \\z — xn\\ + \\xn — xq\\ < hn(l + e)+ l/2n < 1/n
Property (4.1) implies the existence of и € Τκ(ζ) Π Β(υ,ε/2).
Therefore there exist sequences uv converging to и and kp > 0
converging to 0 such that ζ + kpup belongs to K. By (4.4), it also
belongs to xn + [0, hn]B(v, ε) for ρ large enough. Hence there exists
ρ such that ζ + kpup € Q and up € Β{υ, ε). Taking χ := ζ + kpup in
inequality (4.3), we get
((1 - a)hn + kp)\\wn\\ - kp\\up-wn\\
< < \\((1 - a)hn + kp)wn + kp(up - wn)\\
= ||(1 - a)hnwn + kpup)\\ = \\x - xn\\ < (1 - α)/ιπ||^π|| +<5&ρ||ϊχρ|

138
4 - Tangent Cones
and thus, using the fact that \\up — wn\\ < 2e,
Αφ||^π|| < 5Α;ρ||ϊχρ|| + kp\\up — wn\\ < kp(6(l + e) + 2e)
Dividing by kp > 0, we obtain
1-е < \\wn\\ < <5(l + e) + 2e < 3e
Hence e > 1/4, the looked for contradiction of the choice of е. П
4.2 Tangent Cones to Convex Sets
For convex subsets K, the situation is dramatically simplified by the
fact that the Clarke tangent cones and the contingent cones coincide
with the closed cone spanned by К — χ:
Proposition 4.2.1 (Tangent Cones to Convex Sets) Let us
assume that К is convex. Then the contingent cone Тк(х) to К at χ
is convex and
CK(x) = TbK(x) = TK(x) = SK(x)
We shall denote by Тк(х) the common value of these cones, and call
it the tangent cone to the convex subset К at x.
Proof — We begin by stating the following consequence of
convexity:
V ν € SK(x), 3 h > 0, such that V t € [0, h], x + tv € К
since we can write that for any t € [0, h]
x + tv = ί 1 - — ] χ + —(χ + hv)
is a convex combination of elements of K.
It is enough to prove that Sk(x) is contained in Ск(х)- Let
v '■= (y — x)/h belong to Sk{x) (where у € К and h > 0) and let us
consider sequences of elements hn > 0 and xn € К converging to 0
and χ respectively. We see that vn ·.= (y — xn)/h converges to υ and
that
τ. ί hn\ hn
xn + hnvn = [1--г]хп + -гУ € К

4.2. Tangent Cones to Convex Sets
139
since it is a convex combination of elements of K. D
The negative polar cone of the tangent cone Тк(х) to a convex
subset, is called the normal cone to К at χ and is denoted by
NK(x) := TK(x)- = SK(x)-
It can be easily characterized by:
Nr(x) = {p € X* | max < p, у > = <p, χ >}
We observe that
ί г) NK(x) С b(K)
) ii) the "asymptotic cone" Ъ{К)~~ С Тк{х)
Theorem 4.2.2 Any closed convex subset of a Banach space is sleek.
Proof— Let К be a closed subset of a Banach space X. We begin
by proving that the graph of the set-valued map
К э χ --> NK(x)
is closed in Χ χ Χ*, when X is supplied with the norm topology and
X*' with the weak-* topology:
Let us consider sequences of elements xn € К converging to χ
and pn € Νκ(χη) converging weakly to p. Then inequalities
У у € К, <рп,У><<Рп,Хп>
imply by passing to the limit inequalities
V у € К, <p,y><<p,x>
which state that ρ belongs to Νκ(χ). Hence the graph is closed,
so that the set-valued map Τκ{·) is lower semicontinuous thanks to
Proposition 1.1.8. □
It may be useful to characterize the interior of the tangent cone
to a convex subset.

140
4 - Tangent Cones
Proposition 4.2.3 (Interior of a Tangent Cone) Assume that
the interior of a convex subset К С X is not empty. Then
л><Л n J
Furthermore, the graph of the set-valued map
К Э χ ~> Ы(Тк(х))
is open.
For the convenience of the reader, we list in Table 4.3 some useful
formulas of the calculus of tangent cones to convex subsets derived
from convex analysis (see [35, Section 4.1.].) The subsets K, Ki, L,
M, ... are assumed to be convex, the spaces Χ, Υ are Banach spaces
and £(X, Y) denotes the space of continuous linear operators.
Remark — Property
TKlnK2(x) = ТК1(х)ПТк2{х)
is false when assumption 0 € Int(UTi — K-i) is not satisfied. Take for
instance two balls K\ and Кч tangent at a point x. The tangent cone
to the intersection {x} is reduced to {0}, whereas the intersection
of the tangent cones is a hyperplane. This shows that we cannot
dispense with the constraint qualification assumptions in the calculus
of tangent cones to inverse images and intersections. □
We provide examples of tangent cones to some specific convex
subsets:
Examples
— 1. We observe first that an element ν belongs to Tr^ (x) if
and only if V{ > 0 whenever χι = 0.
— 2. We denote by
Sn := 1хеЩ. | f^Xi = ll
the probability simplex.

4.2. Tangent Cones to Convex Sets 141
Table 4.3: Properties of Tangent Cones to Convex Sets.
(1) > If χ € К С L С X, then
Тк(х) С TL(x) & NL(x) c Νκ(χ)
(3) > If Xi € Ki С Xi, (i = 1, ...,n), then
ΓΠ»=ι^(ζι,..·,ζη) = Π?=ι?№;)
(4)a) > If Л € C(X, Y) and ж € if С X, then
TA{K)(Ax) = A{T_K{x))
NA(K)(Ax) = A*-*NK(x)
(4)b) > If Ku K2 С X, χ» € gj, г = 1,2, then
Τκ,+κ^Χ! + X2) = ΤΚ}(χΐ)+Τκ2(Χ2)
Νκλ+κ2{χι +χϊ) = ^κ1(χι)ηΝΚ2(χ2)
In particular, if x\ € К and жг belongs to
a subspace Ρ of X, then
2k+p(xi + S2) = TKl(Xl) + P
Nk+p(xi + X2) = ΝκΙχ^ΠΡ1
(5) > If L С -X" and Μ С Υ are closed convex subsets and
A eC(X,Y) satisfies the
constraint qualification assumption
0 € Int(M - A(L)), then, for every χ € L Π A-1 (M),
Гьпл->(М) = TL{x)nA-lTM{Ax)
NLnA->(M) = NL(x)+A*NM(Ax)
(5)a) > If Μ С У is cZoseci convex and if Л € £(X, У)
satisfies 0 € Int(Im(A) - M), then for χ € Л_1(М)
ΤΑ-4Μ)(χ) = A^TiaiAx)
Na->(M)(x) = A*NM(Ax)
(5)b) > If Κι, K2 С X are closed convex and satisfy
0 € Int(UTi - K2), then for any χ € Κι Π Κ2
TKinK2(x) = ТК1(х)ПТк2(х)
NKlnK2(x) = ΝΚι(χ) + Νκ2(χ)
(5)c) > If Κ{ С X, (г = 1,...,η), are closed and convex,
χ € f)i=i К and if there exists 7 > 0 satisfying
\/xi such that \\xi\\ < 7, f|?-i(-fQ - ^i) ^ 0, then
Τ^ιΚί(χ) = (T=i2kf(x)

142
4 - Tangent Cones
Figure 4.3: Counterexample: Tangent Cone to the Intersection
LnM = {x}
™ $
II
W
II
/ TM{x)
TLnM(x) = {o}
Lemma 4.2.4 The contingent cone Ts"(x) to Sn at χ Ε Sn is the
cone of elements ν G Rn satisfying
^2,Vi = 0 L· V{ > 0 whenever ж,- = О (4.5)
ΐ=ι
Proof— Let us take υ G Tsn(x). There exist sequences hp > 0
converging to 0 and vv converging to υ such that yp ;= χ + /ιρυρ
belongs to Sn for any ρ > 0. Then
η - / η η >
Σϋ* = f Σ 2/ρ.· - Σ жр.·
ο
1 ,υΡ \i = l i = l /
so that J2?=i vi = 0- On *^е other hand, if Xi = 0, then
"Pi
2/Pi.//ip > 0
so that V{ > 0.
Conversely, let us take w satisfying (4.5) and deduce that
у := χ -\- hv
belongs to the simplex for h small enough. First, the sum of the
y,· is obviously equal to 1. Second, y,- > 0 when X{ = 0 because in

4.2. Tangent Cones to Convex Sets
143
this case V{ is nonnegative, and, when χι > 0, it is sufficient to take
h < Xi/\vi\ for having yi > 0. Hence у does belong to the simplex.
□
In the case of general convex cones, we have:
Lemma 4.2.5 Let К С X be α convex cone of α normed space X
and χ € K. Then TK(x) = K + Kx.
Furthermore,
ρ € Nk{x) <=> x € Κ, ρ € K~ к <p,x> = 0 <=ϊ χ € NK- (p)
where NK-(p) := {x € K\ V q € K~, < q - p,x > < 0}.
Assume that po € Int(UT+) and set
S := {x € K\ <p0,x>= 1}
Then
Ts(x) = {veTK(x)\ <Po,v>= 0} (4.6)
Proof — Indeed,
K + Kx = K-R+x с ТК[х)
because, for any ζ € Κ, λ > 0,
V/i<l/A, x + h(z-\x) = (l-h\)x + hz € К
Conversely, ν €. Τκ(χ), being the limit of vn such that x + hnvn €
K, belongs to К + Ra: for all n, because
Vn € K — x/hn с K-R+x
To say that ρ € Nk(x) means that < ρ, χ >= σκ(ρ), which is
equal to 0 if and only if ρ € К", so that the second statement of the
lemma ensues.
Observe that S can be written in the form К Пр^1(1) and the
constraint qualification assumption 0 € Int(po(K) — 1) is satisfied
because po(K) is a cone of R containing 1. We then deduce that
Ts(x) = ТК(х)Про1Т{1}(1)

144
4 - Tangent Cones
i.e., formula (4.6.) D
Remark — If we assume that K~ + {x}~ = X* and that
X is reflexive, then Тк(х) = К + ~Rx thanks to the Closed Range
Theorem 2.4.4. D
The next result plays an important role in control theory for
proving that controllability and observability for closed convex
process are dual concepts:
Theorem 4.2.6 (Polar of a Viability Domain) Let X be α
reflexive Banach space, F : X ~> X be a closed convex process whose
domain is the whole space and К be a closed convex cone. The
following statements are equivalent:
' i) \/x&K, F(x) С TK(x)
<
^ it) VqeK+, F*(q) П TK+(q) φ 0
where K+ := -K~ = {p € X* \ Vχ € К, <р,х>> 0}.
Proof — The first condition is equivalent to
УхЕК, Vq€-(TK(x))-, sup <-g,y><0 (4.7)
yeF{x)
But we know from the latter lemma that q € — (Τκ(χ))~~ = —Νκ(%)
if and only if
x € NK-(-q) = (К-+Щ)- = (К++Щ)+ = (TK+(q))+
On the other hand, since the domain of F is the whole space X,
Proposition 2.6.4 implies that
sup < — q, у > + sup < ρ, χ > = 0
y£F(x) peF*{q)
Therefore, condition (4.7) is equivalent to
Vq€K+, Ух€ (TK+(q))+, sup <p,x>>0 (4.8)
p£F*{q)

4.2. T&ngent Cones to Convex Sets
145
Now, since the images of F*(q) are weakly compact, the
Separation Theorem implies that the intersection of F*(q) and TK+(q) is
not empty, (i.e., that 0 belongs to the closed convex subset F*(q) -
Τκ+(ΐ)), if and only if
Ух EX, 0 < a{F*{q)-TK+{q\x)
It is enough to observe that this statement is equivalent to (4.8),
because
K M' K W } \ +oo if x i (TK+(q))+n
Definition 4.2.7 (Pseudo-Convex & Star-Shaped Sets) A set К of
a normed space X is said to be star-shaped around χ ζ Κ if
ЧуеК, VAe[0,l], x + X{y-x) e К
and pseudo-convex at χ € К if and only if either one of the equivalent
properties
i) TK(x) = S^(x) =: ΰ^ψ
ii) К С χ + Τκ(χ)
holds true.
We observe the following
Lemma 4.2.8 If К С X is star-shaped around χ e K, then it is pseudo-
convex at this point.
Convex subsets are star-shaped around each of their elements and
thus, share with them some properties. For instance:
Proposition 4.2.9 If Χ, Υ are normed spaces, A € £(X,Y) is a
continuous linear operator and if К С X is pseudo-convex at some
point χ of K, then
ATK(x) = TA{K)(Ax)
In particular, when К is convex, we have for every у € A(K)
Π АЩф)) = ТА{к){у)
хекг\А~1(у)

146
4 - Tangent Cones
Proof — Take ν € TA(K) (Ax) and let hn > 0 and i)„ 6 У be
sequences converging to 0 and ν respectively such that
Ax + hnvn € A(K) С Ах + А(Тк(х))
Therefore the vn's belong to А(Тк(х)) and so their limit belongs to
the closure of this cone. □
4.3 Calculus of Tangent Cones
By using the tangent cone Ск(х) instead of sleekness, we can state
the following version of the "pointwise inverse function theorem",
which follows from Theorem 4.1.10 and the proof of Theorem 3.4.10.
Theorem 4.3.1 (Pointwise Inverse Mapping Theorem) LetX
be α Banach space, К be a closed subset of Χ, Υ be a finite
dimensional vector-space f : X ι—> У be continuously differentiable around
an element xq € K. If
f'(xo)(CK(xQ)) = Υ
then the set-valued map у ~> f"1(y) Π Κ is pseudo-Lipschitz around
(f(x0),xo)-
In particular, we single out the following consequence:
Corollary 4.3.2 If К is a closed subset of a finite dimensional
vector-space X, then xq € Int(UT) if and only if Ck(xq) = X.
If X is a Banach space, xq € Int(UT) if and only if there exist
constants a € [0, l[, η > 0 such that
Vx€ BK(x0,η), Βχ С Тк(х) + αΒχ D
4.3.1 Intersection and Inverse Image
We are ready to derive the basic results of the calculus of tangent
cones.

4.3. Calculus of Tangent Cones
147
Theorem 4.3.3 Let X andY be Banach spaces, L с X and Μ С Υ
be closed subsets, f : X ι-> Υ be continuously differentiable around
an element xq € L Π f"1(M).
If Υ is finite dimensional, we posit the pointwise transversality
condition
f(xo)(CL(xo)) - CM(f(xo)) = Υ
Then
Ti(xo)nf'(x0r1(TiI(f(xo))) = Tinrl{M)(x0)
and
CL(xo)nf'(xo)"1(CM(f(xo))) С CLnf-i{M)(x0)
Otherwise, to obtain the same results in the case of any Banach space,
we have to replace the pointwise transversality condition by the local
transversality condition.·
there exist constants с > 0, a € [0, l[ and η > 0 such that
\/χ€ΐΠΒ(χ0,η), \/y€MnB(f(x0),v)
BY с f(x) (TbL(x) П cBx) - TM(y) + aBY
. Furthermore the above assumption yields that the set Lflf~1(M)
is sleek (respectively derivable) whenever L and Μ are sleek
(respectively derivable.)
Proof — Let us prove for instance the inclusion for the Clarke
tangent cones. Consider the closed subset
К := Lf\f-l{M)
and take any sequence of elements xn € К which converges to x. Let
us pick any и € Cl(xq) such that f'(xo)u € Cm(/(^o))· Hence for
any sequences hn > 0 and xn € К converging to 0 and xq
respectively, there exist sequences un and vn converging to и and f'(xo)u
respectively such that, for all n > 0,
xn + Кип € L к f{xn) + hnvn € Μ

148
4 - Tangent Cones
We now apply Theorem 4.3.1 to the subset L χ Μ of Χ χ Υ and the
continuous map / θ 1 associating to any (x, y) the element f(x) — y..
It is obvious that the transversality condition
f'(xo)(CL(xo))-CM(f(x0)) = Υ
implies the surjectivity assumption of Theorem 4.3.1.
The pair (xn + hnun, f(xn) + hnvn) belongs to L χ Μ and
(f Bl)(xn + hnun,f(xn) + hnvn) converges to 0
because / is continuous at x.
Therefore, by Theorem 4.3.1, there exist I > 0 and a solution
(xn, i/n) £ I x Μ to the equation
Cfei)(i„,y„) = о
(i.e., yn = f(xn)) such that
||a;n + hnun - xn\\ + \\f(xn) + hnvn - yn\\
< lhn\\{f(xn + Kun) - f(xn))/h η Vn\\
Hence un := (xn — xn)/hn converges to u, and for all η > 0, we
know that xn + hnun belongs to L Π f"1 (M) because xn + hnun = xn
and f(xn + hnun) = yn. When У is an arbitrary Banach space we
apply Theorem 3.4.3 in exactly the same way. Π
We list now three useful corollaries of this theorem:
Corollary 4.3.4 Assume that Χ, Υ are Banach spaces, Μ с Υ is
a closed subset and that f : Χ η-» Υ is a continuously differentiable
map around an element xq € f"1(M).
When the dimension of Υ is finite, we suppose that
bn(/'(x0)) + CM(f(x0)) = Υ
Then
Tf-4M)(xo) = f'(xo)-1 (TM(f(xo)))
Furthermore
7/-1(m)(*o) = f'(xo)-1 (TbM(f(x0)))

4.3, Calculus of Tangent Cones
149
and
Cf-4M)(xo) D f'ixo)-1 (CM(f(xo)))
Otherwise, to obtain the same result in the case of a Banach space
Y, we have to assume that there exist constants с > 0, a € [0, l[ and
η > 0 such that
( \/χ£Β(χ0,η), VyeB(f(xQ),rinM
\ By С cf(x)(Bx)+TM(y)+aBY
Corollary 4.3.5 Let K\ and K2 be closed subsets of a Banach space
X andx € K\ П Яг- If the dimension of X is finite, we assume that
CKl(x) ~ CK2(x) = X
Then
Ί*Κι{χ)ηΤΚ2{χ) С ТК1пк2(х) & Τ*Κι(χ)ηΊ\2(χ) = TbKinK2(x)
and
СкЛ00) п Ск2(х) С CKlnK2(x)
Otherwise, the same result holds true when X is a Banach space if
we suppose that there exist constants с > 0, a € [0,1[ and η > 0 such
that
Vi eKi Γ)Β(χ0,η), Vy £Κ2Γ)Β(χ0,η)
By С (тьК1(х)ПсВх)-ТК2(у) + аВу
Finally, for a finite intersection, we can state:
Corollary 4.3.6 Let us consider n closed subsets Щ of a Banach
space X and xq € Π£=ι Κί· When the dimension of X is finite, we
assume that
n
\/Vl,...,vneX, f](CKi(xo)-Vi) Φ 0
i=l
Then
pifeCzo) = T^^xo)

150
4 - Tangent Cones
and
η
If the dimension of X is infinite, we assume that there exist constants
с > 0, a € [0, l[ and η > 0 such that
' VxitEKiPl B(xq, η), V V{ <E X,
< 3 wi € X, 3 и € Π?=ι (Гл:{ (zj) - «i - ϊ«ΐ) such that
||u|| < cmaxi=i,...,n ||ϋ;|| & ||ϊ«ϊ|| ^ qmaxi=i1...1n |K||
to get the same conclusions.
Proof — It is enough to apply Theorem 4.3.3 to Μ = {0}, the
subset Hf=1Ki x D, where D :== {(x,.. .,x)}xex, and the map /
defined on this set by f(x, y) = χ — у. □
4.3.2 Example: Tangent cones to subsets defined by
equality and inequality constraints
Consider a closed subset L of a Banach space X and two continuously
different iable maps
g := (gi,...,gp):X ^ W L· h := (hlt..., hq) : X н-> №
defined on an open neighborhood of L.
Let К be the subset of L defined by the constraints
Κ := {χ € L | gi(x) > 0, i = l,...,p k hj{x) = 0, j = 1,...,?}
We denote by I(x) := {i = 1,..., ρ \ gi{x) = 0} the subset of active
constraints.
Proposition 4.3.7 Let us posit the following transversality
condition at a given χ € К:
' 0 ti(x)CL(x) = Hq
< ii) 3 wo € Cl(x) such that h'(x)vo = 0 and
Vi€/(s), <gi(x),v0> > 0

4.3. Calculus of Tangent Cones
151
Then и belongs to the contingent cone to К at χ if and only if и
belongs to the contingent cone to L at χ and satisfies the constraints
Vi€/(s), <gi(x),u>> 0 & Vj = l,...,g, h'a{x)u = 0
Proof— It is clear that for every υ € Тк(х) С Tl(x), equations
h'Ax)v = 0, j = 1,..., q are satisfied and inequalities < д[{х), ν >> 0
are satisfied for all active constraints i € I(x)·
The converse inclusion is a direct consequence of Theorem 4.3.3
with
Υ := Rp χ Rq, f = gxh, Μ := Rp+x {0}
It is sufficient to check that the assumption implies the transversality
condition
(g'(x) χ ti(x)) (CL(x))-TRP+x{Q}(g(x),0) =R"xR«
Indeed, take (y, z) € Rp x R9. There exists a solution υ € Cl(x) to
the equation h'(x)v = ζ by the first assumption. Let
a := mm <д[{х),у0>
and
λ := max(0,yi - < g[(x),v >,...,yp- < g'p(x),v >)/a
We set
ai '■= < 9i(x), V + \V0> -yi
By construction, ai > 0 for any i € /(ж), so that a := (αϊ,... ,αρ)
belongs to the contingent cone to R+ to g{x). Hence w := Xvq + υ
belongs to the convex cone Cl(x) and is a solution to the equations
g'(x)w — a = у and h'{x)w = &, so that
(У, z) € (g'{x) x ti[x))u - TRP+x{0}(g(x), h(x)^j Π
Remark — Observe that the above proof implies that if L = X
in Proposition 4.3.7, then an element и € Ск(х) if and only if
Vi€/(s), <g[{x),u>> 0 & ν;' = 1,...,σ, ^-(s)u = 0 □

152
4 - Tangent Cones
Table 4.4: Properties of Contingent Cones to Derivable Sets
in Finite Dimensional Spaces.
(5) > If L С X and Μ С Υ are closed derivable subsets,
/ € ^(Χ,Υ) is a continuously differentiable map and
χ € L П f~1(M) satisfies the transversality condition
f(x)CL(x)-CM(f(x))=Y,then
Tinf-4M)(x) = П(х) П/'(х^ТмМх))
(5)a) > If Μ с У is a closed derivable subset,
/ € Сг(Х,У) is a continuously differentiable map and
χ € f-^M) satisfies bn(/'(x)) - CW№)) = Y, then
Tf-4M)(x) = fix^TMifix))
(5)b) > If K\ and K2 are closed derivable subsets contained in
Χ, χ € К1ПК2 satisfies Скг(х) — Ск2(х) = X-> then
ТкгпкЛх) = TKl(x)nTK2(x)
(5)c) > If Ki С X, (i = 1,... , n) are closed derivable subsets
and χ € Hi Ki satisfies
Vt/i = 1,..., η, ΠΓ=ι (C*, (s) - «i) ^ 0, then

4.3. Calculus of Tangent Cones
153
4.3.3 Direct Image
Let us consider now two normed vector spaces X and Y, a subset
К С X and a differentiable single-valued map / from X to Y.
We saw that for any у € f(K), we have
D Г(*У?к(х)) С Tf{K)(y)
and
U f'(x)№c№) С T){K){y)
xeKnf-Цу)
It is not that easy to find elegant sufficient conditions implying the
equality
U f'(x)(TK(x)) = Tf{K){y)
xeKnf-^y)
Theorem 4.3.8 Let X be α finite dimensional vector-space, Υ a
normed space, Ω С X be an open subset containing К and f : Ω ι-> Υ
be a single-valued map satisfying
f(K) Э у ~> Г\у)ПК
is pseudo-Lipschitz at {f{xo),xo): there exists a constant I > 0 such
that, for any у € f{K) close to f(xo),
а(х0,ГЧу)ПК) < Z||y-/(x0)||
Then, if f is Frechet differentiable at xq, we obtain the equality
f'(xo)TK(xo) = Tf{K)(f(x0))
Proof— Let υ belong to Т^(^(/(жо))· Then there exist
sequences of elements hn > 0 and vn converging to 0 and ν respectively
such that
/(so) + hnvn = f(xn) € f(K) (4.9)
The point is to choose solutions xn € К to the above equation (4.9)
such that a subsequence of un := (xn — xo)/hn converges to some u.
Such an element и belongs to the contingent cone Tk{xq) and is a
solution to the equation f'{xo)u = v.

154
4 - Tangent Cones
Since the set-valued map
f(K) Э у — К П Г1 (У)
is pseudo-Lipschitz at (f(xo),xo) by assumption, there exist a
constant I and solutions xn € К to the equation (4.9) such that
llzo-Znll < l\\f{xo) - f(xo) - hnvn\\ = ^π||υπ||
Therefore, the sequence of elements un is bounded, so that a
subsequence (again denoted) un converges to some u. □
Remark — Therefore, any sufficient condition implying that
for some xq € K, the set-valued map
1(к)эу ^ кпг\у)
is pseudo-Lipschitz at (f(xo)> ^o) will automatically imply the above
equality between the contingent cone to the image and the closure
of the image of the contingent cone.
One of them is given by the Criterion of Pseudo-Lipschitzeanity
(See Corollary 3.4.8.) □
Actually, when the dimension of X is finite, we have another
criterion:
Proposition 4.3.9 Let Χ, Υ be normed spaces, К С Χ, Α € C(X, Υ)
be a continuous linear operator and yo € Y. If the dimension of X
is finite and if for some xq € К П А~1(уо)
кет(А)ПТк(х0) - {0}
then A(TK(x0)) = ТА{К)(А(х0)).
Proof— Let υ belong to TA(K)(A(xq))■ There exist hn > 0
converging to 0, vn € Υ converging to υ and xn € К such that
A(xQ) + hnvn = A{xn) = A(xQ + hnun)
where we set un := (xn — xo)/hn.
If the sequence un is bounded, being in a finite dimensional
vector-space, a subsequence converges to some u, which belongs to

4.3. Calculus of Tangent Cones
155
Τκ(χο)ι ancl satisfies Au = v. If not, a subsequence (again denoted)
\\un\\ would go to oo. Then a subsequence of elements un = un/\\un\\,
satisfying equations Aun = υπ/||?χπ||, converges to some и in the unit
sphere, which is a solution to the equation Au = 0 and which belongs
to the contingent cone Tk(xq)· This is impossible by assumption. □
When the map is no longer linear and the dimension of X still
finite, we can prove:
Proposition 4.3.10 Let К be α subset of α finite dimensional vector-
space Χ, Υ be a normed space, f : X —> Υ a Frechet differentiable
function and у о € f{K), xq € Κ Π f~1(yo).
Assume that for some ε > 0, the subset f~1(B(yo,e) Π Κ) is
bounded. Then condition
Ух€Г1(Уо)ПК, kei(f'(x))nTK(x) = {0}
implies
__U /ΌΌ(Γ*0Ό) = TfiK)(yo)
xeKnf-i{yo)
Proof— Let ν belong to Tf^(yo) be different from 0. There
exist hn > 0 converging to 0, vn € Υ converging to υ and xn € К
such that yo+hnvn = f(xn), which are then bounded by assumption.
Then a subsequence (again denoted) xn converges to some χ € К
satisfying f(x) = y0.
'We set an := \\xn — x\\ and un := (xn — χ)/απ. Then an
converges to 0 and the sequence un being bounded in a finite dimensional
vector-space, has a subsequence converging to some u, which belongs
to TK(x).
On the other hand,
f(x + OnUn) = f(x) + anf'(x)u + αηε{αη) = yo + hnvn
Hence
f'{x)u + e(an) = hnvn/an
"The left hand side being bounded and vn converging to ν φ 0, we
deduce that the sequence hn/an is bounded, so that a subsequence
converges to some λ. Therefore f'(x)u = Xv. By assumption, λ must
be different from 0. Consequently,
υ = f'{x)u/\ € f'(x)(TK(x)) D

156
4 - Tangent Cones
4.4 Normal Cones
We introduce in this section the dual concept of tangent cones: the
normal cones.
We begin with the following characterization of the polar of the
contingent cone:
Proposition 4.4.1 Let К be a subset of a finite dimensional vector-
space X. Then ρ € (Тк(х))~ if and only
' Ve>0, 3 77 > 0 such that My € Κ Π Β(χ,η),
(4.10)
<p,y-x > < e||y-a;||
Proof — Let ρ satisfy above property (4.10) and υ € Тк(х).
Then there exist hn converging to 0 and vn converging to υ such that
у := χ + hnvn € Κ Π Β(χ, η)
for η large enough. Consequently, inequalities < p, vn >< ε imply by
taking the limit that < ρ, υ >< e for all e > 0. Hence < ρ, υ >< 0,
so that any element ρ satisfying the above property belongs to the
polar cone of Тк(х), even in the case of normed spaces.
Conversely, assume that ρ violates property (4.10): There exist
e > 0 and a sequence of elements xn € К converging to χ such that
(p,xn-x) > er|| Xji X\\
We set hn := \\xn — x\\, which converges to 0, and vn := (xn — x)/hn.
These elements belonging to the unit sphere, a subsequence (again
denoted) vn converges to some v. By definition, this limit belongs to
Tk(x), so that < ρ, υ >< 0. But our choice implies that < p, vn »
e, so that < p,v >> e, a contradiction. Π
Notice that Proposition 4.1.2 implies that when the Banach space
X is smooth, then for any у ф К such that Ик(у) is not empty and
for any ζ € Лк(у), J(y — ζ) belongs to the Τκ{ζ)~.
The most important justification for dealing with dual concepts,
is the availability of the one to one correspondences between closed
convex cones and their polar cones, continuous linear operators and

4.4. Normal Cones
157
their transposes, lower semicontinuous convex functions and their
conjugates. This requires one to deal with convex notions.
Since the Clarke tangent cone is convex, it can then be
characterized by its polar cone, which, by analogy with the case of smooth
manifolds, can be regarded as the normal cone.
Definition 4.4.2 Let χ belong to К С X. We shall say that the
(negative) pofar cone
NK(x) ■= CK(x)~ = {p€X* | Vv€CK(x), <P,v><0}
is the (Clarke) normal cone to К at x.
The normal cone Νχ(χ) is equal to the whole space whenever
Ck(x) is reduced to zero.
The normal cone is related to the closed convex hull of the upper
limit of the polar cones to the contingent cones:
Theorem 4.4.3 Let X be a finite dimensional vector-space and К
be a closed subset of X. Then
NK(x) = со (Limsupj,.^,. (TK(y))~)
This a consequence of Theorem 4.1.10 and the Duality
Theorem 1.1.8. In infinite dimensional spaces, we deduce from
Theorem 4.1.13 its dual version
Theorem 4.4.4 Assume that X is uniformly smooth and that the
norm of X* is Frechet differentiable off the origin. Let К be a closed
subset of X. Then
NK(x) = со (σ - Limsup^^ {T^{y))~)
Unfortunately, since the contingent and intermediate cones are"
not convex, duality may just be a blissful wish, since many problems,
which are not smooth by nature, force us. to*use naturally these larger
tangent cones.
The price to pay in terms of loss of information for playing with
duality, just to be able to conserve some familiar classical
formulation, is indeed too high in many situations.
Therefore, these dual concepts we have presented are
recommended for convex, or more generally, sleek subsets.

158
4 - Tangent Cones
Table 4.5: Properties of Normal Cones In Finite Dimensional
Vector Spaces.
(3) > If Xi € Ki С Χ{, (i = l,...,ri), then
(5) > If L с -X" and Μ С У are cZosed , s € L П /-1(М),
/ € C1^, У) is continuously differentiable and
the transversality condition
}'{x)CL{x) - CM(f(x)) = Υ holds true, then
NLnf-4M)(x) С NL(x) + f'(x)*NM(f(x))
(5)a) > If Μ С У is a closed subset and
if / € C1(X, У) is a continuously differentiable map
such that Im(f'(x)) - CM(f(x)) = Y, then
Nf-4M)(x) С f(x)*NM(f(x))
(5)b) > If K\ and K2 are closed subsets contained in X
and ж € K\ П Яг satisfies (7^ (ж) — Сд:2(:е) = X, then
NKlnib(x) С iV^(a;)+iV^2(a;)
(5)с) > Ιϊ Щ С X, (г = 1,..., η) are closed, χ €.f]{Ki and
V t* = 1,..., n, |T=i (С*< (s) - t/i) φ 0, then

4.5. Other Tangent Cones
159
4.5 Other Tangent Cones
4.5.1 Convex Kernel of a Cone
Let us recall that the Minkowski difference К Q L of two subsets К and
L of a vector space is the subset f\xeL{K — x) of elements ζ such that
L + zCK.
Let Γ be a closed cone. We observe that the Minkowski difference
ΤθΤ := {y<=X\ y + TcT}
is α closed convex cone contained in T.
Indeed, it is obviously a cone, contained in Τ (because 0 £ T) and is
convex: If у and ζ belong to Τ θ Τ, then
y+{z + T) С у + Т с Т
so that у + ζ belongs also to Τ θ Τ.
Definition 4.5.1 When Τ is a closed cone, the Minkowski difference ΤθΤ
is called the convex kernel (or the asymptotic cone) of Τ and is denoted
ЪуТ°°.
The convex kernel provides an explicit convex subcone of T, which may
not be the largest one, but is the intersection of the maximal11 closed convex
subcones of T:
Proposition 4.5.2 The convex kernel of a closed cone Τ is the intersection
of the maximal closed convex cones contained in T.
Proof— Let V denote the family of the maximal closed convex cones
contained in T. Then
T°° С f] Ρ
P£T>
Indeed, if χ belongs to T°° and Ρ ε V, then the closed convex cone
Ρ + Л+х is a closed convex cone containing Ρ and contained in Τ
because χ is in the Minkowski difference. The maximal character of Ρ implies
that it is equal to Ρ + R+x, and thus, that ж £ P.
On the other hand,
\Jp = т
pev
11 We say that a closed convex cone К С Г is maximal if for any other closed
convex cone Ρ satisfying К С Ρ С Г, we have К = P.

160
4 - Tangent Cones
because for any χ ε Τ, the closed convex cone R+x is contained in a cone
Ρ £ V-, thanks to Zorn's Lemma: Indeed, for any family of closed convex
cones Ki С Τ totally ordered for inclusion, (J{ Ki '1Έ> a closed convex
subcone of Γ which is an upper bound of the K{'s.
To prove that f]PevP С T°°, pick any χ ε C\pevp anc^ апУ У G ^·
Since у belongs to some PeP, then
x + y ε P + P С Ρ С Γ π' "
Proposition 4.5.3 -Lei К be α closed cone in α normed space and K°° its
convex kernel. Then
CK(Q) = K°° к 4χβΚ, СК(х) D K°° + Rx
Proof— Fix χ ε Κ. Take any element λχ+y ε Έίχ+Κ°° where у ε Κ°°
and consider any sequence of elements hn > 0 converging to 0 and xn ε Κ
converging to x. Since
xn + hn(Xxn + y) = (1 + Xhn)xn + hny ε Κ
we infer that Xx + y belongs to Ск(х).
Consider now ν ε C^(0) and assume for a moment that it does not
belong to K°°. Then there exists χ ζ. Κ such that χ + ν £ К, and thus,
ε > 0 with
(χ + ν + εΒ) П К = 0
But for h > 0 small enough, we know that there exists e ε εΒ with
Лж + h(v + e) = /1(2; + щ + e) ε if
Since if is a cone, x + v + e belongs to K, a contradiction. □
We can interpret the second statement of Proposition 4.1.6 by saying
that the Clarke tangent cone is contained in the convex kernels of the
contingent cone and adjacent cone.
If one is interested in dealing with convex subcones of the contingent
cone without requiring regularity, it may be advantageous to use the convex
kernel of the contingent cone.
4.5.2 Paratingent Cones
Bouligand had also introduced in the thirties the concept of paratingent
cone Ρχ {x) to К at χ ε Κ, which can be regarded as a symmetric concept
to the one of Clarke tangent cone, in the sense that it is defined as
Κ -χ'
Ρκ(χ) := Limsuph_f0+x,_fifX—-—

4.5. Other Tangent Cones
161
Figure 4.4: The Menagerie of Тал gent Cones: they may be all
different
K- := N~ K+ :=
{{-X,X)}X>Q {(~,~)}n>0
Subset К := K~ U K+
Clarke Tangent Cone СКЩ
Adjacent Cone 7^(0)
Contingent Cone Τκ{ϋ)
Limsupa._t0rj!C(a:)
Paiatingent Cone Pftr(O)
Paiatingent Cone Pg (0)
Paratingent Cone P£~ (0)

162
4 - Tangent Cones
Actually, this cone may be too large to be useful:
For instance, take X := R2 and if:=Rx R+. Then
TK(x,0) = CK{x,Q) = Κ and PK(x,0) = X
Therefore, we are led to introduce a more precise concept.
Definition 4.5.4 (Paratingent Cone) Let К С X be α subset of α normed
space X, L a subset of X and χ £ Lf\K. The Bouligand paratingent cone
Ρχ(χ) to К relative to L at χ is defined by
pk(x) ■= lv\ liminf da{x' + hv)/h=ti\
У /ι->0+, x'-*lx )
When L = K, we set PK(x) := Pg(x).
In other words,
_ К — ν
VxGLnK, P%(x) = Limsuph_>0+iy_>z<x——
One of the current choices of L is the boundary dK of К or the complement
KolK.
Lemma 4.5.5 If К is open, then
ЧхедК, PfcK{x) = Pg{x)
Proof — Since ΡχΚ(χ) С P£(a;), we have to show that any ν ε P# (ж),
i.e., satisfying
3 xn —»·~ x, 3 hn —> 0+, 3 vn —► ν such that xn + hnvn ε Κ
belongs also to ΡχΚ(χ). Since К is open and χ £ К, there exists kn e]0, hn[
such that yn := xn + knVn belongs to dK and converges to χ in dK. Take
In '■= hn — kn > 0, which converges to 0 and note that
Уп ~i~ bnVn = Xn ~f- flnVn
belongs to К for any n. Hence υ does belongs to P^K(x). □
We observe at once that
VselnL, p£(x) = -p£{x)
and in particular, that PK(x) = —Рк(х) is symmetric.

4.5. Other Tangent Cones
163
Proposition 4.5.6 Let L and К be closed subsets of a normed space X.
The graph of the set-valued map P% : К П L ~» X is closed:
VxeKHL, Limsupy^LXP%(y) с Р£(х)
Proof— Consider a sequence of elements (xn,vn) of the graph of
the restriction of P£(·) to Κ Π L converging to (x,v). There are sequences
xmn £ L converging to xn in L, hmn converging to 0+ and vmn converging
to vn such that xmn + hmnVmn belongs to К for every n. We then can
associate with any к > 0 integers m,k and nk such that
max(||a;„fc -x)),))xmknk -s„J|, hmkTlk, \\vnk -v\\,\\vmk7lk -υηκ\\) < 1/k
By taking
Ук := ХткПк-1 "к := hmk7lk, Vk := Vmk7lk
we observe that y^ £ L converges to x, hk > 0 converges to 0, υ к converges
to υ and that ук + hkVk belongs to К for every к > 0. Hence υ belongs to
Pk{x). □
A celebrated theorem of G. Choquet states that the contingent cone
TK(x) and the paratingent cone ΡχΚ(χ) coincide on a residual12 of the
boundary dK.
Theorem 4.5.7 (Choquet's Theorem) Let us consider a separable Ba-
nach space X, К С X and let L be a closed subset of X. Then there exists
a residual R С L of L such that
Vieinfl, TK(x) = p£(x)
Consequently, the graph of the restriction ofTx(·) to Kf\R is closed.
Proof — By definition of the upper limit, we have:
' ρκ(χ) = (\,δ,η>ο UyeBl(,,,) Ue]o,i] ((K ~ y)/h + eB)
= a>0n ^кф^К-у)1ь\+еВ)
= Пп>о Limsupy_^x [Ufce]0,i/n] (K ~ v)/h
12Recall that a residual is a countable intersection of dense open subsets An.
Countable intersections of residuals are residuals. Baire's Theorem states that α
residual of a complete metric space is dense. A property which is true for every
element of a residual is called generic.

164
4 - Tangent Cones
Set -
Sn(y) := - (J (K-y)/h
he]o,i/n]
The set-valued map
Sn : L Э у ~> 5n(y)
is obviously lower semicontinuous with closed values. By Theorem 1.4.13,
there exists a residual An с L such that
\/x<=An, Limswpy^^Sniy) = Sn(x)
Take R := f]n>0 An, which is still a residual. Hence
VzEjR, Limsupy^ LXSn(y) = Sn(x)
so that
Vx<=R, P%{x) = P| Sn{x) = LimBupfc_o+(-K'-a:)/'i = Τκ(χ) ρ
n>0
The paratingent cones Ρχ(χ) are complementary convex, i.e., their
complements are convex, and the contingent cones are generically
complementary convex. This follows from the next subsection.
4.5.3 Hypertangent Cones
We introduce the hypertangent cone to if at a; defined as follows:
Definition 4.5.8 (Hypertangent Cone) Let χ belong to a subset К of
a normed space X. The hypertangent cone Нк{х) to К at χ ζ. Κ is the set
of elements υ £ X such that there exist ε > 0, δ > 0 and η > 0 for which
Βκ{χ,η) + [Ό,δ]{υ + εΒ) С К
In the same way that the Dubovitskij-Miljutin cone (see Definition 4.1.3)
to the complement of К is the complement of the contingent cone to K, it
is easy to check the following
Proposition 4.5.9 The hypertangent cone is an open convex cone
contained in the Dubovitskij-Miljutin cone and satisfying
HK{x)+TvK{x) с DK(x)
Furthermore,
V a; £ дК, the complement of Ρχ(χ) is equal to Hg(x) = —Ηκ(χ)

4.5. Other Tangent Cones
165
This ала Choquet's Theorem 4.5.7 imply the following generic
regularity result:
Theorem 4.5.10 (Shi Shuzhong) Let X be α separable Banach space,
К С X be an open subset. Then there exists a residual R с дК of the
boundary of К such that
VxedK, TK(x) = P%K(x) = -(X\HK(x))
In particular, Τχ(·) is generically complementary convex on dK.
Proof— Indeed, Choquet's Theorem 4.5.7 implies that there exists
a residual of dK on which Τκ(·) and ΡχΚ(·) coincide. Furthermore, К
being open by the last lemma, ΡχΚ(χ) is equal to P§(x) = —P%(x). We
it
then use the fact that Ρϊί(χ) is the complement of Нк{х)· D
it
4.5.4 A Menagerie of Tangent Cones
A. D. Ioffe has introduced a systematic way to classify the four main tangent
cones which we have introduced, i.e., the paratingent, contingent, adjacent
and circatangent cones, as well as the Dubovitskij-Miljutin and hypertan-
gent cones.
Let К and L be subsets of a normed space X and χ £ Κ Л L. We
denote by the notation ]{an —> a) the fact that an = a for all n. The subset
TkRS(K,x) denotes the cone of elements υ £ Χ satisfying
Q(xn -+l x), R(hn -> 0+), S(vn —> v), xn + hnvneK
where Q(-),S(·) £ {V, 3, f} and R{-) £ {V, 3}.
Hence the cones we have introduced can be written
' TK(x)
τνκ{χ)
CK{x)
PkW
DK(x)
. HK(x)
= %^X)
= 4^{K'X)
= Tfa{K,x)
= Tii3^K,x)
= TU{K>X)
= ^ν,ν,νί-^!37)

166
4 - Tangent Cones
The largest of these cones (when L = K) is the paratingent cone
Tif-g 3(if, x) and the smallest is the hypertangent cone T^VV(K, x).
The proofof Proposition 4.1.6 implies that the cones T^ s(K,x) are
convex and are contained in the convex kernel of TqRS(K:x):
T^s(K,x) + T§RS(K,x) с T«RS{K,x)
We also observe the following complementary property: recall that К
denotes the complement of К and set V := 3, 3 = V and f = t· Then
the complement of TqRS(K, x) is equal to Т~~~-(К,х)
In infinite dimensional space, the complexity is compounded by the fact
that (at least for the properties £(·)) we can use either the strong or weak
convergence....
It is clear that we can meet in the proofs of theorems various instances
of these tangent cones, but few of them enjoy useful properties. As far as
the experience of the authors and some of their colleagues is concerned,
the contingent cone plays a major role both in optimization theory and
viability theory, the use of the Clarke tangent cone is necessary when lower
semicontinuity is required or convexity is needed (in this case, it may be
still better to use convex kernels whenever the Clarke tangent cone is
degenerate.) The paratingent cones are the only ones whose graphs are closed,
a property which is quite important indeed, and which can be transmitted
generically thanks to Choquet's Theorem.
The introduction of the cones T^R s(K,x) can also be justified as
particular cases of asymptotic tangent cones to (constant) sequences of subsets.
This is the topic of the next section.
4.6 Tangent Cones to Sequences of Sets
Let X be a normed vector space. We consider a sequence of subsets Kn с X
of X and their upper and lower limits denoted by if" and КУ respectively.
We shall adapt the concepts of contingent cones and intermediate cones
to sequences of subsets Kn in the following way:
Definition 4.6.1 We shall say that the circatangent cone to the sequence
(Kn)n at χ £ КУ is the set С\к Лх) defined by
^(Kn)(x) := ■L,iminfn_).00i/fn3Xn_).x^n_>o+—r

4.6. Tangent Cones to Sequences of Sets
167
IfxGK^, we shall say that
is the paratingent cone to the sequence (Kn)n at x.
Remark — When we consider a constant sequence Kn := K, we
see that circatangent cone C^KnJx) coincides with the Clarke tangent cone
CK{x).
In the case of the constant sequence Kn := K, the paratingent cone
PL· Jx) coincides with the paratingent cone Рк(х)- □
The upper limit of the paratingent cones to a sequence of Kn is
contained in the paratingent cone to the sequence of the Kn's:
Proposition 4.6.2 Let us consider a sequence of subsets Kn С X of X
and their upper limit KK Then
LimsupKnBXn^xTKn{xn) С UmsupKn3Xn^xPKJxn) с PlKn){x)
Proof — It is the same as the one of Proposition 4.5.6. □
We provide a characterization of the polar cone to the paratingent cone
to a sequence of subsets:
Proposition 4.6.3 Let X be a finite dimensional vector-space, Kn be a
sequence of closed subsets of X. Then ρ ε ί PL· Лх)) if and only if there
exists a sequence xn £ Kn converging to χ such that
V e > 0, 3N: η>ϋ such that Vn>N,
V xn £ Kn ΠΒ(χη,η), < ρ, χη — xn > < e\\x
η Xn ||
Proof— Let ρ satisfy above property (4.11) and ν £ PL· Jx). Then
there exist hn converging to 0 and vn converging to ν such that for η large
enough
xn ■= Xn + hnVn £ Kn ηΒ(χη:η)
Prom inequalities < p: vn >< ε we deduce that < ρ, ν >< ε for all ε > 0.
Hence < ρ, ν >< 0, so that
pe (P(W*))~
even in the case of normed spaces.

168
4 - Tangent Cones
Conversely, assume that ρ S PL· Ax) violates property (4.11): There
exist ε > 0 and a sequence xn £ Kn converging to χ such that
VP) 3?n ^n) •*> £ 11^n Xn\\
Set hn := \\xn— Xn\\ (which converges to 0) and vn := (xn — Xn)/hn. These
elements belonging to the unit sphere, a subsequence (again denoted) vn
converges to some υ £ PL· Ax), so that < ρ, υ >< 0. But our choice
implies that < ρ,υη » ε, so that <ρ:υ >> ε, a contradiction. Π
The circatangent cone to a sequence of subsets enjoys the same
properties as the Clarke tangent cones: it is always a closed convex cone and is
actually equal to the lower limit of the contingent cones in finite dimensional
spaces:
Theorem 4.6.4 Let us consider α sequence of closed subsets Kn С X of
α Banach space X and an element xq = linin_юо хп (where xn £ Kn) of
their lower limit K^. Then
1. — The circatangent cone С\к Лхо) is always a closed convex
cone
2. — The lower limit of the contingent cones Тлп(хп) is contained
in the circatangent cone:
Liminfn_>0o,i<:„3a:„-»a:T,Kn(a;Tl) С C^K^{xQ)
and equality holds when the dimension of X is finite.
Proof of the first statement — It is similar to the one of Proposition
4.1.6.
Proof of the second statement — Pick an element
ν £ LiminfA-n3a!n_a:orK-T1(a;n)
with norm equal to one. This means that for any ε > 0, there exists N > 1
such that
Vn > N: Vz £ BKJx0:l/n): Β(υ,ε/2)Γ\ΤΚη(ζ) φ 0 (4.12)
We shall derive a contradiction by assuming that υ does not belong to
C$KJxo). This means that there exist ε £J0, l/4[ and sequences
Xn £ B/cfro.l/Zn), /ine]0,l/4n[

4.6. Tangent Cones to Sequences of Sets
169
with
{xn + hnB(v,e))nKn = 0 (4.13)
Fix N such that (4.12) holds true, η > N and set δ := ε/(1 + ε).
We use Ekeland's Variational Principle (Theorem 3.3.1) in exactly the
same way as in the proof of Theorem 4.1.9 to verify that (4.13) cannot hold.
Proof of the third statement — As for Lemma 4.1.14, the third
statement follows from the second one and:
Lemma 4.6.5 Let Kn be nonempty closed subsets of α reflexive Banach
space and xq = ϋπ^^^ xn (where xn ε Κη). Then
C\Kn){x0) С Liminf^g^
Proof— The proof is analogous to the one of Lemma 4.1.14. Π
When the subsets Kn are convex, we obtain the following relations:
Proposition 4.6.6 Let us consider a sequence of closed convex subsets Kn
of a reflexive Banach space X and its upper and lower limits K^ and K^.
Then for every χ ε Кь:
Ткъ{х) С Ъ]хаЫКпЭХп^хТКп(хп)
and
Τκί{χ) С LimsupKn3Xn^xTKn(xn)
Assume now that X is a Hilbert space and that the lower limit is equal to
the sequentially weak upper limit σ — Limsupn_>00i;srTl and denote by К the
common value. Then
Liminfn^oo^sz^xTVJzn) = TK(x)
Proof
— 1. Let us take χ ε Kb and ν £ SKb(x), the cone spanned by
КУ —x. Hence there exists h > 0 such that χ + hv ε Кь.
Consider any sequence of elements xn belonging to Kn and converging
to x: and let yn denote the projection of x+hv onto Kn. Since yn converges
„to χ + hv, the sequence of elements vn := (yn — xn)/h converges to v.
Furthermore, for hn ε]0, h[
xn + hnvn = ί 1 - -γ J xn + -~yn ε Kn
and we infer that vn belongs to the contingent cone TKn(xn).

170
4 - Tangent Cones
Hence г/, the limit of the vn's, belongs to the lower limit of the tangent
cones TKn(xn)·
— 2. Let χ belong to K$ and υ := (у — x)/h be chosen arbitrarily
in the cone spanned by K^—x. Denoting by xn and yn the projections onto
Kn of χ and χ + hv respectively, we infer that vn := (yn — xn)/h belongs
to the tangent cone Ткп (xn) and converges to v.
— 3. We deduce from Corollary 7.6.5 below that
NK(x) = a-Limsupn_>00ii<Ti3XTi_>xiVA:„(a;n)
By transposition, Theorem 1.1.8 implies that
L\mioiTl^00iKn3Xn^xTKn(xn) = TK{x) □
We shall derive from the Stability Theorem a formula for the circatan-
gent cone of an inverse image.
Theorem 4.6.7 Let X and Υ be two Banach spaces and A £ C(X,Y) be
a continuous linear operator. Consider sequences of closed subsets Ln с X
and Mn С Υ. Let us assume that there exist constants с > 0, a £ [0,1[ and
η > 0 such that
\fxn £ Kn ПΒ(χ0,η), yn ε ΜηΠΒ(Αχ0,η)
BY С A (T*n(хп) ПсВх)- Гм„(у„) + aBY
Let x belong to the lower limit of the sequence Ln Π A~1(Mn). Then
C\Ln){x)nA-'C\Mn){Ax) с СЬпА-чт(х)
Proof — Take any sequence of elements xn ε Ln Г\А-1{Мп) which
converges to χ and any и ε C$L Jx) such that Au ε CiM ΛΑχ). Hence
for any sequence hn > 0, there exist sequences un and vn converging to и
and Au respectively such that, for all n > 0,
xn + h к Axn + hnvn ε Mn
We now apply Theorem 3.4.5 to the subsets Ln χ Mn oiXxY and the
continuous linear operators Α θ 1 associating to any (x: y) the element
Ax — y.
The stability assumptions of this Theorem are obviously satisfied. The
pair (xn + hnun, Axn + hnvn) belongs to Ln χ Mn and
(Α θ l)(xn + hnun, Axn + hnvn) converges to 0

4.7. Higher Order Tangent Sets
171
Therefore, by Theorem 3.4.5, there exist I > 0 ала (xn, yn) £ Ln χ Μη such
that, for η large enough, (Α θ l)(xn, yn) = 0 and
\\xn + hnun — xn\\ + \\Axn + hnvn -yn\\ < lhn\\Aun — vn — 0||
Hence un := (xn — xn)/hn converges to u: and we observe that for all
,n > 0, xn + hnun belongs to Ln П A~1(MTl). □
4.7 Higher Order Tangent Sets
The contingent cones are upper limits of the sets (K — x)/h when h > 0
converges to 0.
We may need more accurate approximations, and consider the sets of
cluster points of mi/l-order differential quotients
К — x — hv\ — ··· — hm~1vm^-i
when h > 0 converges to 0.
Definition 4.7.1 Let χ belong to α subset К of α normed space X and
Vi,... ,vm-i be elements of X.
We say that the subset
m(m), ч τ- K-X-hVx hm~1Vm-l
T^ '{x,v1,...vm-.1) := Limsuph_+0+ —
is the mi/l-order contingent subset of К at (χ,υι,... ,vm_i).
We see at once that Т^-'(х) = Тк(х) is the contingent cone to if at a;
and that if T^(x,Vi,...,vm-i) is not empty, then necessarily,
υ1£Τ£){χ),υ2€Τ£)(χ,υ1), ... .VieT^'ti,!)!,...,»^)
Naturally, we may need more regularity and ask for stronger limits of
the subsets (K—x — hv\ hm~1vm-i)/hm, as in the case of first order
approximations.
Definition 4.7.2 Let χ belong to a subset К of a normed space X and
Vi,.. .,vm-i be elements of X. We say that the subset
mbfmb \ τ- ■ с K — X—hVi km~1Vm-i
T%m)(x,v1,...vm-1) := Limmf^o+ r^ —

172
4- Tangent Cones
Table 4.6: Properties of m -order Contingent Sets
(i)
(2)
(3)
(5)
>
>
>
>
If К С L and χ ζ. Κ, then
If Ki С X, (г = 1,..., те) ала а; £ (J4 ifi, then
Γυ?)1ΛΓ.·(:Ε'υι'···'υ'"-1) = ^еЦх)Тк!](х^1,--
where I(x) := {г \ χ £ ifj}
If Ki С -Xi, (г = 1,..., η) and э^ £ ifi, then
m(77l) /
ГТ" ЛГ ^'Εΐ' * ' ' ' 'Ет1' ^1,1) ■ ■ ■ ι ^Ί,η» ■ ■ " ) ^m—1,1) · ■ ■
С ГК=1Г*7 (xi>vl,ii--->Vm-l,i) _
If Κι С -X", (г = 1,..., η) and χ £ |~|{ i^i, then
Τ^^Ζ,Τ*,...,«„_!) С [^l^Wl,-
■,^m-l),
ι Vm—Ι,η)
■,Vm-l)
is ihe mi/l-order adjacent subset o/ Uf at (χ, νχ,... , um._i) and that
r,(m), ν τ -c K-y-hvx hm~1vm-i
Lfc [x,v1}. ..vm-\) := Limint/l_)o+,s-lKi r^
is the mth -order circatangent subset of К at (x,V\,... ,vm_i).
We shall say that К is mi/l-derivable at x, Vi,..., um_i if the mth-order
contingent and adjacent sets coincide.
They are closed subsets satisfying, for any A > 0,
T^\x, At/X> Α2υ2,..., Ат-Чт_!) = XmT^\x,v1,v2,... ,t/m_i)
Τ^ζ,λ^,λΧ... , Α™"1^-!) = A-^Wi,^... ,t/m_i)
(4.14)
As for the mi/l-order contingent sets,- the adjacent set
T^(m)(a;,vi,...,vm_i)
(respectively the circatangent set C™ (x, Vi,..., um_i)) is empty if one of
the г/j's does not belong to T^°' (x, vi,... ,Vj-\) (resp. C^'(x,Vi,... ,Vj-i))
for j = 1,... , m — 1.)

4.7. Higher Order Tangent Sets
173
We observe that
m—l)
С T^m)(a;,iii + i;i,...,iim_i + i;m_i)
and
, Hi,. .. , _-m_l ,Vl,...,Vm_i)
С C*gl){x,ul + vi,...,um--i+vm-i)
So that, taking into account (4.14), we obtain the following property of
m"1-order circatangent sets:
v/л η ХтС(£1)(х,и1,...,ит„1) + цтС(£1)(х,у1,...,ут^1)
VA,M>0, _____
r(m)( Aui + μν-ί Am~1um_i +μ7η~1υ7η_ι
C °* {X' λ + μ '···' (Α + μ)--1 j
We also observe that
f Cj^l)(a;,i;i,...'ym_i)
[ := Liminf^-*o+,у-*кх,^-*!>!,. .,f^_1-*t>m_1
so that if (^"^(ζ,υι,... ,Vj-i) ^ 0, then 0 ε (^(ζ,υι,... ,vm_i).
Example — Consider for instance the mi/l-order contingent set Тду (ж).
We obtain that for every χ ε Int R™ and all Vi ε Rn
r^i-r.vi,...,^.!) = Rn
For every χ from the boundary of R™ we have
«m ε Tg? (_,«-.■■ ■.«·»--) = T$g\x,v1,...,vm-1)
if and only if
vmi > 0 whenever a^ = vit = · ■ ■ = v^m_1)i = 0
and
vmie'Ri£xi = vli = --- = V(k-!){ =0, vki > 0 for 0 < fc < m - 1 Π
K-y-hv[ hm-1-'
'm-1

174 4- Tangent Cones
Table 4.7: Properties of mth-order Adjacent Sets.
(i)
(2)
(3)
(5)
>
>
>
>
If К С L and χ ζ. Κ, then
If i^ С X,~ (i = 1,..., n) and a; e Ц -Ki. then
where /(ж) := {г | a; £ Ki}
If -FQ С Хг-, (г = 1,..., η) and э^ £ ifj, then
ГТ" AT ''El' ' ' ' '•Еп'г'111' · ' · ι^Ι,η' ■ · · ι^πι—Ι,Ι) · · · >Vm—l,n)
If Ki С X, (г = 1,..., η) and χ £ f|i -K"ii then
The formulas enjoyed by the contingent and adjacent cones can be
easily extended to the mi/l-order contingent and adjacent sets. They are
presented in Table 4.6 and Table 4.7. However, the formulas for the direct
and inverse images involve higher ordeT derivatives and are presented in the
case of second order sets for simplicity.
Proposition 4.7.3 Let Χ, Υ be normed spaces and let д £ C2(X, Y),
К С Χ, χ £ ~Κ and Μ с Υ be given. Then, for any υλ £ TK{x)
\/v2€T^)(x,v1), g'^T^ix^) С rs((2i)(5(a;)l5'(a;)t/i)-^"(a;)(«il«i)
and thus
ί-1(Μ)(^ι) С д'(хГ1(т^(д(х),д'(х)у1)-У"(х)(у1,у1)
lrl(M)1 ' 1J yy ' \ M w^i'V \·υ)"ί) 2
The same results hold true for the second order adjacent sets.
Remark — When g = A £ £(X, Y) is a continuous linear operator,
the higher order derivatives vanish and we obtain the formulas
' А (г£т) (a;, «χ,... ,«„_!)) С 1^jc){Ax,Av1,...,Avn-i)
<
^ A (r£m)(a;,«!,...,«„,_!)) С 3*((™))(Aa;>A«i>...>At/m_i) □

4.7. Higher Order Tangent Sets
175
In order to obtain the converse inclusions, we need to assume a transver-
sality assumption.
Theorem 4.7.4 Let X and Υ be Banach spaces, L с X and Μ С Υ be
closed subsets, f : X >—> Υ be twice continuously differentiable around an
element x0 £ Ln f~l{M).
If Υ is a finite dimensional vector-space, we posit the pointwise transver-
sality condition
?Ы(СьЫ) ~ CM(f(xQ)) = Υ
Then
^'^■«On/'W"1 (^(/(so),/'(:co)ui)- У"(х0)(щ,щ))
С
■42)
nnz-W^i)
and, when Μ is twice derivable, we have the equality instead of the
inclusion. Furthermore
т¥2)(х0,и1)пГ(х0)-1 (T^2)(f(x0)J'(x0)u,) - y"(x0)(uuUl))
= 3ffi}-W*o,«i)
Otherwise, to obtain the same results in the case of any Banach space, we
have to replace the pointwise transversality condition by the local transver-
sality condition:
there exist constants с > 0, a € [0,1[ and η > 0 such that
νχ&ΣΓ\Β{χ0:η), УуеМпВ(/(а;о),ч)
BY С f'(x) (ТЦх) П cBx) - TM(y) + αΒγ
Proof — Let us prove for instance the equality for the second order
adjacent sets. Consider the closed subset К := Ln f~1(M). We already
know from Theorem 4.3.3 that every щ ε TbL{x0) П f {хоУ1 {TbM{f{x0)))
belongs to T^nf_i,Ms(xo). Let us choose such an element u\ and pick any
Щ S t£(2)(:eo,?Xi) such that
f'(x0)u2 £ Ttf (/(zo),/'Mui) - |/"(a;o)(ui,Ui)
Hence for any sequence hn > 0 converging to 0, there exist sequences u2n
and v2n converging to u2 and f'(xo)u2 + \f"{xo){u\,U\) respectively such
that, for all π > 0,
x0 + hnii-L + h\u2n e L к f(x0) + /in/'(a;o)"i + h\v2n ε Μ

176
4- Tangent Cones
We apply now-Theorem 4.3.1 to the subset L χ Μ of Χ χ Υ and the
continuous map / θ 1 associating to any (ж, у) the element f(x) — y. It is
obvious that the transversality condition
f'(xo)(CL(xo»-CM(f(x0)) = Υ
implies the assumption of Theorem 4.3.1. On the other hand
vj(/ θ l)(x0 + hnm + h2nu2n, f(x0) + hnf'{x0)ui. + h2nv2n) converges to 0
because / is continuously differentiable at xq.
Therefore, by Theorem 4.3.1, there exist I > 0 and a solution (xn, yn) ε
L χ Μ to the equation
(/θ 1) (£„,£„) = 0
(i.e., yn = f(xn)) such that
\\x0 + hnU! + h2nu2n - xn\\ + \\f(xo) + hnf'ixo)^ + h2nv2n - yn\\
< lh2n ||(/(a;o + Нпщ + h2nu2n) - f(x0) - /i„/'(a;o)"i)/^ - ^2n||
Hence u2n := (xn — xq — /ι„ϊΧι)//ι^ converges to u2, and for all π > 0,
we know that xq + hnu-i + ^u2n belongs to L Π f~1(M) because
Xq + hnU! + h2nUn = Xn
and f(x0 + НпЩ + h2nun) = yn.
To prove the converse inclusion, fix u2 £ Т^.^^Лх^.щ). Then for
every sequence hn —> 0+, there exists a sequence u2n —> u2 such that
x0 + hnui + h2nu2n £ L7 f(x0 + hnU! + h2nu2n) ε Μ
Thus u2 ε Τ^'^ο,ϊΧι) and since
f(xQ + Ки-^ + h2nu2n)
f{xo) + hnf'(x0)ui + ^hlf"{x0)(uuui) + Η2ηΓ(χ0)η2η + ε{Ηη)Ηΐ
where e(hn) converges to zero with hn, we deduce that
/'Wh2 + J/"W(4i,iii) e Т^2)(/(ж0),/'(а;о)щ) П

4.7- Higher Order Tangent Sets
177
Example: Second order contingent sets to subsets defined by
equality and inequality constraints
Consider a closed subset L of a Banach space X and two twice
continuously differentiable maps
9 ··= (51, · · · ,9p) ■· X ^ RP & h := (hi, ...,hq):X~B?
defined on an open neighborhood of L.
Let К be the subset of L defined by the constraints
Κ := {χ £ L I д{(х) > 0, ί = 1,... ,p &/y(a;) = 0, J = l,...,g}
and let
/(a;) := {»=ll...,p|5i(a;) = 0}
denote the subset of active constraints.
We posit the following transversality condition at a given χ ε Κ:
' i) ti(x)CL(x) = R9
ii) 3 vq £ Cl(x) such that h'(x)vo = 0
& Vie I(x), < g[(x), vq > > 0
By Proposition 4.3.7, υ,χ belongs to the contingent cone to if at a; if and only
if u\ belongs to the contingent cone to L at χ and satisfies the constraints
VieJ(s), <д[{х),щ >> 0 & V j = 1,...,ςτ, /i^(a;)u1=0
Proposition 4.7.5 -Lei us sei
7<2>(a;, ux) ;= {i £ /(a;) | < s<(z), ux > = 0}
Then U2 belongs to the second order contingent set to К at (x, ui) if and
only if v,2 belongs to the second order contingent set to L at (χ, ΐίχ) and
satisfies the constraints
V i<=№(x,ui), <9'г{х),и2>+\д'!(х)(иищ) > О
and Vj = l,...,5, ti-(x)u2 + \ti-{x){ui,ui) = $
Proof — Define
Υ = Β? χ R», / = (5, /ι), M = R^xR«
The proof follows by exactly the same arguments as the ones of
Proposition 4.3.7, after observing that the previous example yields that Μ is twice
derivable and using Theorem 4.7.4. D

Chapter 5
Derivatives of Set-Valued
Maps
Introduction
We need to differentiate set-valued maps as much as we need to
differentiate single-valued maps, for extending the Inverse Function
Theorem to set-valued maps for instance, and for many other reasons.
How should we go about it? The idea is very simple, and goes
back to the prehistory of the differential calculus, when Pierre de
Fermat introduced in the first half of the seventeenth century the
concept of a tangent to the graph of a function:
the tangent space to the graph of a function f at a point (x, y)
of its graph is the line of slope f'(x), i.e., the graph of the linear
function и н-> f'(x)u.
It is possible to implement this idea for any set-valued map F
since we have introduced concepts of tangency for any subset of a
normed space. For instance, we can regard the contingent cone to
the graph of the set-valued map F at some point (x, y) of its graph as
the graph of the associated "contingent derivative" of F at this point
(z,y).
Since the contingent cone is at least... a cone, the contingent
derivative is at least positively homogeneous (or a process). This is
what remains of the familiar, but luxurious, requirement of linearity.
However, when the contingent cone happens to be convex (this
179

180 5- Derivatives of Set-Valued Maps
is the case when the graph is sleek), the contingent derivative is
a closed convex process, i.e., a set-valued analogue of a continuous
linear operator.-
The graphical derivatives associated to the Clarke tangent cone,
which we call here the circatangent derivatives, are also closed convex
processes. They are.presented in the second section together with
the adjacent derivatives, the graphs of which are the adjacent (or
intermediate) tangent cones. Hence the graphical derivatives which are
attached to each such tangent cone correspond to different regularity-
requirements.
It is also possible to define them as adequate limits of differential
quotients. The limits, involving "limsupinf's", that would not have
been proposed by any sensible or even sober person in a right frame
of mind had they not been derived from the "graphical approach."
They are still pointwise limits, which puts them in a different class
than the distributional derivatives1 introduced by Laurent Schwartz
in the early fifties. We shall see in Chapter 7 that they are also
adequate "graphical limits" of differential quotients.
Chain rules are provided in the third section.
With this differential calculus at hand, we are able to
approximate solutions to an inclusion
F(x) э у
by solutions to approximate inclusions
Fn(xn) Э у η
by adapting Lax's principle:
Convergence of the data yn to y, consistency of Fn to F and
stability of the Fn's imply the convergence of some solutions xn's to
the solution x.
Consistency means that (x, y) belongs to the lower limit of the
graphs of the maps Fn. Stability means here that the norms of the
'which were designed to make sense out of Dirac's operational calculus and
solve partial differential equations.
In order to keep the linearity of the differential operators, the limit of
differential quotients is taken in spaces of distributions endowed with topologies
much weaker than the pointwise convergence topology, used ever since Cauchy's
rigorous definition.

5.1. Contingent Derivatives
181
inverses DFn(xn,yn)~l are bounded when the pairs (хп,Уп) range
over a neighborhood of (x,y). We shall deduce that, if the right-
hand side уn converges to y, there exist approximate solutions xn €
-P1-1 (Уп) which converge to x, and we shall provide an error estimate.
When the maps Fn = F are constant, this principle boils down
to the statement of the Inverse-Function Theorem.
We consider also an application to Qualitative Analysis, a field
started in mathematical economics under the name of "comparative
statics" and taken over by computer scientists as a domain of
Artificial Intelligence. For instance, if the vector-spaces X and Υ are
finite dimensional, we are just interested in the possibility of solving
an equation or an inclusion F(x) Э у for prescribed sign vectors. We
provide a dual criterion allowing us to answer this question.
Finally, by using high-order contingent sets to the graph of a
set-valued map F, we define in Section 6 high-order derivatives of a
set-valued map.
We derive in Chapter 10 variational inclusions by linearizing a
differential inclusion, the solution map of which contains the
contingent derivative of the solution map of the original differential
inclusion.
5.1 Contingent Derivatives
By coming back to the original point of view proposed by Fermat, we
are able to define geometrically derivatives of set-valued maps from
the choice of tangent cones to the graphs, even though they yield
very strange limits of differential quotients.
Definition 5.1.1 Let Χ, Υ be normed spaces and F : Χ ~> Υ be a
set-valued map.
The contingent derivative DF(x,y) of F at (x,y) £Graph(F) is
the set-valued map from XtoY defined by
Giaph(DF(x,y)) := TGTaph{F)(x,y)
When F := f is single-valued, we set Df(x) := Df(x,f(x)).

182
5 - Derivatives of Set-Valued Maps
We shall say that F is sleek at (x, y) G Graph(-F) if the map
Graph(F) Э (x',y') ~~> Giwh(DF(x',y'))
is lower semicontinuous at (x,y) (i.e., if the graph of F is sleek at
(z,y))·
It is said to be derivable at (x, y) if Graph(-F) is derivable at that
point.
The set-valued map F is sleek (respectively derivable) if it is sleek
(respectively derivable) at every point of its graph.
Finally, we shall say that F is pseudo-convex at (x, y) G Graph(F)
if and only if
Ух' £ Dom(-F), F(x') С DF(x,y)(x' - x) +y
(i.e., if its graph is pseudo-convex at (x,y)).
Naturally, the contingent derivative is a closed convex process
whenever F is sleek at (x,y).
We can easily compute the derivative of the inverse of a set-valued
map F (or even of a non injective single-valued map): The contingent
derivative of the inverse of a set-valued map F is the inverse of the
contingent derivative:
D(F~l)(y,x) = DF(x,y)-1
The restriction F := f\x of a single-valued map / to a subset
К С X provides an example of a set-valued map defined by
ί\κ(χ)
\ f(x) ifz e К
:~ 10 ifs 0 К
for which we obtain the following formula:
If / is Prechet differentiable around a point χ G K, then the
contingent derivative of the restriction of f to К is the restriction of
the derivative to the contingent cone:
D(f\K)(x) = D(f\K)(x,f(x)) = f(x)\TK{x)
Actually, this follows from the useful

5.1. Contingent Derivatives
Figure 5.1: Graph of the Contingent Derivative

184
5- Derivatives of Set-Valued Maps
Proposition 5.1.2 Let X and Υ be normed spaces, f be a single-
valued map from an open subset Ω С X to Υ, Μ : Χ ~> Υ be a
set-valued map. Define the set-valued map F : X ~> У by:
V χ € X, F(x) := f(x) - M(x)
If f is Frechet differentiable at χ € Ω Π Dom(M), then for every
У € F(x)
DF(x,y)(u) = f'(x)u-DM(x,f(x)-y)(u)
Proof— Let ν belong to DF(x,y)(u). Then there exist hn > 0
converging to 0 and sequences un and vn converging to и and ν
respectively such that for every η
У + hnvn € f(x + hnun) - M(x + hnun)
Since
f(x + hnun) = f(x) + hn(f'(x)u + e(hn))
where e(hn) converges to 0 with hn we get
f(x) - у + hn(f (x)u - vn + e(hn)) € M(x + hnun)
and thus f'{x)u - ν € DM(x, f(x) - y){u).
Conversely, assume that f'(x)u—v belongs to DM(x, f(x)—y)(u).
Hence, there exist a sequence hn > 0 converging to 0 and sequences
un and wn converging to и and f'(x)u — ν such that
f(x) - у + hnwn € M(x + hnun)
Then for some e(hn) -* 0, the sequence vn := f'(x)u + e(hn) — wn
converges to ν and satisfies
y + hnvn € f(x + hnun) - M(x + hnun), □
When we add "state constraints" L, this result becomes:
Proposition 5.1.3 Let Χ, Υ be normed spaces, f : Ω (-► Υ be a
single-valued map, where Ω С X is open, Μ : Χ ~> Υ be a set-valued
map and L С X. Let F : X ~> У be the set-valued map defined by:
f(x) — M(x) when χ € L
0 when χ £ L
F(x) :=

5.1. Contingent Derivatives
185
If f is Freehet differentiable at χ € Ω Π Oom(F), then for every
У € F(x),
ПР(т „ν,Λ г [ fl{-x)u ~ DM(X> f№ ~ M") when u € Tl^
V*ix>yKU)C\ 0 when u£TL(x)
Equality holds true when we assume that either L or Μ is derivable
at χ and Μ is Lipschitz at x.
In particular if Μ is constant, then
\fu € TL(x), DF(x,y)(u) = f'(x)u-TM(f(x)-y)
Remark — The above two Propositions remain true if / is
locally Lipschitz and Gateaux differentiable. Π
Proof— Let ν belong to DF(x, y)(u). Then there exist hn > 0
converging to 0 and sequences un and vn converging to и and ν
respectively such that for all η
χ + hnun € L & у + hnvn € f(x + hnun) - M(x + hnun)
This implies that и belongs to Tl(x)- Since we can write
f(x + hnun) = f(x)+hn(f'(x)u + e(hn))
where e(hn) converges to 0 with hn, we get
f(x) - у + hn(f'(x)u -vn+ e(hn)) € M(x + hnun)
and thus f'(x)u — ν does belong to DM(x, f(x) — y)(u).
Conversely, assume for instance that Μ is derivable, that и
belongs to Tl(x) and that f'(x)u — ν belongs to DM(x, f(x) — y)(u).
Hence, there exist a sequence hn > 0 converging to 0 and sequences
un, й~п and wn converging to и, и and f'{x)u — ν respectively, such
that χ + hnun belongs to L and f(x) — у + hnwn to M(x + hnu~n).
Since Μ is Lipschitz at χ there exists a sequence wn converging to
f(x)u — ν satisfying
-V η > 1, f(x) -y + hnwn € M(x + hnun)

186
5- Denvatives of Set-Valued Maps
Then for some e(hn) —► 0, the sequence vn := f'{x)u + e(hn) — wn
converges to ν and satisfies
У + hnvn € f(x + hnun) - M(x + hnun) □
Our first task is to characterize these contingent derivatives by-
adequate limits of differential quotients. We derive that the
contingent derivative in the direction и is the upper limit of the differential
quotients
Fix + hu1) — у
DF(x,y)(<u) = Limsuph_>0+iJi/_>Ji —
from
Proposition 5.1.4 Let Χ, Υ be normed spaces, F : Χ ~> Υ be a
set-valued map and let (x, y) € Graph(-F). Then
/ pfx -\-hu') — у \
ν € DF(x,y)(u) <=> liminffc^o+.u'-fu^ ( «, ^ —J = 0
If χ € Int(Dom(F)) and F is Lipschitz around x, then
υ € DF(x,y)(u) <^> limiafrffv, (x+ u'y) = Q
/1-Ю+ \ h j
If moreover the dimension of Υ is finite and I denotes the Lipschitz
constant of F at x, then
Oom(DF(x,y)) = X and DF(x,y) is I - Lipschitz
Proof — The first two statements being obvious, let us check
the last one. Let и belong to X and I denote the Lipschitz constant
of F on a neighborhood of x. Then, for all h > 0 small enough and
У € F(x),
y>€ F(x) С F(x + hu) + lh\\u\\B
Hence there exists yh € F(x + hu) such that vn := (уь, — y)/h belongs
to Z||u||J5, which is compact. Therefore the sequence Vh has a cluster
point v, which belongs to DF(x,y)(u).
Fix any Щ, u2 € X and v\ € DF(x,y)(ui). To end the proof
we have to show that there exists v2 € DF(x,y)(u2) such that

5.2. Contingent Derivatives
187
\\vi — щ\\ < ^ ||iii — иг||. Consider hn —> 0+ and v\n —> v\
satisfying
у + hnvin € F(x + hnui)
and let zn € -F(x + /1^2) be such that for all large n,
II^n - 2/ - hnvin\\ < lhn \\m - U21|
(they exist by the Lipschitz continuity of F at x.) Taking a
subsequence and keeping the same notations, we may assume that the
bounded sequence (zn — y)/hn converge to some г/2 € DF(x,y)(u2).
Clearly г/2 satisfies the required estimates. Π
Remark — Domain of the Derivative It is quite useful to relate
the tangent cones to the domain of a set-valued map to the domain of its
derivative.
We always have
Oom(DF(x,y)) с TOom{F)(x)
If F is pseudo-convex at (x,y) £ Graph(F), (and in particular, if F is
convex), then equality
Oom(DF(x,y)) = TOom{F)(x)
holds true, because
Dom(F) = Tr^Graph(F)
where ττ χ is the projection from Χ χ Υ onto X. We thus deduce from
Proposition 4.2.9 that
' Oom(DF(x,y)) = irxGTaph(DF(x,y))
<
= nxTGTa:pbiF)(x,y) = ?;x(Graph(ir))(a;) = TOom{F)(x) Π
Remark — Kernel of the Contingent Derivative The kernel of
the contingent derivative is related to the contingent cone to the inverse
image:
Тр-Цу)(х) С kerDF(x,y) := DF(x,y)-\0)
Equality holds true in this formula whenever F~l is pseudo-Lipschitz around
(2/, ζ).

188
5- Derivatives of Set-Valued Maps
To prove the converse inclusion, let и belong to the kernel of DF(x, y).
Then there exist sequences hn > 0 converging to 0, un and vn converging
respectively to и and 0 satisfying
Vn, y + hnVn e F(x + hnun)
Since F~l is pseudo-Lipschitz around (y,x), there exist Ζ > 0 and an
element x\ 6 F~l(y) such that
\\Xn~ (x + hnun)\\ < l\\y - (y + hnvn)\\ = lhn\\vn\\
Hence, by setting u\ := (x\ — x)/hnt we see that
x + hnuln = x\ £ F~\y)
and that u\ converges to и because ||u^ - un\\ < l\\vn\\ and because vn
converges to 0. Therefore we have proved that и belongs to the contingent
cone to F~1(y) at x. □
Remark — Lower Semicontinuously Differentiable Maps
Definition 5.1.5 (Lower Semicontinuous Differentiability) LetX, Υ
be normed spaces and F : Χ ~» Υ be a set-valued map. We say that F is
lower semicontinuously differentiable at (x,y) &Graph(F) if the set-valued
map
(x,y,u) ε Graph(F) xX-, DF(x,y)(u)
is lower semicontinuous.
Observe that it implies that F is sleek at (xo,yo)· The converse needs
further assumptions. We derive for instance from the "Cross-Convergence"
Theorem 2.3.2 the following criterion:
Proposition 5.1.6 Assume that X and Υ are Banach spaces and that F
is sleek on some neighborhood U of(xo,yo) & Graph(F). If the boundedness
property
Vn el, sup inf \\v\\ < -t-oo
(*,») e wnGraph(F) »е^(*.»)(«)
holds true, then the set-valued map
(x,y,u) ε Graph(F) χ Χ ^ DF(x,y)(u)
is lower semicontinuous on (Uf) Graph(F)) χ Χ.

5.2. Adjacent and Circatangent Derivatives 189
5.2 Adjacent and Circatangent Derivatives
5.2.1 Definitions and Elementary Properties
Naturally, we can also associate with any other concept of tangent
cone a concept of derivative. Since we were led to introduce Clarke
and adjacent (or intermediate) tangent cones, we can introduce two
more graphical derivatives:
Definition 5.2.1 Let Χ, Υ be normed spaces, F : Χ ~> Υ be a
set-valued map and у € F(x).
1. — the adjacent derivative D^F(x, y) is the set-valued map
from XtoY defined by
Graph (D»F(x,y)) := lfcraph(F)(*,y)
2. — the circatangent derivative CF(x, y) is the set-valued
map from XtoY defined by
Graph(CF(x,y)) := CGTaph{F)(x,y)
When F := f is single-valued, we set
Dbf(x) := Dbf(xJ(x)), Cf(x) := Cf(x,f(x))
We see at once that these graphical derivatives are closed
processes, that
Vu, CF(x,y)(u) С DbF(x,y)(u) С DF(x,y)(u)
When / is single-valued, we observe that the circatangent and
adjacent derivatives Cf(x)(u) and D* f(x)(u) are either empty or contain
only one element. In particular,
i) Dbf{x){u) = f'(x)u if / is Frechet differentiable at χ
ii) Cf(x)(u) = f{x)u if / is continuously differentiable at χ
Naturally, the circatangent derivative is always a closed convex
process.

190
5 - Derivatives of Set-Valued Maps
Remark — Since CF(x, y) is a closed convex process, it is
equal to its bitranspose. The transpose
CF(x,y)*:Y*"~> X*'
has been called the codifferential of F at (x, y). It is involved
whenever dual conditions are used.
Instead of founding a "direct" differential calculus on tangent
cones, as we did, it is possible to start with any concept of normal
cone ^Qraph(F) (x> У) *° ^е S^Ph of F at a point (x,y). We can
associate with it the corresponding "codifferential" dF(x, y)* defined
by
ρ € dF(x,y)*(q) if and only if (p, -q) € NGraph(F) (x, y)
When such a normal cone is not convex, it can no longer be
defined as polar of a tangent cone, so that such codifferential,, in
general, is not the transpose of a graphical derivative. □
As for the contingent derivative, a graphical derivative of the
inverse of a set-valued map F is the inverse of the derivative:
' i) D\F~l){y,x) = D^F^y)-1
<
ii) C{F~l){y,x) = CF{x,y)-1
If К is a subset of X and / : X н-> Υ is a single-valued
continuously differentiable around a point χ € К map, then the derivative of
the restriction is the restriction of the derivative to the corresponding
tangent cone:
■ i) D\f\K)(x) := Db(f\K)(x,f(x)) = Г(х)\тьк{х)
<
. H) C(f\K)(x) := C(f\K)(x,f(x)) = f(x)\cK(x)
Actually, these formulas follow from:
Proposition 5.2.2 Consider normed spaces Χ, Υ, a single-valued
map f from an open subset Ω С X to Υ and a set-valued map Μ :
X ~> Υ. Define the set-valued map F : Χ ~> Υ by:
V χ € X, F(x) := f(x) - M(x)

5.2. Adjacent and Circatangent Derivatives
191
If f is Frechet differentiable at χ € Ω Π Dom(M); then for every
У € F(x)
DbF(x,y)(u) = f'(x)u-DbM(x,f(x)-y)(u)
and if f is continuously differentiable at x, then
CF(x,y)(u) = f'(x)u-CM(xJ(x)-y)(u)
The proof is analogous to the one of Proposition 5.1.2 and is left as
an exercise.
Proposition 5.2.3 Let X andY be normed spaces, fa single-valued
map from an open subset Ω С X to Υ, Μ : Χ ~> Υ be a set-valued
map and L С X. Consider the set-valued map F : Χ ~> Υ defined
by:
, χ _ J f(x) — M(x) when χ € L
[X) :~ { 0 when x<£L
If f is Frechet differentiable at χ e Ω Π Dom(F) and Μ is Lipschitz
at x, then for every у € F(x) the adjacent derivative of F at (x, y)
is equal to
n*F(r ,Λί,Λ-ί Г(ф-0ЬМ(хЛх)-у)(и) when и€ТЦх)
DF{x,y){u)-^ 0 ^η uiTbdx)
The same formula holds true for a map f continuously differentible
at x, the circatangent derivatives of F and Μ and the Clarke tangent
cone to L.
The proof is the same as the one of Proposition 5.1.3 and these
statements remain true when / is Lipschitz at χ and Gateaux
differentiable. □
5.2.2 Limits of Differential Quotients
In order to characterize adjacent and circatangent derivatives in
terms of limits of differential quotients, we need the concept of "lim
sup inf" (or Γ-convergence) of functions of two variables introduced
independently by several authors (de Giorgi, Rockafellar, ...):

192
5- Denvatives of Set- Valued Maps-
Definition 5.2.4 (Lim sup inf) LetL and Μ be two metric spaces
and φ : L χ Μ l·-* Л be a function. We set
lim sup^/^a; ϊτιίυΐ->υφ(χ', у') := sup inf sup inf ф(х', у')
e>0 V>0 χ'βΒ{χ,η) y'eB(y,e)
We write (x', y') -±p (x, y) to denote that (x1, y') converges to (x, y) _
while remaining in the graph of F.
Hence, by translating the definition of the intermediate and the
Clarke tangent cones, we obtain the following characterizations:
Proposition 5.2.5 Let (x, y) belong to the graph of a set-balued map
F : X ~> У from a normed space X to another Y. Then
ν belongs to V*F(x,y)(u) if and only if
<
lim sup^0+ Mu^ud (v, Р^')~У) = 0
and
ν belongs to CF(x,y)(u) if and only if
<
k lim sup/l_>0+i vtf)^F{Xiy) miu>->ud (υ, Εί^ψ}^ = 0
If F is Lipschitz around χ € Int(Dom(-F)); then the formulas become
much simpler:
υ belongs to D^F(x,y)(u) if and only if
and
ν belongs to CF(x,y)(u) if and only if
Um/l_>0+[ (x>,y>)->F{x,y) d (v, n*'+hv)-y'^ = 0
Remark: Kernel of the Derivative As for the contingent
derivative, the kernels of other derivatives are linked to the associated tangent
cones to the inverse image:
Тр-г(у)(х) С kerDbF(a;,y) := D*F(x,yrl(0)

5.2. Adjacent and Circatangent Derivatives 193
ала equality holds true in this formula whenever F""1 is pseudo-Lipschitz
around (y,x). In this case, we also have:
kerCF(x,y) с Ср-цу)(х) □
The proofs are left as an exercise.
Proposition 5.2.6 Let us assume that the images of F are convex
and that F is Lipschitz around x. Then for any (x, y) € Graph(-F)
the images of the adjacent derivative D F(x, y) are convex and
DbF(x,y)(0) = TF{x)(y)
DbF{x,y){u) + D^F{x,ym = D*F(x,y)(u)
Proof— Let vi and V2 belong to DF(x,y)(u). Then, for any
sequence hn > 0 converging to 0, there exist sequences u\n and U2n
converging to и and sequences v\n and V2n converging to v\ and V2
respectively such that
V n, у + hnVin € F(x + hnUin) (i = 1,2)
Since F is Lipschitz around x, there exists I > 0 such that for all η
large enough,
y + hnV2n € F(x + hnUin) + lhn\\u2n — Uin\\
so that we can find another sequence V3n converging to V2 such that
у + hnV3n С F(x + hnu\n)
Now, F(x + hnu\n) being convex, we deduce that for all λ € [0,1],
у + hn{\vin + (1 - λ)υ3π) € F(x + hnum)
Since Xvin + (1 — \)vsn converges to Xvi + (1 — λ)г/2, this element
belongs to D]'F(x,y)(u).
By Proposition 5.2.5, we see that ν € DbF(x,y)(0) if and only
if d(v, (F(x) — y)/h) converges to 0, i.e., ν belongs to the adjacent
tangent cone to F(x) at y. Since F(x) is convex, it coincides with
the tangent cone.

194
5- Derivatives of Set-Valued Maps
Since 0€DbF(x,y)(0) we obtain that
Vuj DbF(x,y)(u) С DbF(x,y)(u) + DbF(x,y)(0)
To prove the opposite inclusion fix
ν € DbF(x,y)(u) & w € DbF(x,y)(0)
Let hn —► 0+, vn —► υ be such that
V n, у + /ιη^π € -F(:c + /inu)
By convexity of -F(:e), there exist wn -* w such that for η large
enough, у +y/h^wn € -F(a;). Then, by the Lipschitz continuity of F,
for all large η and for some w'n, we have
у + VK.w'n £ -Р(ж +/ιπω); ||и4-гоп|| < Z\AnlMI
Thus
(1 -y/K)(y + hnvn) + y/K(y +y/h^w'n)
' = у + hn(vn + w'n) -y/Khnvn = у + hn(v + w) + hne(hn)
€ F(x + hnu)
where e{hn) converges to 0. Hence
r a- л. ( , F(x + hnu)-y\
lim dist [v + w, — - = 0
n->DO V К J
This ends the proof. D
5.2.3 Derivatives of monotone operators
Let X be a Hubert space (identified with its dual.) We recall that a
set-valued map F : X ~> X is monotone if and only if
V(z,p), (y,q) € Graph(-F), <p-q, x-y>> 0
(see Chapter 3 for an introduction to monotone and maximal
monotone maps.) We also recall that when F is monotone, its resolvent
J := (1 + -F)-1 is single-valued and Lipschitz (with constant equal
to 1) on its domain. Therefore, we can easily compute derivatives of
F in terms of the derivatives of its resolvent.

5.2. Adjacent and Circatangent Derivatives
195
Proposition 5.2.7 Let X be α Hilbert space identified with its dual
X*. We supply X* with the weak-* topology. If F : X ~> X*
is monotone, then the contingent derivative DF(x,p) at every pair
(x, p) € Graph(F) is semi positive-definite in the sense that
V(u,r) € Gr&ph(DF(x,p)), <r,u>>0
Furthermore, the following statements are equivalent:
( a) r € DF(x,p)(u)
[ b) и € DJ(x+p)(r + u)
All the above statements remain true for the adjacent and
circatangent derivatives.
Proof — The first claim is obvious, since < г, и > is the limit
of a sequence < rn, un >, where ρ + hnrn € F(x + hnun), (because
un converges to и strongly and rn converges to r weakly) and since
h\ < rn, un > = < χ + hnun - x,p + hnrn - ρ > > 0
For proving the second statement, we observe that ρ belongs to
F(x) if and only if χ = J (χ + ρ), because χ + ρ € (1 + F)(x) and
because J is the inverse of (1 + F). We also deduce from the latter
that the two following statements are equivalent:
ii) r + u € D(l + F)(x,x+p)(u)
ii) и € DJ{x + p,x)(r + u) = DJ{x+p){r + u)
On the other hand, Proposition 5.1.3 implies that
D(l + F)(x,x+p)(u) = u + DF(x,p)(u)
We have obtained the formula we were looking for. Π
Since the cone-valued map Νκ associating with any χ € К the
normal cone Νκ(χ)~^Ό a closed convex subset is maximal monotone
(because the normal cone is the subdifferential of the indicator of K),
and since its resolvent is the projector of best approximation onto
K, we deduce the following corollary:

196
5 - Derivatives of Set-Valued Maps
Corollary 5.2.8 Let К be α closed convex subset of α Hilbert space
and ρ belong to the normal cone Nk(x) to К at some χ € К. Denote
by Пк the best approximation projector onto K. Then,
-q € DNK(x,p)(u) <=> и € DTlK(x+p)(u + q)
5.3 Chain Rules
We derive from the calculus of tangent cones the associated calculus
of derivatives of set-valued maps. We begin naturally by the chain
rule for computing the composition product of a set-valued map G :
Χ ~> Υ and a set-valued map Η : Υ ~> Ζ.
We shall need the following result:
Proposition 5.3.1 Let Χ, Υ be normed spaces, F : Χ ~> Υ be a
set-valued map and К be a subset of X. Assume that F is Lipschitz
around some χ € К. Then, for any у € F(x), we have
DbF(x,y)(TK(x)) С TF[K){y)
As a consequence, we deduce that when Μ is a subset of Υ and
у € Μ, then
ТРщм){х) С DbF(x,y)+1(TM(y)) (5.1)
Proof — Take и in TK(x) and ν € DbF(x,y)(u). Then there
exist sequences hn > 0 converging to 0, u\n and U2n converging to и
and vn converging to ν such that
χ + hnuin € К & y + hnvn € F(x + hnU2n)
Since F is Lipschitz around χ with a Lipschitz constant I, we deduce
that
y + hnvn € F(x + hnuin) + lhn\\uin - U2n\\
so that there exists another sequence υ* converging to ν such that
y + hnv*n € F(x + hnuin) С F{K)
This implies that ν belongs to the contingent cone to F{K) at y.

5.3. Chain Rules 197
Consider now К := F+1(M). Since F(F+1(M)) is contained in
M, we deduce that
ДьЯя>2/)(?>+1(м)(:е)) С TF(F+4M))(y) С Гм(2/)
from which formula (5.1) ensues. Π
Remark — Naturally, we can show in the same way that for
a Lipschitz map F the formula
DF(x,y)TbK(x) С TF{K){y)
is also true whenever у € F(x). Π
We begin by the following simple result:
Theorem 5.3.2 Let us consider normed spaces Χ, Υ, Ζ, a set-
valued map G : X ~> У and a set-valued map Ή. :Υ ^* Ζ.
1. — Let us assume that Η is Lipschitz around y, where
у € G(x). Then, for any ζ € H{y), we have
( DbH(y,z)oDG(x,y) С D(HoG)(x,z)
\ DH(y,z)oD]>G(x,y) С D(HoG)(x,z)
2. — If G := g is single-valued and Frechet differentiable at
x, we obtain
Vz€H(g(x)), D(Hg)(x,z)(u) С DH(g(x),z)(g\x)u)
and the equality holds true when Η is Lipschitz around g(x).
Proof — We apply Proposition 5.3.1 to equality
Graph(tf о G) = (1 χ Я) (Graph(G))
for proving the first statement. The second one follows from
Graph(tfoff) = (g χ l)-1 (Graph(tf))
ТгЧк)(х) С /Ή"1 СВД(х))) □

198"
5- Derivatives of Set-Valued Maps
Remark — Let us assume that G is Lipschitz around x. Then,
for all у € G(x) and ζ € (HU\G)(x), we have
D(HnG)(x,z) С DH(y,z)nDbG(x,y)
because we can apply Proposition 5.3.1 to Μ := Graph(fT) and
because
Graph(tfDG) = (G χ 1)+1 (Graph(tf))
We state now a more powerful result.
Theorem 5.3.3 Let Χ, Ζ be Banach spaces and Υ be a finite
dimensional vector-space. Consider set-valued maps G : Χ ~> Υ
and Η : Υ ~> Ζ. Fix x0 € Dom(G), y0 € G(x0) П Oom(H) and
z0eH(y0).
If G and Η are closed and satisfy the transversality condition
lm(CG(x0,y0))-Oom(CH(y0,z0)) = Υ
then
' i) DbH(y0,z0)oDG(x0,y0) С D(HoG)(x0,z0)
< ii) DbH(yo,z0)oD]'G(xo,yo) С Db(HoG)(xQ,zQ)
^ Hi) CH(y0,z0)oCG(x0)y0) С C(HoG)(x0,z0)
Proof — If we denote by ω the continuous linear operator from
XxYxYxZ to Υ associating to (x, yi, y2, ζ) the element y\ — уг
and by πχχζ the canonical projection from Χ χ Υ χ Υ χ Ζ onto
Χ χ Ζ. Observe that
Graph(tf ο G) = πχχΖ ((Graph(G) χ Graph(tf)) η иГ^О))
We apply Theorem 4.3.3 with L = Graph(G) x Graph(#), / = ω
and Μ = {0}.

5.3. Chain Rules 199
Consequently, we deduce that, for instance,
(Graph(L>bG(z0,yo)) x Graph(L>btf (yo,zQ)j) П ω~ι(0)
= feraphtGJ^'IO) X ГСгарЬ(я)(2/o^o)) П^(0)
(5.2)
= ^raphiGjxGraphiff)^»'У°'У°'z°) nω_1(°)
= Г(СгарЬ(С)хСгарЬ(я))пш-1(о)^0'Уо'Уо'z^
By applying the projection ж χ χΖ to both sides of these equalities,
we prove that
' Graph (ubtf (yo, z0) о D*G(x0, yQ))
= πχχζ ((СгарЬ(£>ьС(г0, Ы) * Graph(L>btf (j/q, z0))) Π uT^O))
= πχχζ (^r(Graph(G)xGraph(ff))no;-i(o)^0'2/0'2/0'2;o)y
C r7rxx2r((Graph(G)xGraph(H))no;-i(o))^0' z^>
..= rGraph(ffoG)(S0'z°) = Graph(ub(FoG)(o;o^o))
The proof of the other statements is similar and left as an exercise.
D
Remark — We recall that the norm of a process (positively
homogeneous set-valued map) A:Y^>X is defined by
||A|| := sup inf \\x\\
IMI<i хЕА(у)
If Υ is any Banach space, we have to assume that there exist constants

200
5 — Derivatives of Set-Valued Maps
с > 0, а £ [0,1[ ала η > О such that
' V(s,yi) £ Graph(G)nB((:co,2/o),77),
V(ifc,z) £ Graph(H)nB((y0,*)),q),
<
By с Jm(DbG(x,yi))ncBY- Oom(DH(y2,z)) + αΒγ and
. \\DbG(x,yi)\\<ck \\D{H^){z,y2)\\<c
to prove the above formulas.
Indeed, the above assumption implies the second transversality
condition of Theorem 4.3.3. For any υ £ Υ, we can find v\ £ lm(DbG(x, yi))
and v2 £ Dom(DH(y2,z)) such that
ν = vi — v2 + e, \\v\\\ < с||г?||, ||e|| < α\\υ\\
Hence ||г/2|| < (с + 1 + а)||г/|| and there exist
u £ i^Gfa yO-^vO & ω £ DH(y2,z)(v2)
such that ||?x|| < c||t/i|| and ||ω|| < ο||υ2||.
Therefore, ν = uj(u,vi,v2,w) + e where (u,vi,v2,w) belongs to the
contingent cone to the product of the graphs of G and Η and e £ «By.
Consequently, we can derive properties (5.2) and conclude the proof. □
We obtain converse inclusions under more severe assumptions:
Proposition 5.3.4 Let Χ, Υ, Ζ be normed spaces. Consider set-
valued maps G : X ~> У', Η :Υ ~» Ζ and assume that G is pseudo-
convex at (жо)Уо) € Graph(G) and Η is pseudo-convex at (yo,zo) 6
Graph(#). Then
D(HoG)(x0,z0) с DH(yQ,zQ)oDG(x0,y0)
Proof— Let ω and πχχζ be the operators denned in the proof
of Theorem 5.3.3. When the graph of G is pseudo-convex at (x, y)
and the graph of Η is pseudo-convex at (y, z), we derive the above
property from Proposition 4.2.9 applied in the following situation:

5.3. Chain Rules
201
' Graph(.D(tf о G)(x0, zq)) = rGraph(HoG)(s0>2b)
= cl [πχχΖ (^r(Graph(G)xGraph(H))no;-i(o)(:cO'2/o>2/o>2;o)JJ
С cl {πΧχΖ ((Graph(L>G(z0)2/o)) x Graph(UF(y0,^))) Πω"1^)))
= d(Graph(^(yo,*o)°£G(xo,yo)))
Remark — We can deduce the conclusions of this proposition
from any other criterion implying that the contingent cones to the
images by πχχζ axe the closures of the images of the contingent
cones. Π
Remark — Let X be a normed space and K, L be subsets of X.
Recall that the Bouligand paratingent cone Ρχ(χ) to К relative to L at
χ £ L Π Κ is defined by
P%(x) := {v\ liminf dK(x' + hv)/h = 0}
and that we set Рк(х) := Рк(.х)·
•We can obtain upper estimates of the composition product by using the
pamtingent derivatives defined by saying that the graph of the paratingent
derivative PF(x, y) of F at (x, y) is the paratingent cone to the graph of F
at (x,y):
Graph(PF(a;)2/)) := PGraph(F) (x> v)
Hence, for any и £ X, the paratingent derivative in the direction и is equal
to
Fix' + hu') — y'
PF{x,y){u) = Limsuph_>0+iU,_>Ui(x,i2/,)_>j;,(Xi2/)
We can also define the lop-sided paratingent derivatives PxF(x,y) and
PyF(x, y) in the following way:
' i) P*F(x,y)(u) = LimSuph_t0+|M,_tMiy,_tF(x)/^yb^
<
^ ii) pyF(x,y)(u) = Umsuph^Q+iU^UiX^F_i(v)XF^'+hh^--y

202
5 — Derivatives of Set-Valued Maps
Theorem 5.3.5 Let X,Y be normed spaces. Assume that G : Χ -^ Υ is
Lipschitz around x. If Υ is a finite dimensional vector-space and G(x) is
bounded, then
D{HoG){x,z) С (J P*H(y,z)oP*G(x,y)
yeG(a;)
Proof— Let w belong to D(H о G)(x,z)(u): there exist sequences
hn > 0, un and wn converging to 0, и and w respectively such that
V n, z + hnwn £ H(yn) where yn ε G(x + hnun)
Since G is Lipschitz around x, there exist I > 0 ала elements y° ε G(x)
such that, for η laxge enough,
У П У ΤΙ .· η II II Til II
νη := —;: satisfies \\υη\\ < l\\un\\
hn
Furthermore, G{x) being relatively compact, a subsequence (again denoted)
y° converges to some y. We can also extract a subsequence (again denoted)
vn which converges to some v, since this sequence is bounded and the
dimension of Υ is finite.
Prom the relations
yn + hnvn ε G(x + hnun) к z + hnWn ε #(y° + hnvn)
we infer that w belongs to PzH(y, z)(v) and υ to PxG(x,y)(u). Π
It is not difficult to check that for all у ε G(x), и ε Dom(CG(a;,y)),
P(HDG)(x,z)(u) С (PH(y,z)nCG(x,y))(u)
Indeed, let и ε Oom(CG(x, у)) and w belong to P(HoG)(x,z)(u). Hence
there exist a sequence hn > 0 converging to 0 and sequences of elements
(.Χη,Ζη) ε Graph(#DG), un and wn converging to (x, z), и and w
respectively such that
Vn>0, zn + hnwn ε f] H(y)
у £ G(xn+hnun)
The set-valued map G being Lipschitz, there exists a sequence of
elements yn ε G{xn) converging to y. By definition of the square product, we
know that zn ε H{yn) (because zn ε (HoG)(xn).)
Take now any υ in CG(x,y)(u). Since G is Lipschitz around x, there
exists a sequence of elements vn converging to ν such that
V η > 0, уn + hnvn ε G(xn + hnun)

5.4. Inverse Set-Valued Map Theorem
203
Therefore,
V η > 0, zn+ hnwn £ H{yn + hnvn)
so that we infer
w ε PH(y,z)(v)
Since this is true for every element υ of CG(x, y)(u), we deduce that
w ε Π PH(y,z){v) = (PH(2/^)nCG(^l2/))(u) Π
veCG(x,y)(u)
5.4 Inverse Set-Valued Map Theorem
5.4.1 Stability and Approximation of Inclusions
Let Χ, Υ be Banach spaces and F : X ~> Υ and Fn : Χ ~> Υ be
set-valued maps with closed graphs. We are ready to extend Lax's
Principle to the approximation of a solution to inclusion
find xq € X such that F(xq) Э yo
by solutions to the approximate inclusions
find xn € X such that Fn(xn) Э yn
Definition 5.4.1 We shall say that the set-valued maps Fn are
consistent at (xo,yo) € Graph(F) if there exist sequences of
(xon,yon) € Graph(Fn)
such that XQn converges to xq and y$n converges to y$.
We shall say that a family of set-valued maps Fn is stable around
(x, y) if there exist constants с > 0, α£ [0,1[ and η > 0 such that
* {xn> Уп
) e Gr&ph(Fn)nB((x0,y0),v), Mv e Y,
< 3un € X, 3wn <E Υ with ν £ DFn(xn,yn)(un)+ wn
and \\un\\ < c\\v\\ & ||гоп|| < a\\v\\

204
5 - Derivatives of Set-Valued Maps
Theorem 5.4.2 Let us consider Banach spaces X and Υ, a
sequence of closed set-valued maps Fn : X ~» У, yo &Y and a solution
xq to the inclusion F(xq) Э vq.
Assume that the set-valued maps Fn are consistent at (xq, vq) and
that the family of Fn 's is stable.
Then there exists a constant I such that, for any sequence of
elements (xonfyon) € Graph(i1Tl) converging to (xo,yo) and any
sequence yn converging to yo, we have for η large enough
d\xon,F~1(yn)j < 1\\уоп-Уп\\
As a consequence, we obtain the following error estimate: for any
sequence yn converging to yo,
^(^Oi-FirHyn)) < I (d((xo,yo), Graph(.Fn)) + ||yo-yn||) .
so that xq can be approximated by solutions xn G F~l(yn).
Proof — We apply Theorem 3.4.5 with X replaced by X x Y,
Kn by СгарЬ(^п), fn by the projection ΐΐγ from Χ χ Υ onto Υ. We
have to prove that the stability assumption implies transversality
assumption (3.9) of Theorem 3.4.5, i.e., that for all ν € Υ, there
exist (un, vn) in the contingent cone Tqj.^^p ,(ιπ, yn) and wn G Υ
satisfying
ν = vn + wn, max(\\un\\,\\vn\\) < c\\v\\, \\wn\\ < α\\υ\\
This information is provided by our stability assumption since the
contingent cone to the graph is the graph of the contingent derivative
and the norm of vn = ν — wn is smaller than or equal to (1 + α)||υ||·
D
When the set-valued maps Fn are all equal to F, we obtain:
Theorem 5.4.3 (Inverse Set-Valued Map Theorem) Consider
α closed set-valued map F : X ~> У', an element (xq, yo) of its graph
and let us assume that there exist constants с > 0, a G [0,1[ and

5.4. Inverse Set-Valued Map Theorem
205
η > 0 such that
' V(s,y) € Graph(F) П ^((zo.yo), ту), V υ e У,
13u € -X", Зги € ysuch that υ € DF(x,y)(u) + w
and ||u|| < с||г>|| & HI < a||v||
Then г/о belongs to the interior of the image of F and F'"1 is pseudo-
Lipschitz around (yo,xo).
We can extend Theorem 3.4.10 to the case of set-valued maps by
introducing an adequate definition of uniformly sleek maps, which
are the set-valued analogues of the continuously differentiable maps.
Definition 5.4.4 (Strongly Sleek Maps) We shall say that F is
uniformly sleek at a point (xq, 2/0) G Graph(F) if its graph is
uniformly sleek at this point, i.e., if
sup d((u,v),Grapb(DF(x,y)))
(u,v)eGTeLph(DF(xQ,yQ))r)(BxB)
converges to 0 when (x,y) converges to (xo,yo) in Graph(F).
With this definition, we can state a natural set-valued version of
Graves' Theorem
Theorem 5.4.5 Let us consider Banach spaces X and Υ, a closed
set-valued map F : X -^ У and an element (xo,yo) of its graph. Let
us assume that
DF(xq, г/о) is surjective
1. — If F is uniformly sleek at (xo,yo), then 2/0 belongs to
the interior of the image of F and F'"1 is pseudo-Lipschitz around
(Уо,хо)-
2. — // the dimension of Υ is finite, it is sufficient to
assume that F is sleek at (xo,yo) to get the same conclusions.
The proof follows from Theorem 3.4.10. The very same proof
yields

206
5 — Derivatives of Set-Valued Maps
Theorem. 5.4.6 Consider Banach spaces X and Y, a closed set-
valued map F : Χ ~> Υ and an element (xo,yo) of its graph. Let us
assume that the dimension of Υ is finite and
Cir(xo)2/o) is surjective
Then ?/o belongs to the interior of the image of F and FT1 is pseudo-
Lipschitz around (yo,xo).
5.4.2 Localization of Inverse Images
In this section we provide some estimates of the inverse image F'"1 (г/*).
Theorem 5.4.7 Let F be a set-valued map from a normed space X
to a normed space Υ and (x*,y*) belong to its graph.
1. — Assume that the dimension of X is finite. Then, for
any closed cone Ρ satisfying
kei(DF(x*,y*))nP = {0}
there exists ε > 0 such that?
F-1(y*)n(x* + e(PnS)) = {x*}
In particular ifker(DF(x*,y*)) — {0}, then
F-\y*)t\B{x\e) = {x*}
2. — Assume that F~1(y*) is convex^. Then
F'^y*) С x* + kei(DF(x*,y*))
In particular ifkei(DF(x*,y*)) = {0}, then F~1(y*) = {x*}-
2If i1-1 is pseudo-Lipchitz at (j/, ж), it is possible to show that conversely, the
property
P"V)n(i'+£(Pnfl)) = {χ*}
implies that кег(Л^(а;*, у*)) П Int(P) = 0.
3or even, pseudo-convex at x*.

5.4. Inverse Set-Valued Map Theorem
207
Proof — Assume that the first statement is false. Then there
exists a sequence of elements xn G F_1(y*) converging to x* such
that xn — χ* Φ 0 belong to P. Setting hn := \\xn — x*\\ which
converges to 0+ and un := (xn — x*)/hn belonging to the unit sphere
and to the cone P, we infer that a subsequence (again denoted) un
converges to some element и £ Ρ οι norm 1 because the dimension
of X is finite.
On the other hand, inclusions
y* + hn0 € F(xn) = F(x* + hnun)
imply that 0 € DF(x*,y*)(u), i.e., that и belongs'to the kernel of
DF(x*,y*). This is impossible. By taking Ρ = X, we obtain the
local uniqueness of a solution x* to y* G F(x).
The second statement follows readily from the definition, since
we observe that the convexity of the inverse image implies that
F'^y*) С χ* + 2>-1(у.)(а:*) С х*+kei(DF(x*,y*)) D
5.4.3 The Equilibrium Map
Let us consider two finite dimensional spaces X and Λ and a set-
valued map
F:Ax X ^ X
We denote by Ε : Λ ~»· X the equilibrium map defined by
νλ€Λ, E(\) := {xeX\0€F(\,x)}
which associates with any parameter λ e Λ the set of equilibria of
F(\, ·). We can derive some information on this equilibrium map
from the knowledge of its contingent derivative.
Proposition 5.4.8 Let Λ and X be two finite dimensional vector-
spaces and F : Α χ Χ ^ Χ be a closed set-valued map. Assume
that the circatangent derivatives CF(\, x, 0) are surjective for every
(X,x) belonging to the graph of the equilibrium map E. Then
1. — The contingent derivative of the equilibrium map is the
equilibrium map of the contingent derivative:
0 € Ι?ί,(λΙχΙ0)(μ,υ) <^> ν e DE{\,x){u)

208
5 — Derivatives of Set-Valued Maps
and
0 € CF(X,x,0)fa,v) =D- υ € CE(X,x)(p)
2. — // for any μ £ A, there exists an equilibrium υ of
CF(X, χ, 0)(μ, ·), then the equilibrium map is pseudo-Lipschitz around
(X,x).
3. — Set
Q{X,x) := {v€X\0€DF(\,x,0)(0,v)}
Then, for any equilibrium χ £ E(\) and any closed cone Ρ satisfying
Ρ Π Q(X,x) = {0}, there exists ε > 0 such that
E(\) П (x + ε(Ρ П B)) = {χ}
and in particular, an equilibrium χ £ E(\) is locally unique whenever
0 is the only equilibrium of DF(\,x, 0)(0, ·).
4- If E(\) is convex, then E(X) С χ + Q(\,x).
Proof — We observe at once that the graph of the equilibrium
map Ε is the inverse image of the graph of F by the continuous linear
operator π € £(Λ χ Χ, Λ χ Χ χ Χ) defined by
V(A,i)eAxI, π(Χ,χ) := (λ,χ,Ο)
Because the circatangent derivative CF(\,x,0) is surjective, the
transversality condition holds true:
1т(тг)-ССт&рЪ{р)(тг(\,х)) = AxX-Graph(CF(\,x,0)) = AxXxX
Therefore Corollary 4.3.4 yields
Graph(DE(X,x)) = π"1 (GTaph(DF(\,x,0)))
and
Giaph(CE(\,x)) D π"1 (Giaph(CF(\,x,0)))
which are the formulas we were looking for.
The second statement follows from Theorem 5.4.6 applied to E"1.
We then apply Theorem 5.4.7 to the inverse of Ε to derive the
localization properties of E(\). Π

5.4. Inverse Set-Valued Map Theorem
209
5.4.4 Local Injectivity
Definition 5.4.9 Let Χ, Υ be norrned spaces and F : Χ ~»· Υ be a
set-valued map.
We shall say that it is locally injective around x* if and only if
there exists a neighborhood N(x*) such that
Vn ^i2e N(x*), F(xi)nF(x2) = 0
It is said to be (globally) injective if we can take for neighborhood
N(x*) the whole domain of F.
Theorem 5.4.10 Let F be a set-valued map from a finite
dimensional vector-space X to a norrned space Υ and (x*,y*) belong to
its graph. Assume that there exists 7 > 0 such that F(x* + ηΒ) is
relatively compact and that F has a closed graph.
If for all у G F(x*) the kernels of the paratingent derivatives
PF(x*,y) of F at (x*,y) are equal to {0}, then F is locally injective
around x*.
Proof — Assume that F is not locally injective. Then there
exist sequences of elements х\пф 3>2n, converging to x* and yn
satisfying
Vn>0, yn € F(xln) П F(x2n)
Let us set hn := \\х\п — X2n\, which converges to 0, and
Un ■= (Xln ~ X2n)/hn
The elements un do belong to the unit sphere, which is compact.
Hence a subsequence (again denoted) un does converge to some и
different from 0.
Then for all large η
yn € F(xln) Π F(x2n) = F(x2n + hnun) Π F(x2n) С F(x*+-yB)
we deduce that a subsequence (again denoted) yn converges to some
у € F(x*) (because Graph(F) is closed.)
Since the above relation implies
V η > 0, yn + hn0 £ F(x2n + Кип)

210
5 — Derivatives of Set-Valued Maps
we deduce that .
0 € PF(x*,y)(u) _ ......
Hence we have proved the existence of a non zero element of the
kernel of PF(x*, y), which is a contradiction. □
5.5 Qualitative Solutions
This application deals with problems motivated by economics in the sixties
(comparative statics) and, recently, by a domain of Artificial Intelligence
known under the name of "qualitative simulation" or "qualitative physics."
It concerns "sign solutions" (or confluence solutions, to adopt the
terminology used in Artificial Intelligence) of nonlinear equations and inclusions.
We shall provide a criterion involving adequate linearizations of the sign
problem, which allows to obtain sign solutions for both the original problem
and its linearization.
Let us consider a closed set-valued map F : Rn ~> Rm and the problem
find χ £ Rn such that F(x) Э у
We associate with Rn the η-dimensional confluence space 1Zn defined by
Kn := { -, 0, + }n
whose elements are denoted by a := (αϊ,..., an)·
Given a multi-sign b ε TZm, does there exist a "qualitative solution"
a £ 7гп, defined by:
3 χ &y € F{x) such that sign (ж) = a h sign(y) = b (5.3)
We provide a criterion4 for answering this question.
Before stating our first theorem, it is convenient to introduce the
notations
Qn(a) := R£ := {ν ε Kn \ sign of (Vi) = a;}
Qn(a) := аЩ := {ν <= Kn \ sign of fa) = a; or 0}
4As in numerical analysis, which deals both with approximating infinite-
dimensional problems by finite-dimensional ones and with solving them, problems
of qualitative analysis arise at two levels: the passage from "quantitative to
qualitative" and algorithms for solving the latter. Only the first aspect is investigated
here in the framework of solving this inclusion.

5.5. Qualitative Solutions
211
The cone <2*(a) is defined by:
(Vi > 0 if a; = +
Vi < 0 if di = -
Vi £ R if aj = 0
Observe that <2n (a) is the closure of Qn (a) and that the positive polar cone
of Qn{a) is the cone Q*n{a).
Theorem 5.5.1 Let α ε Пп, bell™, F : Rn ~> Rm be α closed set-valued
map. Consider xQ &Oom(F) П Qn(a) satisfying F(xQ)r\Qm(b) φ%. If the
criterion
{(0,0) is the only solution (p, g) to ρ ε CF(xQ,yQ)*(q)
(5.4)
satisfying ρ ε Q*(a) & g ε -Q^(b)
holds true, then a solves the qualitative inclusion
3x ε Qn{a) such that F(x) nQm{b) φφ (5.5)
Proof— Pick τ/ο ε i^o) П Qm(b), then (a;o,yo)_belongs to the
intersection of the graph of F and the closed convex cone Qn(a) χ Qm(b),
so that {{xo,ya), (ха,Уа)) is a solution in Graph(F) χ (Qn(a) χ Qm{b)) to
the equation
(zi,2/i)-(z2,2/2) = °
Let leR" denote the unit vector 1 := (1,..., 1) and set
αϊ := (аД)^!,...,!
We shall prove that for some ε > 0 there exists a solution
(zi.yi) ε Graph(-F), (za.ito) 6 Qn(a) xQm{b)
to the equation
(»i,J/i)-(a;2,2/2) = е(а1,Ы)
So that Zi = 2Γ2+εα1 belongs to Qn(a) and j/i = y2+ebl belongs to Qm(b)
and to F(x1).
For that purpose, we apply Theorem 5.4.6, which states that a solution
tothe above equation does exist provided the assumption
CGraph(F)fro, i/b) - CqM Ы x C^m(b)(yQ) = R" χ Rm
is satisfied.

212
5- Derivatives of Set-Valued Maps
Since xq belongs to Qn(a) := a72™, the cone C-q гаЛхо) coincides with
the tangent cone, which contains Qn (a) (recall that when_ Q is a_ convex
cone, TQ(x) = cl(<3 + a;R) D Q.) In the same way, С^(Ь)(у0) D Qm(b).
The above assumption would follow from
СгарЦСЩздо)) - (QM x Qm(b)) = R" x Rm
By polarity, this is equivalent to the condition
(G™ph(CF(xQ,yQ))rn(Qn(a)xQm(b))+ = {0}
which is nothing other than condition (5.4.) Π
In the single-valued case, we obtain:
Corollary 5.5.2 Lei_a ε ΊΙη, b ε 1lm, F : Rnj-> Rm be a closed set-
valued map. Let xq ε Qn(a) be such that f(xo) ε Qn{b) and assume that f
is continuously differentiable at xq. Then the criterion
0 is the only solution q to q ε —Qm{b) & f'(xa)*Q 6 Qn(a) ;
implies that a solves the qualitative equation
3 χ such that sign(x) = a & sign(/(a;)) = b
Remark — When A is a closed convex process, we can apply the
above theorem with
(xQ,yQ) = (0,0) ε Graph(A)n(Qn(a)xgm(b))
Since CA(Q, 0) = A (because the tangent cone to the closed convex cone
Graph(A) at the origin is equal to this closed convex cone), we deduce from
the above theorem the following consequence:
Corollary 5.5.3 Let α ε 72.™, b ε 1Zm and A be a closed convex process5
fromBJ1 toRm. If
{(0,0) is the only solution (p, q) to ρ ε A*(q)
such that ρ ε Q*n{a) h q ε -Q*m{b)
then a solves the qualitative inclusion
3i£ Qn(a) such that A(x)nQm(b) φ 0
5Closed convex processes provide a way to represent uncertainty for linear
operators analogous to "sign matrices." They possess better mathematical
properties than sign matrices. In particular, since we cannot extend "sign matrix"
to nonlinear maps, we can not say that a sign matrix is the derivative of a sign
operator. Observe that this corollary provides the same criterion for both the
initial inclusion and its set-valued linearization.

5.5. Qualitative Solutions
213
Therefore, criterion (5.4) implies the existence of a qualitative solution
to both problem (5.5) ала the lineaxized problem
3u ε Qn(a) such that CF(xa,ya)(u) Π Qm(b) /|D
Let us consider the case when F : Rn ~> Rm is the set-valued map
defined by:
T?( \ = I f(x) ~ M№ when x e L
[X) : \ 0 when χ i L
where / : Rn i-+ Rm is continuous, L is a closed subset of Rn and Μ :
Rn ~> Rm is a closed set-valued map.
Corollary 5.5.4 Let a <=Tln, b <= Um, Μ : Kn ^ Rm be a closed set-
valued map and L с Rn be a closed set. Assume that xq and j/q satisfy
xQ £ Qn(a)nL & yo ε {f{xQ)-M{xQ))nQm{b)
that f is continuously differentiable at xq, Μ is Lipschitz at xq and that
CL{xQ) -Dom{CM{xQ,yQ - f{xQ))) = Rn
The criterion
(0,0) is the only solution (p, q) to
■ Ρ e f'{xo)*{q) + CM(xa,ya - /{χ0))*(-9) + Νφο) (5-6)
_ such that ρ ε Ql(a) & q ε -Q*m{b)
implies that a solves the qualitative equation
3i£ Qn(a)ilL & у ε Qm{b) such that у ε f(x) - M{x)
It follows from Theorem 5.5.1 and Proposition 5.2.3.
The particular case when Μ is constant yields
Corollary 5.5.5 Let α ε Пп, b ε 1lm and L с Rn, Μ С Rm be closed
sets. Assume that xq and j/q satisfy
xQ ε α»η£ & уо ε (/Ы-м)пдт(Ь)
Assume that f is continuously differentiable at xq. Then the criterion
(0 is the only solution q to
q £ -Q*m(b)nNM(f(xa)-yQ) & /'Ы*«ед;(а)-%Ы
implies that a solves the qualitative equation
3x ε (3„(a)ni & у ε <Эт(Ь) such that /(ж) ε y + М

214
5 — _ Derivatives of Set-Valued Maps
Another example is provided by qualitative solutions to variational
inequalities.
Example Variational Inequalities
Let us consider a closed convex subset К С Rn and a C1-map / from
a neighborhood of К to Rn.
An element yQ ε Rn being given, we recall that an element xQ ε Κ is
a solution to the variational inequalities
V χ ε -ЙГ, < f(xQ) - 2/0, xa - χ > < 0
if and only if xq is a solution to the inclusion
2/o € f{xa) + NK{xQ)
where Nk(xq) is the normal cone to К at xq.
Theorem 5.5.6 Let α ε Пп, bell™ and let xQ ε Qn(a) ПК, yQ ε Q^b)
satisfy
Vχ ε Κ, < f(xQ) -ya,xo -χ > < 0
The criterion
(0,0) is the only solution (p, q) to
■ Ρ e f'(xQ)*(q) + CNK(xQ,yQ-f(x0))*(-q) (5.7)
_ satisfying ρ ε Q*n(a) & q £ -Q*m{b)
implies that a solves the qualitative variational inequalities
(3x ε Qn{a)nK & у ε Qm{b) such that
\fx ε Κ, <f{x)-y,x-x>< 0
Proof — This a consequence of Proposition 5.1.2 with M(x) :=
-NK{x) and Theorem 5.5.1. Π
Remark — Recall that Proposition 5.2.7 provides a characterization
of the circatangent derivative of the normal cone map К Э χ ~> Νκ (x)
through the circatangent derivative of the best approximation projection
πχ to the closed convex subset K:
q ε CNK{x,p){u) <£=^ и ε CirK(x + p)(u + q)
which can be used to check criterion (5.7.)

5.6. Higher Order Derivatives
215
5.6 Higher Order Derivatives
Naturally, by taking mi/l-order contingent sets to a graph of a set-valued
map F : X ~> Y, we obtain concepts of high-order derivatives.
Definition 5.6.1 Let Χ, Υ be normed spaces and F : Χ ~> Υ be a set-
valued map.
The mi/l-order contingent derivative DF(x, y) ofF at (x, y) £ Graph(.F)
is the set-valued map from X to Υ defined by
Graph(D(m)-F(a;,2/,Ui, i>i,..., Jim-i,Vi))
··= T^)a:ph{F)(x,y,u1,v1,...,um-1,vm-1)
We say that it is m—th derivable if its graph is m—th derivable.
This definition implies right away that
D(m)(F-l){y,x,vi,Ui,...,vm-i,um-i)
= (D(m)F(x,y,Ui,vl,...,um-i,vm-i)yi
For simplicity, we shall restrict ourselves to second order contingent
derivatives of set-valued maps.
If if is a subset of X and / is a single-valued map which is twice
continuously differentiable around a point χ ζ. Κ, then the second order
contingent derivative of the restriction of f to К is given by the formula:
D<V{f\K){x,Ul){u2) = fl(24/l*)fo/(a0,ui,/'(a0ui)(u2)
= f'{x)u2 + \f"{x){ui,ui) whenever u2 eT^'(x,Ui)
It is empty when u2 ^ T^\x,Ui).
The proof of this fact is similar to the proof of the following proposition.
Proposition 5.6.2 Let X and Υ be normed spaces, f be a twice
continuously differentiable single-valued map from an open subset ilcX to Y,
Μ : X ~~y γ be a set-valued map and L с X. Let F : Χ ~> Υ be the
set-valued map defined by:
■pi \ — j f(x) ~ M(x) when χ ε L
[x) '— 1 0 when x<£L

216
5- Derivatives of Set-Valued Maps
Then, for each χ £ nnDom(.F), у £ F(x) and V\ £ DF(x,y)(ui), the
second order derivative D^ F{x,y,u\,Vi)(u2) in the direction^ £ T^ '(x,Ui)
is contained in
/'(aO«2 + |/'Ή(«ι,«ι) - D^M(x,f(x)-y,u1,f'(x)u1-v1)(u2)
Furthermore D^F(x,y,Ui,Vi)(u2) is empty whenever u2 £ Т^'(х,щ).
Equality holds true when we assume that either L or Μ is twice derivable
and Μ is Lipschitz.
Proof— Let v2 belong to D^F(x,y,ui,vi)(u2). Then there exist
hn > 0 converging to 0 ала sequences U2n and V2n converging to u2 and v2
respectively such that χ + hnU\ + h^u2n belongs to L and
Vn>0, y + hnvi+hlv2n £ f(x + hnui + h2lu2n)-M(x + hnui + h2lu2Tl)
This implies that u2 belongs to Tl(x,ui). Since / is twice continuously
differentiable
(f(x + hnui + hlu2n)
= f(x) + ft™/'(:c)ui + h2n(f'(x)u2 + £/"(aO(ui,Ui)) + e(hn))
where e(hn) converges to 0 with hn. Then
f{x) -y + M/'(aOui - "ι) + h2n{f'(x)u2 + \f"{x){uuщ) -v2 + e{hn))
£ M(x + hnui + hlu2n)
and thus f'(x)u2 + \f"{x)(u\,u-\) — v2 does belong to
D^M(x,f(x)-y,ul,f'(x)ul-vl)(u2)
Conversely, assume for instance that Μ is twice derivable, that u2
belongs to T"l{x,U\) and that
f{x)u2 + ^f"{x){uuul)-v2 £ D^M{x,f{x)-y,ul,f'{x)u1-vl)(u2)
Hence, there exist a sequence hn > 0 converging to 0 and sequences u2n
and w2n converging to u2 and f'{x)u2 + ^f"(x)(ui,Ui) - v2 respectively
such that
χ + hnui + h\u2n £ L
and
f{x)-y + hn(f'{x)ui-v1) + hlw2n £ M(x + hnui + h2nu2n)

5.6. Higher Order Derivatives
217
(we have used the Lipschitz continuity of M). Then for some e(hn) —> 0,
the sequence
V2n ■= f'{x)u2 + ^f"{x)(ui,u1)+£{hn)-w2n
converges to v2 and satisfies
у + hnvi + hlv2n e f(x + hnui + h2nu2n) - M(x + hnui + hlu2n) □

Chapter 6
Epiderivatives of
Extended Functions
Introduction
When we consider real-valued functions, the order relation on R
is involved in optimization (for rinding a minimum) or in Lyapunov
functions (which decrease along solutions to differential equations
or inclusions.) These are reasons why we have to associate with a
real-valued function V the set-valued map Vf defined by
VT(s) := F(a;)+R+
whose graph is the epigraph of V and consider not only the
contingent derivative of V, but also the contingent derivative of Vj. We
shall check in Section 1 that its graph is the epigraph of a lower
semicontinuous function, which is called the contingent epiderivative
D^V(x) of V at x. For instance, the Fermat rule1 states:
if χ minimizes V, then DiV(x)(v) > 0 for all ν £ X,
'which should be called Oresme-Fermat Rule, Nicholas Oresme (1323-1382),
Bishop of Lisieux, 300 years ahead of his time, anticipated the great Fermat
himself by observing in his treatises De Proportionibus Proporiionum and Algo-
rismus Proporiionum that near a maximum, the increment of a variable quantity
becomes 0. (He also anticipated the coordinate system in another book Tractatus
de Latitudine Formarum, Copernicus' views and was only one step to the
logarithms.) He did not hesitate to attack repeatedly the extravagant presumptions of
astrologers (called at the time mathematicians!..., the mathematicians were then
219

220
6- Epideriv&tives of Extended Functions
so that necessary conditions in optimization and optimal control can
be derived from this simple rule, provided that a rich and exhaustive
calculus of contingent epiderivative is available.
Section 3 is thus devoted to such an epidifferential calculus.
As for derivatives of set-valued maps, an epidifferential calculus
involves not only the use of contingent epiderivatives, but of other
epiderivatives2 corresponding to various tangent cones. So, adjacent
D\V{x) and circatangent С|У(ж) epiderivatives are presented in the
second section, as well as convex epiderivatives.
We study in Section 4 dual concepts: the subdifferential doV(x)
and the generalized gradient dV(x) of an extended function defined
respectively by
d0V(x) := {p € X* | Vν € Χ, <ρ,ν>< ΌΊν(χ)(ν)}
dV(x) := {peX*\Vv€X, <p,v>< C}V{x){v)}
They naturally coincide when V is convex or when it is continuously
differentiable (in this case, they are made up of the gradient V'(x).)
The subdifferential is always contained in the generalized
gradient, and even more, in the case of finite dimensional vector-spaces,
the upper limit of the subdifferentials doV(xn) when xn converges to
χ is contained in the generalized gradient dV(x).
Again in finite dimensional vector-spaces, the subdifferential
coincides with the (local) subdifferential d-V(x) (used in the concept
of viscosity solutions of Hamilton-Jacobi equations), which is the
subset of ρ G X* such that
liminf VW-VM-<P,*-E2L > о
X '•E 11 uif JLi 11
This still increases the population of our menagerie with species
doomed to disappear through Darwinian evolution for lack of duality,
since their use does not allow recovery of the original information ου
epiderivatives.
called geometers), whose persuasive arguments influenced affairs of the state, and
he even criticized the Pope. A clear thinker, he translated Aristotle's books into
French and commented on them with extremely original views.
2This procedure is also known under the name of the epigraphical approach.

6.0. Introduction
221
The use of generalized gradients for functions is then
recommended in a "convex environment, i.e., for convex, and more
generally, episleek functions (with convex epigraphs.)
However, there are other reasons than the exploitation of duality
for using these dual concepts.
One is their familiarity with more classical concepts. For usual
functions on Hilbert spaces, there is a canonical identification
between, say, a derivative of a differentiable function and its gradient,
and it became traditional to formulate many results in terms of
gradients and normal cones. Think of the traditional role of the Lagrange
multipliers in constrained optimization, for instance, which appeals
to generalizations stated in a dual form (Kuhn-Tucker rule,
maximum principles in optimal control.)
Another reason lies in the fact that the first attempts to
generalize the concept of gradients used limiting procedures. Since it
seems easier to take limits of elements (the gradients, for instance)
than limits of functionals (the associated directional derivatives, for
example), many generalizations of the concept of gradient dealt with
set of limits or cluster points of gradients taken in various ways.
These approaches are still pursued (in the framework of "proximal
analysis", for instance), but we do not dwell on them in this book.
When we deal with convex functions, which are the extended
functions with convex epigraphs, all these epiderivatives coincide.
Furthermore, lower semicontinuous convex functions enjoy duality
properties: We recall in Section 5 what the Fenchel conjugate of a
real-valued function is and that this Fenchel conjugacy is a bijective
correspondence between lower semicontinuous convex functions on a
Banach space and its dual.
It plays a role similar to the one played by the Legendre
transform, since the inverse of the subdifferential map of a convex function
is the subdifferential map of the conjugate function.
These facts are at the origin of duality theory in optimization.
We also recall here the Fenchel Theorem which provides a criterion
for the existence of solutions to convex minimization problems, from
which one can derive a formula for computing the conjugate of the
composition of a function with a continuous linear operator.
We finally point out that the subdifferential map of a convex

222
6 - Epideriv&tives of Extended Functions
function is a maximal monotone map. Its Yosida approximation is
the gradient of a continuously differentiable convex function, called
the Moreau-Yosida approximation.
We conclude this chapter with a presentation of high order epi-
derivatives, defined from the high-order derivatives of the set-valued
map Vf associated with an extended function. We shall also
estimate the second-order contingent epiderivative of the Moreau-Yosida
approximation of a convex function.
6.1 Contingent Epiderivatives
6.1.1 Extended Functions and their Epigraphs
Let us consider an extended function V : X i-> R U {±00} whose
domain
Dom(F) := {x € X \ V(x) φ ±οο}
is not empty. (Such a function is said to be proper in convex and non
smooth analysis. We shall rather say that it is nontrivial in order to
avoid confusion with proper maps.)
Any function V defined on a subset К С X can be regarded as
the extended function Vk equal to V on К and to +00 outside of K,
whose domain is K.
An extended function is characterized by its epigraph
£p(V) := {(χ,λ) € Χ χ R I V(x) < X}
An extended function V is convex (resp. positively homogeneous) if
and only if its epigraph is convex (resp. a cone.)
We also observe that any positively homogeneous extended
function is non trivial whenever V(0) φ —oo. In this case, V(0) = 0.
The hypograph of a function W : X —> R U {±00} is defined in a
symmetric way:
Hp(W) := {(χ,λ) ε Xxn\W(x) > A} = -£p(-W)
Such a function is concave if and only if — W is convex, i.e., if and only if
its hypograph is convex.

6.1. Contingent Epiderivatives
223
The main examples of extended functions are the indicators φκ
of subsets К defined by
. , , /θ if χ € К
ФкМ ■■= { +oo if not
The indicator φκ is lower semicontinuous if and only if К is closed
and ψ χ is convex if and only if К is convex. One can regard the sum
V + φκ as the restriction of V to K.
We recall the convention inf(0) := +00.
Lemma 6.1.1 Consider a function {:Ih RU{±oo}. Its epigraph
is closed if and only if
\/v € Χ, δ(ν) = liminf<5(?/)
V1—*V
Assume that the epigraph of δ is a closed cone. Then the following
conditions are equivalent:
i) \fv € Χ, δ(ν) > -oo
ii) <5(0) = 0
Hi) (0,-1) $ Ερ{δ)
Proof — Assume that the epigraph of δ is closed and pick
ν € X. There exists a sequence of elements vn converging to ν such
that
lim δ(νη) — liminf <5(г/)
η—юо υ'—*ν
Hence, for any λ > lim inf„/_>„ δ(υ'), there exist iV such that, for all
η > Ν, δ(νη) < λ, i.e., such that (г;п, λ) € ερ(δ). By taking the
limit, we infer that δ(ν) < λ, and thus, that δ(ν) < liminfi;'_+i;5(?;')·
The converse statement is obvious.
Suppose next that the epigraph of δ is a cone. Then it contains
(0, 0) and 5(0) < 0. The statements ii) and Hi) are clearly equivalent.
If i) holds true and <5(0) < 0, then
belongs to the epigraph of δ, as well as all (0, —λ), and (by letting
λ —> +oo) we deduce that <5(0) = —00, so that г) implies ii).

224
6 - Epidenvatives of Extended Functions
To end the proof, assume that <5(0) = 0 and that for some v,
δ(ν) = -oo. Then, for any e > 0, the pair (v, —l/ε) belongs to the
epigraph of δ, as well as the pairs (εν, —1): By letting e converge
to 0, we infer that (0, —1) belongs also to the epigraph, since it is
closed. Hence <5(0) < 0, a contradiction. □
6.1.2 Contingent Epiderivatives
We can also regard an extended function V as the set-valued map
V : X ~> R defined by
V(x) :
j V(x) if ж € Dom(V)
0
if χ i Dom(V)
so that we can define in the usual way the contingent derivatives of
V at ж € Dom(V):
BV(i)(ti) := DV(x,V(x))(u)
= {v | liminf/^o+y-ni \V(x + hu') - V(x) - hv\/h = 0}
However, minimization problems and Lyapunov functions involve
obviously the order relation on R. Hence, when dealing with such
problems, it is convenient to consider two new set-valued maps V|
and Vj defined in the following way:
i) VT(x) := {
ii) Vi(x) :
V(x) + ~R+ if ж € Dom(V)
0 if V(x) = +oo
R if V(x) = -oo
V(x) - R+ if ж € Dom(V)
0 if V(x) = -oo
R if V(x) = +00
We see at once that
Graph(VT) = Sp{V) L· Graph(V|) = Hp(V)
Therefore, we are led naturally to associate with these two set-valued
maps Vf and Vj their contingent derivatives. We observe that the

6.1. Contingent Epiderivatives
225
values of the contingent derivative of Vj are obviously half lines in
the sense that
VA > V(x), Vii € Dom(DVT(a;,A)),
£>VT(:c,A)(?x) = DVT(:c,A)(?x)+R+
Definition 6.1.2 Let V:Ih>RU {i°°} be α nontrivial extended
function and χ belong to its domain. We shall say that the function
D-\V(x) from X to RU {±00} defined by
Vu €l, D}V(x)(u) := inf{i;|7; € DV^x, V(x))(u)}
is the contingent epiderivative ofV at χ in the direction u.
The function V is said to be contingently epidifferentiable at χ if
its contingent epiderivative never takes the value —00.
It is said to be episleek (at x) if its epigraph is sleek (at (x, V(x))).
Actually, the contingent epiderivative can be characterized as a
limit of differential quotients:
Proposition 6.1.3 Let V : Χ i-> RU{±oo} be a nontrivial extended
function and χ belong to its domain. Then
η лп и ^ ν ■ f V(x + hu')-V(x)
D-tVixjiu) = hmmf — -1 ^-J-
1 ' /ι->0+,ιι'->ιι h
The function V is contingently epidifferentiable at χ if and only if
ΌΊν(χ)(0) =0.
Proof — Assume first that D^V(x)(u) < +00. By the very
definition of the contingent epiderivative, for any λ > D^V(x)(u),
there exists some ν € DV^(x, V(x))(u) smaller than λ. Therefore,
we can associate with any e > 0 elements h €]0, e[ and (u1, v') €
B((u, v), e) satisfying V(x) + hv' > V(x + hu'). So that
V(x + hu') - V(x)
η
Hence the liminf of the differential quotients (V(x + hu') — V(x))/h
is smaller or equal to D^V(x)(u).

226 6 - Epiderivatives of Extended Functions
Conversely, let hn > 0 converging to 0, un converging to и and
λη > V(x + hnun) be such that the differential quotients
vn '·= (K-V(x))/hn
converge to
.. . c V(x + hu')-V(x)
ν := hmmf — -1 ^-L
h—>0+, и'—щ h
Then the pairs (x + hnun, V(x) + hnvn) belong to the graph of Vj,
so that ν € DV^(x)(u). Hence
.. . , V{x + hu')-V{x) ,,
hmmf -f ^ > D}V{x)(u)
Consider now the case when D^V(x)(u) = +oo, i.e., the case
when the subset DVT(i, V(x))(u) = 0. Then for any ν € R, for any
hn > 0 converging to 0 and any sequence un converging to u, we have
V(x + hnun) > V(x) + hnv
for η large enough, so that the liminf of the differential quotients is
equal to +oo. □
We define in a symmetric way the contingent hypoderivative DiV(x)
from X to R U {±oo} of V : X н-> R U {±00} at a point χ of its domain by
D{V{x){u) := -DT(-V)(s)(u) = limsup У(ж + hu') ~ V(x)
h-*a+,u'—>u "■
We could also have defined the contingent epiderivative of a
function by taking the contingent cone to its epigraph:
Proposition 6.1/4 Let V : X (-► RU{±oo} be a nontrivial extended
function and χ belong to its domain.
Then the contingent cone to the epigraph of У at (ж, V(x)) is the
epigraph of the contingent epiderivative of V at x:
εΡ(ΌΊν(χ)) = T£p{v)(x,V(x))

6.1. Contingent Epideiivatives
227
Figure 6.1: Epigraph of the Contingent Derivative

228
6 - Epiderivatives of Extended Functions
Consequently, the epigraph of the contingent epiderivative at χ is
a closed cone. The contingent epiderivative D^V{x) is then lower
semicontinuous and positively homogeneous whenever V is
contingently epidifferentiable at x.
Furthermore,
Vw > V[x), T£p{v)(x,w) С TOom{v)(x) χ R
and
Vw > V(x), Oom(DxV(x)) χ R С T£p{v)(x,w)
If the restriction of V to its domain is upper semicontinuous, then
for all w > V(x),
T£p{v)(x,w) = TOom{v)(x) χ R
Proof
1. — The first claim follows from the very definition of the
contingent epiderivative. Fix w > V(x). Let us assume that (u, v)
belongs to Тщу){х, w). We infer that there exist sequences un, vn
and hn > 0 converging to u, ν and 0 such that
w + hnvn > V(x + hnun)
We thus deduce that и belongs to the contingent cone to the
domain of V at x, and thus, that T£p(y){x, w) С Tj)om,v,(x) χ R.
2. — If и belongs to the domain of the contingent
epiderivative of V at x, if w > V(x) and if ν is any real number, we check
that (u,v) belongs to T£p(yj(x,w).
Indeed, there exist sequences of elements hn > 0, un and vn
converging to 0, и and D-\V(x){u) respectively such that
(x + hnun, V{x) + hnvn) € Sp(V)
But we can write
(x+hnun,w+hnv) = (x+hnun,V(x)+hnvn)+(0,w-V(x)+hn(v-vn))
Since w — V(x) + hn(y — vn) is strictly positive when /j,T1 is small
enough, we infer that (x + hnun, w + hnv) belongs to the epigraph of
V, i.e., that (u,v) belongs to the cone T£p(y)(x,w).

6.1. Contingent Epiderivatives
229
3. — Let w be strictly larger than V{x) and и belong to
T-£)0m/V\(x)■ Then there exist sequences un and hn > 0 converging
to и and 0 such that V(x + hnun) < +oo for all n.
When V is upper semicontinuous on its domain, for all
0 < e < (w-V(x))/2
there exists η > 0 such that, for all /ιπ||ϊχη|| < η, we obtain
V{x + hnun) < V{x) + ε < w - ε
Let ν be given arbitrarily in R. Then, for any hn > 0 when ν > 0 or
for any hn €]0, 73jj[ when ν < 0, inequality ги — e < ги + hnv implies
that V(x + hnun) < w + hnv, i.e., that the pair (u, v) belongs to
TSp(y)(x,w). Π
Proposition 6.1.5 LetV : Χ ι—» RU{±oo} be an extended function
and χ belong to its domain. Take any
и € Oom(D^V(x))nOom(DiV(x))
Then
{ВтПх)(и), !>№)(«)} С DV(x)(u) С [ВтК(х)(и),и^(х)(и)]
Proof— Since Graph(V) = £p(V) П Wp(V), we deduce that
the inclusions
TGraphooO^H) С V)^^^^^^^1))
can be translated into
Graph(DV(a;)) С £p(Z>TF(s)) П Ир(В^(х))
from which the inclusion
DV(x)(u) С [Βτν(α:)(«),.Ο^(α0(«)]
follows. Since the contingent epiderivative of У at ж in the direction
и is equal to
,ч/ч ,. η VYx + ftu') - V(x)
DrV(x)(u) := liminf — ^ ^
1 Ч /i-+0+,u'-ni /l

230
6 - Epiderivatives of Extended Functions
we see that D^V(x)(u) is the limit of a subsequence of differential
quotients (y(*+fcO-y(*?j wnen fc_ 0+ and thus, that ΌΊν(χ)(η)
belongs to DV(x)(u). The same is true for the contingent
hypo derivative. □
Theorem 6.1.6 Let V : X i-> R U {±00} be an extended function
which is continuous on a neighborhood of χ € Dom(V). Then for
every
и €' DomCUTV(s)) aBom{DiV{x))
we have
DV(x)(u) = [D1V(x)(u),DlV(x)(u)}
Proof— By Proposition 6.1.6, it remains to show that when V is
continuous on a neighborhood of x, the inclusion
ΊΈχν)(χ>ν(χ)) ПГНр(у)(х,К(х)) С TGTaph(y)(x,V(x))
holds true. Pick any (u, v) in the intersection: There exist sequences
h^ > 0 and h!j^> 0 converging to 0+, and sequences of pairs (u£, ?4)
and (uj^, v^) converging to (u, v) such that
V{x + hjui)-V{x) T V{x + hjui)-V{x) ,
hi ~ Vn & Л*
We claim that it is enough to show that for every η large enough,
there exists λη € [0,1] such that, setting
К ··= (l-\n)hl + \nhi
and
(l-Xn)hlul + \nh^n (l-An)ftTt4 + Awft*4
Un ;= & ^ ;=
we have the following equality
V(x + frn"n) - V(x)
hT
= Vn
Indeed hn > 0 converges to 0+ and (un, vn) converges to (u, v). So,
(u, v) belongs to the contingent cone to the graph of V.

6.1. Contingent Epiderivatives
231
To find such λπ, consider for all large η the continuous functions
ψη : [0,1] —> R defined by
[ν(χ + (ΐ- λ)4«τ + λ/44) - (v(x) + (i - \)hlvl + xh&i)
and observe that
ψη(0) < 0 & ψη(1) > 0
Since ψη is continuous, there exists λπ € [0,1] such that φη{λη) = 0.
Then λπ satisfies.:the required property. □
The contingent epiderivative coincides with the directional
derivative < V'(x),u > when V is Frechet differentiable.
If V is Frechet differentiable at a point χ € AT, then the contingent
epiderivative of the restriction is the restriction of the derivative to
the contingent cone:
The formulas become much more simple when V is Lipschitz: the
contingent epiderivative coincides with the lower Dini derivative :
Proposition 6.1.7 Let us assume that V:IhRU {±oo} is
Lipschitz at a point χ of its domain. Then
V(x + hu) — V(x)
D-[V(x)(u) = liminf — (the lower Dini derivative)
h—*0+ fl
and satisfies for some i>0:
Vu e X, \D<tV(x)(u)\ < l\\u\\
Remark — There are other intimate connections between contingent
cones and contingent epiderivatives.
Let фк be the indicator of a subset K. Then it is easy to check that
Di(1>k)(x) = VTk(x)
Therefore we can either derive properties of the epiderivatives from
properties of the tangent cones to epigraphs, as we did, or take the opposite
approach by using the above formula. D

232
6 - Epiderivatives of Extended Functions
There is also ал obvious link between the contingent cone and the
contingent epiderivative of the distance function to К since we can write for
every χ 6 K: —
TK(x) = {υ £ X | D}dK{x){v) = 0}
We also mention the following estimate of the contingent epiderivative
of the distance function to K:
Proposition 6.1.8 Let К be α closed subset of α normed vector-space and
Пйг(у) be the set of projections of у onto K, i.e., the subset of ζ £ Д"
such that \\y — z|| = dfc(y)- Then when Т1к(у) Φ 0> we have the following
inequalities:
D^dK{y){v) < d(v,TK(UK(y)))
Proof — We begin by proving this inequality when у belongs to K.
Indeed, for all ω £ X, inequality
di((.y + hv) < ак(у + hw) + h\\v — w\\
implies that
D^dfc(y)(v) = liminf djc{y + hv)/h < \\v — w\\ when w £ Тк{у)
h—*Q+
Assume next that у £ K. Choose ζ £ TiK(y). Then
dK(y + hv)-dK(y) < \\y-z\\ +dK(z + hv)-dK(.y) = dK(z + hv)
Prom the first part of the proof, we deduce that Dj άκ(ζ)(ν) < d(v, Τκ(ζ)),
and consequently, that D-\dK(y)(v) < d{v,TK{z).) Π
6.1.3 Fermat and Ekeland Rules
Since we can define the contingent epiderivative of any extended
function V:XhRu {+oo}, we can extend the "Fermat rule" to
any minimization problem.
Theorem 6.1.9 (Fermat Rule) Let V : X н-> R U {+00} be a
nontrivial extended function on a normed space X and χ € Dom(V)
a local minimizer ofV on X.
Then χ is ά solution to the variational inequalities:
Vu € X, 0 < D}V{x){u)

6.1. Contingent Epiderivatives
233
Proof — The proof is naturally obvious: We write that for all
v! € X and all h > 0,
0 < (V(x + hu')-V(x))/h
and we take the lim inf when h converges to 0 and v! to и. П
The converse statement is not true without further assumptions
such as convexity or, more generally, pseudo-convexity, or second
order conditions:
Definition 6.1.10 An extended function V : X i-> R U {±00} is
called pseudo-convex atx€. Dom(V) if its epigraph is pseudo-convex
at (x, V(x)), i.e., if
Vy € Χ, ΌΊν(χ)(ν-χ) < V(y)-V(x)
We infer easily from this definition the converse of the Fermat Rule:
Proposition 6.1.11 Let V : X н-> R U {±00} and χ € Dom(V)
satisfy 0 < D-[V{x){u) for all и € X. If V is pseudo-convex at x,
then χ achieves the minimum ofV.
What is not obvious is the use of this Fermat rule for more and
more general problems, when the function V is built from other
simpler functions and involves constraints.
The search for necessary conditions for a minimum requires quite
a rich calculus of contingent epiderivatives which provides estimates
of D^V(x)(u). In particular, when constraints (of the type χ € К)
are involved, the fact that the epiderivative of the restriction to К
is the restriction of the epiderivative to the contingent cone TV (ж),
allows one to write necessary conditions using also contingent cones
to constraint sets (or in the dual form, using gradients and polars of
the contingent cones.)
In the same way, it is easy to derive an epidifferential version of
Ekeland's Variational Principle:
Theorem 6.1.12 Let X be a Banach space, V : X (-► R+ U {+00}
be a nontrivial lower semicontinuous bounded from below function

234
6 - Epiderivatives of Extended Functions
and xq € Dom(V) be a given point of its domain. Then, for any
ε > 0, there exists a solution χε € Dom(V) to:
(i) V{xe) +e\\xe-xo\\ < V{xq)
(6-1)
it) Vu € Χ, Ο < ΌΊν(χε)(η) + ε\\η\\
6.1.4 Elementary Properties
We present below some formulas concerning epiderivatives which
allow exploitation of the Fermat and Ekeland rules.
1. — Sum and composition
Proposition 6.1.13 Let us consider two Banach spaces X, Y, a
single-valued map f : X н-» У and two extended functions V and W
from X and Υ to RU {+00} respectively.
Let xq belong to the domain of the function V+Wof and assume
that f is Frechet differentidble at xq. Then
D}V{x0){u)+D}W{f{x0)){f'{x0)u) < D}{V + W о f){x0){u)
In particular, if К is a subset of X and if xq € К nDom(V), then
Vu € TK(xo), D^V(x0){u) < ΌΊ(ν\κ)(χο)(η)
2. — Supremum of Functions
Let us consider now a family of functions
Vi-.X ^ RU{±oo}, (»€/)
and let us associate with it the function U defined by
U(x) := supVi(x)
We set I(x) := {i € / | Ц,(х) = U(x)}. The following estimate is
obvious whenever I(x) is not empty and χ € Dom(i7)·.
У и € X, sup D}Vi(x){u) < D^U{x){u) (6.2)
i£l{x)

6.1. Contingent Epiderivatives
235
because, for all г € I(x),
Vi(x + hu) - Vj{x) U(x + hu) - U(x)
h ~ h
3. — Marginal Function
Let us consider normed spaces X and Υ and an extended function
U:X хУн->Ни{±оо}.
We associate with it the marginal function I^iIhRu {±°o}
defined by
V(x) := mfU(x,y)
Let π denote the projection from XxYxHtoXxH. We observe
that:
kEP(U) С Ep(V) С irEp(U)
The first inclusion is obvious. The very definition of the infimum
implies that for every e > 0 and every (χ, λ) € Ep(V), there exists
ye € Υ such that (x, ye, λ + e) belongs to Ep(U). Taking projections
and passing to the limit, the second inclusion ensues.
Let xq € X be given. Suppose that there exists yo €. Υ which
achieves the minimum of U(xq, ·) on Y:
V(xq) = U(x0,yo) φ ±οο
By taking contingent cones to the above inclusions, we obtain the
inequality
Vu € X, D-tV(xo)(u) < liminf finf D-tU(xo,yo){u',v)]
u1—>u \v& J
because
-KSp^D^Uixo.yo)) = тгТ£р^(х0,у0,и(хо,уо))
С Τπ{ερ{υ))(χ0,ν(χο)) = T£p{v)(xq,V(xq)) = Sp(D}V{xQ))
can be easily translated into the desired inequality. Π
In order to obtain the equality sign in some of the above formulas,
we need further assumptions, such as the functions are episleek or
pseudo-convex, which actually require the introduction of adjacent
and circatangent epiderivatives. These are the topics of the next
section.

236
6 - Epiderivatives of Extended Functions
6.2 Other Epiderivatives
We recall the definition of "lim sup inf" (or Γ-convergence) of
functions of two variables introduced in Definition 5.2.4:
]imsupa!,_KBiiifB#_n,0(a:/Jy/) := sup inf sup inf Mx\y')
e>0 J7>° i'eB(i,ij) v'€B(y,e)
In the same way, we set
lim in£xi_>x sup4i_>4<j>(x', y') := inf sup inf sup ф(х',у')
ε>0 η>0 χ'£Β{χ,η) у'еВ{у,е)
6.2.1 Adjacent and Circatangent Epiderivatives
We can associate with the adjacent and Clarke tangent cones the
adjacent and circatangent derivatives of V at χ G Dom(V) of an
extended function V : Χ i-> R U {±00} regarded as the set-valued
map V : X ~> R taking empty values outside of the domain of V:
' DW(x)(u) := DW(x,V(x))(u) =
<
{v I lim sup/l_>0+ 'miui->u \V(x + hu') — V(x) — hv\/h = 0}
and
' CV(i)(u) := CV(x,V{x)){u) =
<
{v I lim sup^o+,χ'-χ nuv^ \V{x' + hu') - V{x) - hv\/h = 0}
We are also naturally led to associate with the two set-valued maps
Vf and Vj, defined in the preceding section, their adjacent and
circatangent derivatives at points (x,V(x)).
Definition 6.2.1 (Epiderivatives) Let V : X 1-» R U {±00} be а
nontrividl extended function and χ belong to its domain. We shall
say that the functions D^V(x) and C^V(x) from X to R U {±00}
defined respectively by
' i) Vuel, D\V{x){u) := mi\v\v£D]'V}{x,V{x)){u)}
<
w ii) Vuel, C}V{x){u) := inf {υ \υ€ CVt(i,V(i))(u)}

6.2. Other Epiderivatives
237
are the adjacent and circatangent epiderivatives of V at χ in the
direction u.
These epiderivatives can also be characterized as limits of differential
quotients.
Proposition 6.2.2 Let V : X i-> RU{±oo} be a nontrivial extended
function and χ belong to its domain. Then
ь , w ч ι- · r V(x + hu')-V(x)
D]V{x)[u) = limsup^_t0+
and
V(x' + hu') — λ'
CiV(x)(u) = limsuph^0+iXl^xy(xl)<x^V(x)mfu>^u
We define in a symmetric way the adjacent and circatangent hypoderiva-
tives D\V(x) and CiV{x) from X to RU {±00}
D\V(x)(u) = -D\{-V){x){u) := lim infh_0+ supu,^uV{x+^ ~V{x)
and
CiV(x)(u) = -C^-V){x){u)
:= lim rciih-+Q+,x>->x,v{x')>\'-*v(x) sup^^V^a;' + hu') - X')/h
Naturally, when V is contingently epidifferentiable, the adjacent
and circatangent epiderivatives are lower semicontinuous and
positively homogeneous, and the circatangent epiderivative is moreover
convex. They coincide with the directional derivatives < V'{x), и >
when V is respectively Frechet and continuously differentiable.
Remark — It is also possible to derive another interpretation
of the epiderivatives in terms of epilimits of the following differential
quotients
r, τ,/ w ч V(x + hu)-V(x)
VhV(x) : и н-> VhV(x)(u) := \ __ _ ; K-L _;_
as we shall see in Chapter 7. □

238
6 - Epiderivatives of Extended Functions
Let К С Χ, χ € Κ and V be Frechet differentiable at x. Then
the adjacent epiderivative of the restriction V\k is the restriction of
the derivative to the adjacent cone.
When V is continuously differentiable at a point χ G K, the
similar statement holds true for the circatangent derivative:
л nt/,π \r \r \ f <V'(x),u> tiu€lt(x)
,) Щтк)(*)(и) := |+0oH' ifnot
«0 Cf №)(«)(.) := {^'W'°> !ίη„ΓΚΜ
The formulas become much more simple when V is Lipschitz.
Proposition 6.2.3 Let us assume that V:1hRU {i°°} is
Lipschitz around a point χ of its domain. Then
ρτ/γ„.ν„Λ — 4m sup - — -
/ι->0+
UtV(a;)(u) = limsup
1 ■ " ■ h
is the upper Dini derivative andr
плп \< \ ν V{x' + hu)-V{x')
C^V(x)(u) = limsup — j- —L
/i->0+,a:'->3: "·
Furthermore, for some neighborhood U of x,
i) the map (y,u) gM χ Χ ι-> C^V(y)(u)
is upper semicontinuous
ii) the map и ι—» C^V(x)(u) is Lipschitz on X
{ Hi) Vu€X, C<t(-V)(x)(u) = CfV{x)(-u)
We also observe the following relations between tangent cones to
epigraphs and epigraphs of epiderivatives.·
Proposition 6.2.4 Let V : X i-> RU{±oo} be a nontrivial extended
function and χ belong to its domain.
'Used also with the notation V°(x, u) by Clarke and VT (x, u) by Rockafellar.

6.2. Other Epiderivatives
239
Then the tangent cones to the epigraph ofV at (x, V(x)) are the
epigraphs of the corresponding epiderivatives ofV at x:
Sp{D\v{x)) =Ί*εήν){χ,ν{χ)) к Sp{C}V{x)) = CSp{v){x,V{x))
and, when χ G К С Χ,
D\0>k)(x) = Ψτ*κ(χ) & βΐ(Ψκ)(χ) = ФсК{х)
The relations between tangent cones and epiderivatives of the
distance functions are given in:
Proposition 6.2.5 Let К be a subset of a normed space X, у G X
апаИк(у) be the set of projections of у onto K. Then ifUfciy) φ 0,
i) D\dK{y){v) < d(v,T\{IiK{y)))
(6.3)
ii) C}dK{y){v) < d(v,CK(nK(y)))
Proof — The proof of the first statement is analogous to the
proof of Proposition 6.1.8. For proving the second inequality, we take
ζ G Ик{у) and w G Ck{z)· We observe that when у <£ K,
VzeUK(y), У x€K, \\z-x\\ < 2\\y-x\\
Hence
s^Ph<a, \\ν-χ\\<β(άκ(ν + hv) - dK{y))/h
< sup^<Qi \\ζ-χ\\<2β dK{z + hw)/h + ||v - w\\
so that for every w G Ck{z), С-\ак{у){у) < ||υ—го||. Since ζ G Πκ(?/)
and w G С к (ζ) were chosen arbitrarily, our proof ensues. Π
The adjacent derivatives enjoy
Proposition 6.2.6 Let V : X ^> Ru{±oo} be an extended function
continuous on a neighborhood of χ G Int(Dom(V)). The values of the
adjacent derivative at χ are convex.
Furthermore, for any
и G T>om{D\V{x))f\-Dom{D\v{x))
we have
DbV(x)(u) = [D\V{x){u),D\V{x){u)\

240
6 - Epiderivatives of Extended Functions
Proof — As for the contingent case, we can check that for any
и e Oom(D\V{x)) Π Oom(D\V{x)), we have
{D\v(x)(u),D\v{x){u)} С DbV{x){u) С [D\V{x)(u),D\v{x){u)}
Let us take now two elements vi and V2 in D^V(x)(u). Pick any
λ € [0,1]. We shall show that (1 - λ)υι + λυ2 belongs to DbV{x){u)
and thus, that D^V(x)(u) is an interval.
By definition of the adjacent derivative, we know that there exist
sequences u\h and и^н converging to и when h —> 0+ such that
У(х + hulh) - V(x) V(x + hu2h) - V{x)
/ι h
Since У is continuous on a neighborhood of x, for h small enough,
it maps the interval χ + h[uih,U2h] to a connected subset, i.e., an
interval which contains V(x + huih) and V(x + hu^h)· Then there
exists ги^ € [ϊχι^, U2h\ such that
(l-A)V(s + ftui/l) + AF(s + u2/l) = V(x + hwh)
Hence Wh converges to и and
hm —^ ^-^ = (1 — λ)υι + Лг/г
This ends the proof. D
6.2.2 Other Convex Epiderivatives
Convexity of the circatangent epiderivative counts among its
attractive features. But, when the function is not episleek, there are other
positively homogeneous convex lower semicontinuous functions lying
between the contingent epiderivative and the circatangent
epiderivative, among which one can choose whenever convexity is a
prerequisite to apply duality arguments.
Definition 6.2.7 Let 1A;IhRU {^°°} be α nontrivial extended
function and χ belong to its domain. We shall say that a lower
semicontinuous convex positively homogeneous function
6V{x) :X ^ RU{±oo}

g.2. Other Epiderivatives
241
satisfying
D^V{x){u) < 6V{x){u) < C^V{x){u) (6.4)
is a convex epiderivative. It is said to be minimal if there is no other
strictly smaller convex epiderivative.
All these convex epiderivatives coincide when V is episleek.
Convex epiderivatives exist whenever V is contingently epidiffer-
entiable at χ and do not exist if the circatangent epiderivative at χ
takes the value —oo (or, equivalently, CfV(x)(0) < 0.)
The epigraphs of the convex epiderivatives are the closed convex
cones lying between the Clarke tangent cone and the contingent cone
to the epigraph of У at ж.
Among the possible choices of such convex epiderivatives, we can
single out the convex epiderivative Όψν{χ) whose epigraph is the
convex kernel of the epigraph of the contingent epiderivative. It
satisfies
Vuel, D^V{x){u) < D^°V{x){u)
= suPveOom{DiV{a:)№V(xKu + v)-DlV(x)(v)) ^ fyV(x)(u)
Denote by Vy(x) the family of minimal convex epiderivatives 6V(x) of
V at x. Since the convex kernel of the contingent cone is the intersection of
its maximal closed convex subcones (by Proposition 4.5.2), we deduce that
D^V{x) is the supremum of the minimal convex epiderivatives:
VueX, DfV{x)(u) = sup oY(x)(u)
8V(x)eVv(x)
Also, the convex kernel of a cone Ρ being equal to the Clarke tangent cone
to Ρ at the origin (Proposition 4.5.3), we obtain the formula
VueX, D^V(x)(u) = CT(DTy(^))(0)(u)
Remark — Paratingent epiderivatives
We can also define the paratingent epiderivative P^V(x) of an extended
function У:1н RU {±oo} at χ € Dom(y) by saying that the epigraph of
the paratingent epiderivative is the paratingent cone -P-pD,7?.. {x>V{x)).

242
6 - Epiderivatives of Extended Functions
One can then check that
P^V(x)(v) = hminf —i r1 —
h-*0+,y-*x,V(y)-+V(x),u'-*u h
We deduce from Choquet's Theorem 4.5.7 the following result due to
Shi Shuzhong:
Theorem 6.2.8 Assume that V : X н-> R is continuous. Then there
exists a residual R on which the contingent and paratingent epiderivatives do
coincide.
Proof — Since Graph(V) is closed, Choquet's Theorem implies that
there exists a residual 1Z of the graph of V on which the contingent and
paratingent cones coincide. Let R denote the projection Ώ.1Ζ of the residual
11 С Graph(y) С Χ χ R to X. The continuity of V implies that the
restriction of the projector Π to the graph of V is bicontinuous, so that R
is a residual of the domain of V. D
6.3 Epidifferential Calculus
Is the epiderivative of the sum of two functions equal to the epi-
derivative of the sum? We answer here this question and study as
well the epiderivative of the composition product.
Theorem 6.3.1 Let us consider two finite dimensional vector-spaces
X and Υ, a continuous single-valued map f : X >—» Υ and two
extended lower semicontinuous functions V and W from X and Υ to
R U {+oo} respectively. Let xq belong to the domain of the function
U := V + Wof
We assume that f is continuously differentiable around xq, that V
andW are contingently epidifferentiable at xq and f(xo) respectively
and that the following transversality condition:
Dom(CTW(/(s0))) - /'Ы (Dom(CTF(s0))) = Υ
holds true. Then the epiderivatives ofU satisfy the estimates:
( Ϊ) D}U{xQ){u) < D\V(x0)(u) + DiW(f(xoW(x0)u)
ii) D\U(x0)(u) = D\V(xQ)(u) + D\W(f(x0))(f'(xo)u)
{ Hi) CTl7(so)(«) < CiV(x0)(u) + C7TW(/(so))(/'(so)«)

q.3. Epidifferential Calculus 243
In particular, if К is a closed subset of X and if xq G К П Dom(V)
satisfies
Dom(CTT/(z0)) - CK(xQ) = X
then, if V is contingently epidifferentiable at xq,
■ i) Vu G TbK(xQ), Di(V\K)(xo)(u) < DiV(x0)(u)
< ii) Vug 2k(so), D\{V\k){xq){u) = D\V{x0){u)
. Hi) Vug Ck{xo), Ci(V\k)(xo)(u) < CiV{x0){u)
Proof — We shall prove the formula only in the case of cir-
catangent epiderivatives.
If we set
' К := Sp(V) x Sp{W) xR С IxRxFxRxR
< G(x,a,y,b,c) := (f(x)-y,a + b-c)
H(x,a,y,b,c) := (x,c)
we can write
Sp{U) = H{K П (Г1 (0,0))
We shall use Theorem 4.3.3 to estimate tangent cones to КпС_1(0,0).
We first observe that the assumptions of our theorem imply the
corresponding transversality conditions of Theorem 4.3.3.
Let us set
z0 = (xo,V(xo)J(xo),W(f(x0)),U(x0))
Then we deduce that
CK(zo)nG>(z0)-1(0,0) с CKnG- Чо^о)
It remains to show that this inclusion implies the desired inequality.
Fix и G Όοτα(ΟΊν{χ0)) such that f'{x0)u G Dom(CTW(/(z0))) and
define
A = CiV(x0)(u), μ = CiW(f(xQ))(f'(xo)u)

244
6 - Epiderivatives of Extended Functions
Hence
(η,\,ί'(χο)η,μ,λ + μ) € СкЫ ^G'(zq)-1 (0,0)
and thus, it belongs to the Clarke tangent cone to К П G_1(0,0) at
z0.
This means that for any sequence hn > 0 converging to 0 and
any sequence
zn ■= {хп,ап,Уп,Ьп,Сп) € АГП(?-1(0,0)
converging to zq, there exist elements wn := (un, \n, υη, μη, un)
converging to (u, λ, f'(xo)u, μ, λ + μ) satisfying
Vn>0, zn + hnwn e finG_1(0,0)
Since ζπ and ζπ + /ιπ^π belong to G~1(0,0), we infer that
j[xn) = Уп> J(xn "i" h>nun) = Уп "Ь 11"пУп1 Cn = ап "Ь 0π, ί'π = λπ +'μη
and using that ζπ + /ιηιοη belongs to if, we deduce that
У(жп + hnun) -an W(yn + /ιπυπ) - 6n
7 s An к s μ-η.
tin hn
Consequently,
U(xn + hnun) - Cn
7 S ^Π
/in
Since (un, vn) converges to (u, λ + μ), we finally obtain
CiU(x0)(u) < \ + μ = CiV(x0)(u) + CiW(f(xoW'(xo)u) □
Remark — If Χ, Υ are Banach spaces, the conclusions remain true
when we replace the transversality assumption by the following stability
assumption: there exist constants с > 0, a £ [0,1[ and η > 0 such that, for
all n,
г) \fx G Dom(y)nB(a;o,,7), Vt/ ё Dom(W) Π Β{/{χ0),η)
Βγ С /'(го) (Dom(2^V(aO) П сВх\ - Dom(DTW(y)) + aBy
й) raP1IeDom(B}v(*))li?TV(aO(u)l/llull ^ с
"Ό suPueDomc/wO/)) Ι^Τ^ΜΜΙ/ΝΙ ^ с
(6.5)

6.3. Epidifferential Calculus 245
For that purpose, we have to check that the second transversality
assumption of Theorem 4.3.3 is satisfied, i.e., that there exists a constant
d > 0 such that, for all n, for all
(x,a,y,b,c) € К close to zq = {xa,V(xQ),f(x0),W{f{x0)),U{xQ))
for all (z, A) G Υ x R, there exist (ω, μ, υ, ι/, δ) € Τκ(χ, α, y, b, c) and e such
that
!i) ζ = f'(x)u-v + e & A = μ + ν - δ
и) ||е|| < а(И + |λ|) & И + ||V|| + |μ| + \V\ + \δ\ < c'())z\) + |λ|)
Assumptions (6.5) imply right away that there exist и € Oom(D^V(x)),
ν £ Oom(D^W(y)) and e satisfying
(г) ζ = f'{x)u - ν + e,
ii) ||e|| < (φ|| & ||U|| < φ|| & N < {l + a + c\\f'(x)\\)\\z\\
Let us now take
μ := с||и||, г/ := с)Н|, 5 := μ + ι/-λ
We deduce from (6.5)m) that (u, μ) belongs to £p(Db~V(x)), that (v, u)
belongs to Ep(D^W(y)) and that
D\V{x){u) + D^W{y){v) < с(И + И|) = μ + ν = λ + δ
Consequently
1*1 < (|A| + c(|N + ||«||)) < c'(N| + |A|) D
We now provide formulas for computing the epiderivative of the
supremum of a finite number of extended functions.
Let us consider a finite family of functions
V- :X ι-» RU{±oo}, (i G/)
with which we associate the function U defined by
U(x) := max.Vi{x)
We set
I{x) :={»€/ | Ц(х) = U{x)}

246
6 - Epiderivatives of Extended Functions
Proposition 6.3.2 Consider α finite dimensional vector-space X_-,
and η extended functions VI : X >—» R U {+00}. If the following^
transversality assumption at xq € Dom(i7) holds true
η
V щ e Χ, Π (Вот(СтИ(:со)) - щ) φ 0
i=l
then
η
У и e Π DomCDlfViCxo)), £>bl7(so)(«) = max 1>Ы(а:о)(и)
ί=ι ' ie/(x0)
and
η
У и € Π Dom(CTK-(^o)), CU{xq){u) < max CTVi(so)(«)
i=i ie/(xo)
Proof— Since the dimension of X is finite and since the epigraph
of U is the intersection of the epigraphs of the η functions Vi, we can
use Corollary 4.3.6, stating that if for all pairs (щ, \i),
η
f](c£p{Vi)(x0,U(xo))-(ui,\i)) φ 0
г'=1
then
Т*£р{и)(хо,иЫ) = (\Ttp{Vi)(xo,U(xo))
The former property follows immediately from the assumption
η
f|(Dom(CTVi(so))-«i) φ 0
i=l
The left-hand side of the latter formula is the epigraph of the adjacent
epiderivative of U at xq. Since
Tb£p{V{)(xoMxo)) D T^om{Vi)(x0)x-R
whenever Vi(xo) < U(xq), i.e., whenever i £ I{xo), we deduce that
η
Vu€ i]Oom(D\Vi(x0)), DbU(x0)(u) < max D\Vi(x0)(u)

6.3. Epidifferential Calculus
247
The proof of the second statement is analogous. D
We now complete formulas for the epiderivatives of marginal
functions: Consider two normed vector spaces X and Υ and an extended
function
U :X xY ь-* Ru{±oo}
with which we associate the marginal function V : X t-^ILU {+00}
defined by
V(x) := inf U(x,y)
Proposition 6.3.3 Let us consider two normed vector spaces X and
Υ, an extended function [/;Xx7hRU {i°°} a^d its marginal
function V. Let xq G Dom(V) and suppose that there exists yo € Υ
which achieves the minimum of U(xq, ·) onY:
V{x0) = U(x0,yo)
If U is pseudo-convex at (xo,yo), then
У и £ X, О^У(хо){и) = liminf ( inf D-[U{xQ,yQ){ul,ν)
u'—*u \v£Y
' Proof — Let π denote the projection from XxYxRto.X'xR.
We recall that:
kEP{U) С Sp(V) С ttSp{U)
We deduce these statements from any criterion implying that the
contingent cones to the images are the closures of the images of the
contingent cones. This is the case of the epigraph of a pseudo-convex
function, which is pseudo-convex. Then, equalities
{тгЕр(0^и(х0,Уо)) = πΤερψ)(χα, yo,U{x0,y0))
= Τπ{ερ{υ))(χ0,ν(χ0)) = TSp{v)(x0,V{xo)) = ερ(ΌΊν(χ0))
can be easily translated into the desired result. □

248 6- Epiderivatives of Extended Functions-
6.4 Generalized Gradient
6.4.1 Subdifferentials arid Generalized Gradients
We devote this section to dual concepts of epiderivatives of extended
functions.
When a function V is differentiable at x, its gradient V'(x), being
a continuous linear functional, is therefore an element V'(x) € X* of
the dual of X:
Vv€X, <V'(x),v>= DfV(x)(v)
When V is no longer different iable, but contingently epidifferen-
tiable, we can still introduce subgradients of У at ж, which are those
continuous linear functionals ρ € -X"* satisfying
V«el, <p,v > < DfV(x){v)
which constitute the (possibly empty) closed convex subset
dQV(x) := {p£X*\Vv£X, <ρ,υ> < ΌΊν(χ)(υ)}
Definition 6.4.1 (Subdifferential) Let V : X ■-» R U {±00} be
a nontrivial extended function and χ belong to its domain. Assume
that V is contingently epidifferentiable at x. The subset
d0V{x) С X*
is called the subdifferential ofV at χ (or contingent generalized
gradient) and its elements are called the subgradients.
We observe that when the contingent epiderivative D^V(x)(-) is
convex, it is the support function of the subdifferential doV(x).
When it is not convex, it is convenient to regard any convex
epiderivative 6V(x)(·) (see Definition 6.2.7) as the support function
Vv€X, 6V{x){v) = a(dsV(x),v)
of the closed convex subset dsV(x) defined by
d6V{x) := {p£X*\ Vv€X, <p,v>< 6V{x)(v)} (6.6)

6.4. Generalized Gradient
249
Definition 6.4.2 Let ViIhRu {±°o} be α nontrivial extended
function and χ belong to its domain. Assume that V is contingently
epidifferentiable at x.
The (Clarke) generalized gradient of V at χ is defined by
dV{x) := {p£X*\ V«el, <P,u>< СтУ(ж)(и)}
More generally, we say that the closed convex subset dsV(x)
associated to a convex epiderivative 6V(x) through formula (6.6) is the
δ-generalized gradient of У at x.
Naturally, we observe that
d0V(x) С d6V{x) с dV{x) (6.7)
Another way to state that ρ G doV(x) is a subgradient is to say
that the pair (p, — 1) belongs to the polar cone to the contingent cone
to the epigraph of V at (x, V(x)):
(p,-l) € (T£p{v)(x,V(x))Y
In the same way, to state that ρ € dV{x) amounts to saying that
the pair (p, — 1) belongs to the (Clarke) normal cone N£p(V)(x, V(x))
to the epigraph of V at (x, V(x)).
•Remark — Recall that Oy{x) denotes the family of minimal
convex epiderivatives 6V(x) of У at i. The 5-generalized gradients dsV(x)
associated with minimal convex epiderivatives 6V(x) axe the minimal δ-
generalized gradients containing the subdifferential daV(x).
If we denote by 9оо^(а;) the generalized gradient associated with the
convex kernel of the contingent epiderivative, which is the supremum of the
minimal convex epiderivatives, we obtain the formula
dQV(x) с dooVix) = со (J dsV(x) ) с dV(x)
Naturally, when V is Prechet differentiable at x, then
ΌΊν{χ){ν) =<V'{x),v> < 6V(x)(v)
so that the subdifferential 8qV{x) is reduced to the gradient V'{x).

250
6 - Epiderivatives of Extended Functions
When V is continuously differentiable at x, the circatangent epi-
derivative coincides with the directional derivative < V'(x),· >, so
that the «^-generalized gradients are reduced to the only gradient:
dsV(x) = {V'(x)} when V is continuously differentiable at χ
If У is continuously differentiable around a point χ G K, then
the generalized gradient of the restriction is the sum of the gradient
and the normal cone:
d{V\K){x) = V'(x) + NK(x)
We also note that the generalized gradient of the indicator of a
subset is the normal cone:
θφκ(χ) = NK(x)
The Fermat and Ekeland rules can be formulated in a ^-generalized
gradient way:
Theorem 6.4.3 (Fermat Rule) Let V : X i-> RU{+oo} be a non-
trivial extended function defined on a normed space and χ £ Dom(V)
be a local minimizer of V on X. Then it is a solution to the
inclusions:
0 e d0V{x) С d5V{x) С dV(x)
for any convex epiderivative 6V(x) ofV at x.
Theorem 6.4.4 (Ekeland Rule) Consider a Banach space X, a
nontrivial lower semicontinuous bounded from below function
1^:IhR+u {+oo}
and let xq G Dom(V) be a given point of its domain. Assume that
V is contingently epidifferentiable and let 6V(x) denote any convex
epiderivative of V at χ.
Then, for any ε > 0, there exists a solution χε £ Dom(V) to:
i) V(xe) +ε\\χε - x0\\ < V(xq)
(6.8)
ii) 0 € θδν{χε)+εΒ* С dV{xe)+£B*

6,4. Generalized Gradient
251
Proof— Indeed, Theorem 6.1.12 implies the existence of χε € X
satisfying (6.8) г) ала
Vv€X, 0 < DTV(se)(t/)+e||t/||
Therefore, for any convex epiderivative 6V(xe), we deduce that
0 < 6V(xe)(v)+e\\v\\
= σ{δδν{χε),ν) + σ{εΒ*,ν) = σ(θδν(χε) + eS*,υ) D
6.4.2 Limits of Subdifferentials and Gradients
Not only the subdifferential doV(x) is contained in the Clarke
generalized gradient dV{x), but, in a finite dimensional vector-space, the
upper limit of subdifferentials doV(xn) when also a subset
of this generalized gradient:
Theorem 6.4.5 Let X be α finite dimensional vector-space and
^:IhRu {+oo}
an extended lower semicontinuous function contingently epidifferen-
tiable in a neighborhood of χ £ Dom(V). Then
Limsup^,^,))^^)) d0V{x') с dV{x)
Proof — Indeed, let xn be any subsequence converging to χ
such that V(xn) converges to V(x) and ρ be the limit of a
subsequence of subgradients pn G doV(xn). Since
(Pn,-1) € {Tsp(v)(xn,V{xn)))
we deduce from Theorem 4.4.3 that [p, —1) belongs to the (Clarke)
normal cone to the epigraph of V at (x, V(x)). This means that ρ
belongs to the generalized gradient dV{x). □
In Hubert spaces and some Banach spaces, one can prove that
the limits of the gradients at points where V is differentiable belong
to the generalized gradient: . _

252
6 - Epiderivatives of Extended Functions
Theorem 6.4.6 Consider α uniformly smooth Banach space X such
that the norm of X* is Frechet differentiable off the origin and an
extended lower semicontinuous function V : Χ \—► RU {+00}
contingently epidifferentiable on. a neighborhood of χ £ Dom(V).
Let a sequence of elements {xn)n>i converging to χ be such that
V is Frechet differentiable at xn and V(xn) converges to V{x). Then
σ - Iimsupn_too{V'/(sn)} С dV{x)
Proof— We claim that the pair (V'(xn), —1) belongs to the
polar cone to the weak contingent cone Τξ ,vJxn, V(xn)), defined in
Section 4.1.5. by
(u, v) € T£ ,Vs{xn, V(xn)) <=$■ (u, v) is a weak cluster point of
<
(uh,Vh) € (Sp(V) - (xn, V(xn))) /h when ft-»0+
Indeed, fix (u,v) € T^p,vJxn,V{xn)). Since V is Frechet
differentiable at xn,
V{xn + huh) = V(xn) +h< V\xn),uh > +he(h) < V(xn) + hvh
where e(h) converges to zero with h. Since continuous linear func-
tionals are weakly continuous, we deduce that
<V'(xn),u>-v = ({V'{xn),-l),(u,v)) < 0
Now, assume that ρ is the weak-* limit of a subsequence of the
gradients V'(xn). Then the pair (p, — 1) belongs to the sequentially weak
upper limit of the polar cones to the weak contingent cones:
(p,-l) € ^-Limsupn_too(2fp(vj(snjF(sn)))
By Theorem 4.4.4, it is contained in the normal cone N£p^(x, V(x)).
This means that ρ belongs to the generalized gradient dV(x). □
6.4.3 Local Subdifferentials and Super differentials
In finite dimensional vector-spaces, the subdifferential coincides with
the local subdifferential d-V{x) defined in the following way:

6.4. Generalized Gradient
253
Definition 6.4.7 Let X be α normed space, V : X i-> R U {±00}
and xq € Dom(V). We shall say that the subset d-V(xa) of elements
ρ G X* satisfying
^^ΥΜζΙψζ^φ^ι^^ > о (6,9)
z-zo If-zoII
is the local subdifferential4 of V at жо·
In a symmetric way, the local superdifferential d+V(xo) ofV at
xq is defined by
d+V(x0) := -d-(-V)(x0)
It is the subset of elements ρ € X* satisfying
V{x)- V{xq)~ <p,x-x0>
limsup—- r. -. < 0
1—no IF -EoH
Clearly super- and subdifferential are closed (possibly empty)
convex sets. When V is Frechet differentiable, they coincide with the
gradient of V.
Proposition 6.4.8 Let X be a normed space, V:IhRU {±00}
be a nontrivial extended function and xq G Dom(V). Then the local
subdifferential d-V(x) is contained in the subdifferential doV(x).
These two subsets coincide when the dimension of X is finite.
Proof— Let us take ρ € d~V(xo) and υ € X. There exist
sequences of elements hn > 0 and vn £ X converging to 0 and υ
respectively such that (V(xo + hnvn) — V(xo))/hn converges to D^V(xq)(v).
Therefore
' D^V(x0)(v)- <p,v>
< = Нтп_>00(У(жо + hnvn) ~ V{xq) -hn<p, vn >)/hn
k > \\v\\]iminix->X0(V(x)-V(xQ)-<p,x-x0>)/\\x-x0\\ > 0
4This concept, used by Crandall & P.-L. Lions for defining viscosity solutions
to Hamilton-Jacobi equations, has been called "subdifferential." We add here
the adjective "local" to avoid confusion with the concept of subdifferential (or
contingent generalized gradient), although these two concepts are equivalent in
finite dimensional vector-spaces.

254
6 - Epidenvatives of Extended Functions
so that < ρ,υ >< DjV(xo)(v) for every υ € X.
. Conversely, let ρ satisfy < p,v >< D^V(xq)(v) for every υ G X
and assume that X is finite dimensional. There exists a sequence of
elements xn converging to xq such that
{]iminfx->XQ(V(x) - V(xq)- <p,x- xq >)/\\x - xq\\
= linin-foo(V(a;n) - V(xq)- <p,xn- xo >)/\\xn - xo\\
Taking a subsequence if needed, we may assume that the sequence
of elements vn :— (xn — xo)/\xn — xq\\ of the unit sphere (which is
compact) converges to an element v. Then for hn := ||a;n — xq\\, we
have
Нтп_>00(У(а;п) - V(x0)- <p,xn- x0 >)/||a:n - x0\\
' — Итп_>оо(У(ж0 + hnvn) - V(xq) -hn< p,vn >)/hn
> D}V{xQ){v) - <ρ,υ>> Ο
so that ρ satisfies property (6.9.) □
6.4.4 Remarks
The table of formulas on support functions allows one to translate
the properties of the circatangent epiderivatives into corresponding
properties of the generalized gradient and vice-versa.
For instance, Proposition 6.2.3 can be restated in the following
form.
Proposition 6.4.9 When V: Jf hRU {^°°} zs locally Lipschitz
on the interior of its domain, then the generalized gradient satisfies:
• the function (x,u) € Χ χ Int(Dom(V)) (-► a(dV(x),u) is
upper semicontinuous, so that dV(-) is upper hemicontinuous
J «Vie Int(Dom(V)), dV(x) is nonempty bounded closed convex
^.VxS Int(Dom(l/)), dV{x) = -d{-V{x))

6.5. Convex functions
255
Remark — When a function V is continuously differentiable at x, its
gradient V'(x) being a continuous linear functional, it is both an element
V'(x) e X* of the dual and the image V'(a;)*(+1) of +1 by the transpose
V'(x)* of V'(x), a linear operator from R to X*.
When V is no longer continuously differentiable, the generalized
gradient remains intimately connected with the transpose of the circatan-
gent derivative CV^(x,V(x)), (which, shall we recall, is the codifferential
CVf(x, V{x)Y) of the set-valued map VT at (x, V{x)). Namely:
Proposition 6.4.10 Let V : X i-> R и {±00} and χ € Dom(V). The
generalized gradient is the value at 1 of the codifferential o/Vj at (x, V{x)).
Proof — Indeed, the codifferential is a closed convex process from R to
X* which, being positively homogeneous, needs to be defined only at the
points —1,0 and +1.
We obtain from the definitions
i) CVi(x,V(x))*(-l) = 0
ii) CVT(a;,V(aO)*(0) = (Dom(CTV)(a;))~ =: d°°V(x) (6.10)
Hi) CVT(a;,V(aO)*(+l) = dV{x) □
6.5 Convex Functions
Convex functions enjoy further properties. We already mentioned
that an extended function is convex (respectively lower semicontin-
uous) if and only if its epigraph is convex (respectively closed.) A
convex function defined on a finite dimensional vector-space is locally
Lipschitz on the interior of its domain and a convex lower semicon-
tinuous function on a Banach space is also locally Lipschitz on the
interior of its domain.
Proposition 6.5.1 When the function V:IhRU {+00} is
convex, the contingent, adjacent and circatangent epiderivatives coincide
and are equal to
D}V{x){u) = ШЫ(Ы^^±^Щ
1 Ч М ' u'-m \h>0 h J
Furthermore, when χ belongs to the interior of the domain of V,
Vii £ I, DiV(x)(u) = mf — j- K-± is finite

256
6 - Epiderivatives of Extended Functions
When V is convex, the generalized gradient coincides with the subdif-
ferential introduced by Moreau and Rockafellar for convex functions:
dV{x) = {p € X*\ V у £ X, <p,y-x> < V(y) - V(x)}
There is more to that: lower semicontinuous convex functions
enjoy duality properties. In the same way that we associated with cones
their polar cones, with closed convex processes their transposes, we
can, following Fenchel, associate with lower semicontinuous convex
functions conjugate functions for the same reasons, and with the
same success.
Definition 6.5.2 Let V : X —> RU{+oo} be any nontrivial extended
function defined on a Banach space X. We associate with it its
conjugate function V* : X* -»RU {+°°} defined on the dual of X
by
Vp € X*, V*{p) := sup{<p,x>-V{x))
xeX
Its biconjugate V** :IhRU {ioo} is defined by
V**(x) := sup {<p,x> ~V*(p))
pe x*
For instance, the conjugate function of the indicator ψ χ of a
subset К is the support function αχ.
We deduce from the definition the following convenient inequality
Vxel, ρ € X\ <p,x>< V(x) + V*(p)
known as Fenchel's Inequality. The epigraphs of the conjugate and
biconjugate functions being closed convex subsets, the conjugate
function is lower semicontinuous and convex and so is its
biconjugate when it never takes the value -oo. We observe that
Vx £ X, V**(x) < V{x)
If equality holds, then V is convex and lower semicontinuous. The
converse statement, a consequence of the Hahn-Banach Separation
Theorem, is the first basic theorem of convex analysis:

6.5. Convex functions
257
Theorem 6.5.3 A nontrivial extended function V : X —> RU{+oo}
is convex and lower semicontinuous if and only if it coincides with
its biconjugate. In this case, the conjugate function V* is поп trivial.
So, the Fenchel correspondence associating with any function V
its conjugate V* is a one to one correspondence between the sets of
nontrivial lower semicontinuous convex functions defined on X and
its dual X*. This fact is at the root of duality theory in convex
optimization.
We deduce at once the following characterization of the subdif-
ferential:
Proposition 6.5.4 Let V : X —> RU{+oo} be a nontrivial extended
convex function defined on a Banach space X. Then
ρ € dV{x) <=> <p,x>= V{x) + V*(p)
If moreover the function V is lower semicontinuous, then the inverse
of the subdifferential dV(-) is the subdifferential dV*(·) of the
conjugate function:
ρ € dV{x) Φ=» χ € dV*(p)
Since - V*(0) = inf-cgx V(x), the Fermat rule becomes:
Theorem 6.5.5 Let K:1h»RU {+°°} be a nontrivial convex
extended function defined on a normed space X. Then χ £ X achieves
the minimum ofV on X if and only if
0 € dV(x) or, equivalents, V и € X, 0 < D}V^c){u)
If moreover the function V is lower semicontinuous, then dV*(0) is
the set of minimizers of V.
The second fundamental theorem of convex analysis, also a
consequence of the Separation Theorem, provides a criterion (weak
constraint qualification) for computing the conjugate function of the sum
and the composition product of convex functions:
Theorem 6.5.6 (Fenchel) Let X and Υ be reflexive Banach spaces,
A € C(X,Y) a continuous linear operator and V:I-»RU {+00}

258
6 - Epiderivatives of Extended Functions
and W : Υ —> R U {+00} be convex lower semicontinuous functions.
Let U :=V + WoA.
We posit the weak constraint qualification assumption5: - -
0 € lnt(A(Dom(V)) - Oom(W))
Then
Dom([T) = Dom(l/*) + A*Oom(W*)
and, for every ρ G Dom(i7*)J there exists a solution q G Y* to the
minimization problem
U*{p) = {V*(p - A*q) + W*{q) = inf {V*{p - A*q) + W*(q))
When ρ = 0, we deduce from the fact that
t/*(0) = -inf I7(x)
and from the above theorem the duality relation: for any χ € X, q €
inf (F(s) + W(Ax)) + inf (V^f-A*?) + W*{q)) = 0
We observe that the subdifferential map dV(·) of a nontrivial
lower semicontinuous convex function is obviously monotone. When
X is a Hubert space, we can prove that dV(-) is maximal monotone.
This is a consequence of Minty's Theorem 3.5.8 and the following
Moreau Theorem:
Theorem 6.5.7 (Moreau) Let X be a Hilbert space identified with
its dual and
V :X ^ RU{+oo}
be a nontrivial lower semicontinuous convex function. For every λ >
0, define the Moreau-Yosida approximation V\ by
Vx € X, Vx{x) := mf (v(y) + ^||s - y||2) < V(x)
5We observe that U is non trivial if and only if 0 e A{Oom{V)) - Dom(W)·
In this case, inequality U*[p) < inf,ey* (V*(p — A*q) + W*(q)) is always true.

6.6. Higher Order Epiderivatives
259
This minimization problem has a unique solution J\x:
Vx(x) = V(Jxx) + ±\\x - Jxxf
which is the resolvent of the maximal monotone map dV(-).
Furthermore, V\ is convex, continuously differentiable, and
converges pointwise to V on Dom(V) when λ —> 0+.
The Yosida approximation defined by A\x := (x- J\x)/\, is both
equal to the gradient ofV\ at χ and to a subgradient of V at J\x:
Axx = V{{x) к Axx € dV(Jxx)
Finally, this implies that the domain of dV(-) is dense in Dom(V).
6.6 Higher Order Epiderivatives
Let us consider a normed space X and a nontrivial extended function
V ;X h+ Ru{+oo}
We can define m^-order epiderivatives by taking mt/l-order derivatives
of the set-valued map Vj.
We observe that vm belongs to
jD(m)VT(a;,2/,Ui,wi,... ,um-i,vm-\){um)
if and only if
vm > liminf^_>o+)li/ji_>lim -^
(V{x + hUi+... + hT^Um-i + hmu'm) -y-hVi-...- hr^Vm-l)
Denote by D^'V(x,ui,... ,Uj) the infimum of the subset
i?u)VT (a;, V{x), UuD^Vix)^),..., D^~l)V{x, uu ..., ty-Ofa)) Ы
which is equal to
D\jS>V(x, Ui,...,uj) := lim inffc_0+i u/_Uj. £
<
(V{x + Atii + ... + ft^S-i + A^) - ΣίΖα tiD\l)V{x,Ul,...,u,))

260 - 6- Epidenvatives of Extended Functions
By taking successively у = V{x);-V\ := D^V(x,Ui), etc., we can check
recursively that for any j,
' 2?ϋ)ντ (x,V(x),u1,D]V(x){u1),...,D<f~1)V{x,ul,...,Uj-1)) (Uj)
D\3)V{x, uu..., uj), +001
Definition 6.6.1 Let us consider α normed space X and a nontrivial
extended function F:Xb»RU {+00}. Let χ € Dom(y) and
U\ € Dom(£)jV(x)),..., Urn—ι e Dom(l)jm 1V(a;,wi,...,wm_2,·))
be given. Then
D^V(x,ui,...,um) ■= liminf^_>o+,u'm-^Um t^
<
_ (y{x + hu1 + ...hm-1um-1+hmu^)-YZ^hlD\l)V{x,u1,...,ui))
is called the mt,l-order contingent epiderivative of V at χ in the directions
Ml, U2, ..., Um.
We define in the same way т*к-отает adjacent and circatangent epi-
derivatives
D\{m)V{x,Ui,...,um) h C^m)V(x,uu...,um)
using the corresponding m—th order derivatives of set-valued maps.
For m = 2, we obtain
D^V(x,uuu2)
:= limЫк-*о+!и'2->и2 {V(x + hui + h2u'2) - V{x) - hD^V{x){ul)) /h?
We observe right away that
£>}т)^(а;,«1,...,«т-1,·) = Ψτ™[Χί^ „_,)(·)
and that, for every χ G К
) '^=> D(fn'dK(x,Ui,...,um-i:um) = 0
The mt/l-order contingent epiderivative enjoys the same kind of calculus
as those of first-order.

6.6. Higher Order Epiderivatives
261
Theorem 6.6.2 Let us consider two finite dimensional vector-spaces X
and Y, a continuous single-valued map f : X \-^ Y, and two nontrivial
extended functions V and W from X and Υ to R U {+00} respectively.
Let хй belong to the domain of the function U := V + W о f. We
assume that f is twice continuously differentiable around xq, that V and W
are contingently epidifferentiable at xQ and f(xQ) respectively and that the
following transversality condition:
Oom(CiW(f(x0))) - f'(x0) (Oom(CiV(xQ))) = Υ
holds true.
Then the second-order epiderivatives of U satisfy the estimates:
Df)U{xu,Ui,u2) < Db}{2)V{xu,ui,u2)
+ D\2)W(f{xQ)J,{xQ)u1,f,(xQ)u2 + £/"(:βο)(«ι,«ι))
and
Dbi{2)U(xa,u1,u2) = D^2)V{xu,uuu2)
+ Ό){2)]νϋ(χ0),Γ(χ0)^,Πχ0)^ + |/"(a;o)(«i,«i))
In particular, if К is a closed subset of X and ifxQ € K(lOom(V) satisfies
Оот(СтУЫ) - CK{xQ) = X
then, ifV is contingently epidifferentiable at xq,
i) VueTf^ctti), D{2)(V\K)(x0,Ul,u2) = D{2)V(xQ,Ul,u2)
[ ii) УиеТ%2)(х0,щ), d\{2){V\k){xu,uuu2) = D\(2)V{xQ,uuu2)
Proof— The proof is exactly the same as the first-order Theorem 6.3.1,
but we use Theorem 4.7.4 instead of Theorem 4.3.3. □
We can make more precise the Fermat rule by using mt/l-order
contingent epiderivatives.
Indeed, we saw that if V : X —> R U {+00} achieves its minimum at
some xq G Dom(^), then for every u\ e X, we have DfV(xQ)(ui) > 0. We
can now distinguish the directions u\ at which D^V{xq){u\) = 0. Then we
see that for such directions, Di 'V{xq){u\,u2) > 0, and so on.

262
6- Epiderivatives of Extended Functions
Proposition 6.6.3 Let us consider a~ normed space X and a nontrivial
extended function V : X j—> R U {+00}. Assume that xq achieves the
minimum of V. Then
Vtti,...,Um-i such that V j — 1,.. . ,m — 1, D^ V(xq,u\, ... ,Uj) = 0,
<
we have VtimeX, 0 < D^m'V(xa,ui:... ,um-i,um)
We can prove that second order conditions are sufficient in finite
dimensional vector-spaces:
Theorem 6.6.4 Consider a normed space X and a nontrivial extended
function 1/:IhRU {+00}.
Assume that xq e Dom(y) satisfies
( i) \/u € X, 0 < D^V{xa)(u)
{ (6.11)
{ ii) \/u^Q such that D^V{xu){u) = 0, D^]V{xu, щи) > О
Then xq is a weak local minimizer of V in the sense that for every и in
the unit sphere S, there exists eu > 0 such that, for any h £ [0, eu] and any
e € euB,
V{xQ) < V{xQ + hu + h2u + h2e)
Proof — Assume the contrary: there exist и € S and sequences of
hn > 0 and en ζ Χ converging to 0 such that
V(xQ + hnu + h2nu + hlen) - V(xQ) < 0
By letting η —► oo, we deduce that D^V(xQ)(u) < 0. Together with
assumption (6.11) г), this implies that D^V(xQ)(u) = 0. We then infer from
the definition of the second-order contingent epiderivative that
Df]V{xQ,u,u) < 0
which is a contradiction of assumption (6.11) ii). D
Remark — A local minimizer is always a weak local one.
Stronger conditions on the second order epiderivative imply stronger
conditions. □

6.6. Higher Order Epiderivatives
263
6.6.1 Second Order Epiderivatives of Moreau-Yosida
Approximations
Let X be a Hilbert space identified with its dual and
V :X h+ Ru{+oo}
be a nontrivial lower semicontmuous convex function. We recall that it is
the pointwise limit when A —> 0+ of C1 convex functions V\, the Moreau-
Yosida approximations, defined by
Ух eX, Vx(x) := mf (v(y) + ±\)x - yf^j
This minimization problem has a unique solution 3\x:
Vx(x) = V(Jxx)+±-\\x-Jxxf
Furthermore, A\x := (x — J\x)/X is both equal to the gradient of V\ at χ
and belongs to a subgradient of V at J\x:
Αχχ = V{(x) & Αχχ e dV{Jxx)
Finally, we associate with any ρ £ dV(x) the normal cone NgV^{j>) to the
subdifferential of V at χ at some ρ € dV(x), equal to
{v e X\ <p,v>= D-[V{x){v)}
For instance, when V = ψκ is the indicator of a closed convex subset
K, then
Vx{x) := d2K{x)/2\ & Jxx = -κκ{χ)
is the projection of best approximation of a; to Χ and
ΝΝκ(πκχ)(χ-πκχ) ■= {ν € Τκ(πκχ)\ < χ - πκ(χ),υ > = 0}
Theorem 6.6.5 Let X be a Hilbert space identified with its dual and V :
Χ η-► R U {+oo} be a nontrivial lower semicontinuous convex function.
Then, for any χ £ X, A > 0, for every щ, ui, V2 & X and
Ы e NdV{Jx(x))(Ax{x))
we have
О^Ух{х,иии2) < D^V{Jxx,v1,v2)+<Axx,U2-V2>+ — \\ui-vl\\2

264 6- Epiderivatives of Extended Functions
In particular, by taking V = ψκ we obtain
Corollary 6.6.6 Let К be α closed convex subset of α Hilbert space X.
Then, for any U\, u2, v\ £ Тк{кк{х)) satisfying
< χ — πκ{χ), ν\ > = 0
and V2 £Tj( (x,vi), inequality
D2d2K(x,Ui,u2) < 2<x-nK(x),u2-v2> + )\щ - vi\\2
holds true.
Proof — We observe that
V\{x + hui + h2u2)
< V{J\x + hvi + h2v2) + \\x - J\x + /i(wi - vi) + h2{u2 - υ2)\\2/2λ
<
= V{J\x + hvi + h2v2) + \\x - Jxx\\2/2X + h < {x - J\x)/X, щ - vi >
+h2 (< (x - Jxx), u2 - v2 > /A + ))щ - Vl ||2/2A + e(h))
where e(h) converges to 0 with h.
We now take into account that
Αχ = (1-Λ)/λ & D^Vx(x)(Ul) =<Axx)Ul>
and that
D\V{J\x){vi) =< Αχχ,νι > whenever νλ € NdV(Jx{x))(Ax(x))
Therefore, we obtain
' (Vx(x + hu, + h2u2) - Vx(x) - hDiVx(x)(Ul)) /h2
- < (V{Jxx + hvi+h2v2)-V{Jxx)-hD]V{Jxx){vl))/h2
+ <Axx,u2-v2>+\\u1-v1))2/2X + e(h)
We then take the liminf when h —> 0+ in the above inequalities to obtain
the formula. □

Chapter 7
Graphical & Epigraphical
Convergence
Introduction
Here are some basic notions on graphical limits of single-valued
and/or set-valued maps and epigraphical limits (or epilimits) of
extended functions.
It is one of the basic themes of this book to regard maps, and
more generally, set-valued maps, not only as maps from one space to
another, but as graphs, in order to characterize them in an intrinsic
and symmetric way.
This point of view, which goes back to the prehistory of analysis
with Fermat and Descartes dealing with curves rather than functions,
has been let aside for a long time. In particular, as far as limits of
functions and maps are concerned, generations of mathematicians
have been accustomed to deal with many concepts of convergence of
functions, from pointwise to uniform, but all based on the fact that
a map is a map, and not a graph.
When dealing with limits of maps, either single-valued or set-
valued, it is quite advantageous to replace pointwise convergence by
graphical convergence:
Instead of studying (more or less uniform) limits of images of
maps, we consider the limits of their graphs1. Then the graphs of
1This point of view, regarding maps as graphs, has already been used in
265

266
7- Graphical L· Epigraphical Convergence
the upper and lower graphical limits of a sequence of set-valued maps
Fn are the upper and lower limits of the graphs of the FnS.
One of the main reasons for doing so is to treat on the same
footing a map and its inverse. This is quite important in approximation
theory, where the problem is to derive pointwise convergence of the
inverses from the pointwise convergence of the maps.
Indeed, we already proved an extension of Lax's principle to the
general case of inclusions (Theorem 5.4.2) stating that stability,
convergence of the data and consistency imply the convergence of the
solutions:
Consistency of set-valued maps Fn at (хо,Уо) means simply that
{x0> Уо) belongs to their graphical lower limit.
Hence, a first problem is to investigate the connections between
graphical and pointwise convergence.
We then relate in Section 3 the concepts of graphical
convergence of set-valued maps to the concepts of epigraphical limits of
functions. These concepts have recently been successful in
overcoming the failure of pointwise convergence in many problems of calculus
of variations, optimization, stochastic programming2, etc.
The use of this concept is mandatory whenever the order
relation of the real line comes into play, as in optimization or Lyapunov
stability for instance. In such cases, a function is replaced by the
set-valued map obtained by adding to it the positive cone R+ (for
minimization), whose graph is thus the epigraph of the function.
Therefore, the convergence of the graphs of such set-valued maps is
the convergence of the epigraphs of the associated functions.
We present just a selection of issues on epiconvergence, among
which are some formulas dealing with epilimits of sums and
composition products of functions (Section 4) and conjugates of epilimits
(Section 5.)
We answer in Section 6 the problem of the graphical convergence
of gradients of differentiable functions and generalized gradients of
lower semicontinuous functions. In the last section, we study the
"asymptotic derivatives" of a sequence of functions.
Chapter 5 to build a differential calculus of set-valued maps based on
"graphical derivatives."
2see for instance the book [30] and its bibliography.

7.L Graphical Limits
267
7.1 Graphical Limits
7.1.1 Definitions
We shall use the concepts of upper and lower limits of sets to define
graphical convergence of set-valued maps.
Definition 7.1.1 (Graphical Convergence) Let us consider
metric spaces Χ, Υ and a sequence of set-valued maps Fn : X ~> У. The
set-valued maps ΙΛπτπ_>0οί1π and Lim n-»oo-Pn from X to Υ defined
by
(i) Graph (Lim8 п-юо-Рп) := LimsupTl_>00Graph(i;,Tl)
ii) Graph(LimbTl_>oo.FTl) := LiminfTl_>ooGraph(FTl)
are called the (graphical) upper and lower limits of the set-valued
maps Fn respectively.
If the graphical lower and upper limits coincide, it is called the
graphical limit.
To say that (x,y) belongs to the graphical lower limit of set-valued
maps Fn amounts to saying that they are consistent at (x,y) (see
Definition 5.4.1.)
We provide a more explicit characterization of these graphical
upper and lower limits, which follows immediately from the
definitions:
Proposition 7.1.2 Consider metric spaces Χ, Υ and a sequence of
set-valued maps Fn : X ~> У. Then
у € (UiJn^oaFn^ (i)
if and only if у is the limit of a subsequence of elements yni G F{xni)
where xn/ converges to x. Furthermore
у € (Limbn_>00i1n)(a;)
if and only if у is the limit of a sequence of elements yn € F(xn),
where xn converges to x.

268
7 Graphical <fe Epigraphical Convergence
The pointwise limit / of single-valued maps fn (when it exists) is
a selection of the graphical upper limit of the fn. The latter is equal
to f when fn remains in an equicontinuous subset: Indeed, in this
case, any limit of elements (xn, fn(xn)')--bemg of the form (z,/(z))
belongs to the graph of /.
We point out two useful formulas:
Proposition 7.1.3 Let us consider a sequence of set-valued maps
Fn:X^Y. Then
(Lim^eJ^Xz) D n£>oLimsuPn->oo-FM£(z,e))
(Limbn_>00i1n)(z) С D£>0Liminfn->ooFn(B(x,e))
These formulas can be regarded as relating graphical convergence
with some kind of "almost pointwise convergence." But can we
compare the graphical convergence of Fn and the "pointwise
convergence" of Fn, i.e., the upper and lower limits of the subsets Fn{x)l
The following statement provides the easy answers.
Proposition 7.1.4 Let us consider metric spaces Χ, Υ and a
sequence of set-valued maps Fn '. Χ ~> Υ■ Then the relations
{LuJn^ooFnj (z) = Limsupn_>00)In_>I Fn(xn)
and, setting Kn '■= Oom(Fn),
(Limbn_00.FnJ (z) D Limmfn-.oo,!,,-»^! Fn(xn)
B(Fn(xn),e))
hold true.
The missing equality holds true under more assumptions:
Theorem 7.1.5 Let X and Υ be Banach spaces. Consider a
sequence of closed set-valued maps Fn '■ X ~~>Y and its lower graphical
limit Lim n->oo-Fn· Let
?/o € (Limbn_>ooFn)(z0)

7.1. Graphical Limits
269
and 'assume that there exist constants с > 0 and η > 0 such that
f V(znjy„) € Graph(-Fn)nJ3((zo,2/o),?7),
[ ||£>-Fn(zn,2/n)ll := snpueXMveDFn{Xniy7i){u)\\v\\/\\u\\ < с
Then, for any sequence xn converging to xq, we have
?/o € Liminfn_>00Fn(a;n)
Consequently, whenever this assumption is satisfied for every г/о €
(Limbn_>00i;in)(:Eo)) then
(Limbn_>00i;in)(2;o) = Liminfn_»Oo,xn_»l0i<1n(3:n)
Proof — This is just Theorem 5.4.2 of Chapter 5 applied to
the set-valued maps F~l with a = 0. □
7.1.2 Graphical Convergence of Closed Convex
Processes
Let us consider a sequence of closed convex processes Fn from a Banach
space X to a Banach space Υ and their transposes F£ : Y* ^ X*.
Graphical'lower and upper convergences are exchanged by transposition:
Proposition 7.1.6 Let Χ, Υ be Banach spaces and Fn : Χ ~> Υ be closed
convex processes. We supply X* and Y* with the weak-* topologies. Then
the graphical lower limit Lim n-*ooFn of the sequence (Fn)n is the transpose
of the sequentially weak graphical upper limit of the transposes F£:
Limb n^ooFn = [cr-Lia^n^ooF^j
Proof— It follows from Proposition 1.1.8 applied to the closed convex
cones Kn := Graph(Fn):
The lower limit of the jFfn's is equal to the negative polar cone of the
sequentially weak upper limit of the cones K~. This can be translated by
saying that the graph of the graphical lower Umit is the negative polar of
the graph of the sequentially weak graphical upper Umit of the transposes
F*. Π

270
7 Graphical <fe Epigraphical Convergence
7.1.3 Monotone and Maximal Monotone Maps
Let X be a Hubert space, the dual of which is identified with X and F :
X ^л X be a monotone map.
If the graphical lower limit Lim n-*oaFn of a sequence of monotone
maps Fn is maximal monotone, then it is actually the graphical limit of
this sequence:
Proposition 7.1.7 (Convergence of Monotone Maps) Let X be a
Hubert space and let its dual X* be supplied with the weak-* topology. Consider
monotone set-valued maps Fn : X ~> X* and a maximal monotone map
F:X^X*.
If F is contained in the graphical lower limit Lim n-*ooFn °f the Fn 's,
then F is actually the graphical limit of the Fn 's.
Proof— We have to prove that the graphical upper limit Lim"n_>00.FTl
of the set-valued maps Fn is contained in F.
Let ρ belong to (Lim"n_>00FTl)(a;). Hence the pair (x, p) is a cluster
point of a sequence of elements (xn,Pn) of the graph of Fn.
Take now any pair (y, q) in the graph of F. Since
Graph(F) С Graph (Liml'n_00.Fn)
we know that there exists a sequence of elements (г/„, qn) of the graph of Fn
converging to (г/, q). The monotonicity of the set-valued maps Fn implies
the inequalities
<Рп-Цп , xn-yn>> ϋ
Going to the limit, we deduce that
V (г/, q) G Graph(F), <p-q, x-y>> 0
Therefore ρ belongs to F(x) because of the maximality of the graph of F
among monotone graphs. D
7.2 Convergence Theorems
We present in this section two versions of a Convergence Theorem
which plays a key role in the study of differential inclusions (see
Chapter 10 below.)
Let α(·) be a measurable strictly positive function from an open
subset Ω С Rn to R+. We denote by L1(fi; У, a) the space of classes
of measurable functions integrable for the measure a(u>)du>.

7.2. The Convergence Theorem
271
Theorem 7.2.1 (Convergence Theorem I) Let X be a
topological vector space, Υ be a finite dimensional vector space and Fn be
a sequence of nontrivial set-valued maps from X to Υ. We assume
that for any χ € Oom(F), there exists a neighborhood V of χ such
that \Jn>i Fn(V) is bounded.
Let us consider measurable Junctions xm and ym from Ω to X
and Υ respectively, satisfying:
for almost all ω € Ω and for all neighborhoodU ofO in the product
space X x Y, there exists Μ := Μ(ω,Κ) such that
Vm > M, {хт{ш),ут{ш)) € Graph(.Fm)+W (7.1)
If we assume that
i) xm{·) converges almost everywhere to a function x(-)
' H) Ут(·) € Ll(£l;Y, a) and converges weakly in L1(Q;Y, a)
to a function ?/(·) 6 L1(D,;Y,a)
then for almost all ω € Ω, у (ω) € coF^(x(uj)).
Proof — Let us recall that in a Banach space ^(Ω,',Υ,α) in
our case), the closure of a convex subset (for the normed topology)
coincides with its weak closure for the weakened topology3
σ (l\Q; Y, a), L°°(fi; Y\ a"1))
We apply this result: for every m, the function y(-) belongs to the
weak closure of the convex hull co({yp(-)}p>m.) It coincides with the
3By definition of the weakened topology σ(Χ, Χ*), the continuous linear
functional and the weakly continuous linear functional coincide. Therefore, the
closed half-spaces and weakly closed half-spaces are the same. The Hahn-Banach
Separation Theorem, which holds true in Hausdorff locally convex topological
vector spaces, states that closed convex subsets are the intersections of the closed
half-spaces containing them. Since the weakened topology is locally convex, we
then deduce that closed convex subsets and weakly closed convex subsets do
coincide. This result is known as Mazur's Theorem.
When X is not reflexive, closed convex subsets of the dual supplied with the
weak topology σ(Χ*, X) may be different from their closure for the weak topology
σ(Χ\Χ**).

272
7 Graphical <fe Epigraphical Convergence
(strong) closure of co({?/p(-)}p>m). Hence we can choose functions
CO
%(') := J2 атУр(·) € С0({?/р(-)}р>ш)
p=m
(where the coefficients a^ are positive or equal to 0 but for a finite
number of them, and where ]C^Un am = 1) which converge strongly
to ?/(·) in ^(Ω-,Υ,α). This implies that the sequence a(-)vm(·) con-
verges strongly to the function a(-)y(·) in L1 (Ω; Υ"), since the operator
of multiplication by α(·) is continuous from ^(Ω,-,Υ,α) to L1(fi;Y).
Thus, there exists another subsequence (again denoted by) wm(·)
such that4
for almost all ω € Ω, а(ш)ут(ш) converges to a(u))y(ω)
Since the function α(·) is strictly positive, we deduce that
for almost all ω € Ω, vm(u>) converges to у (ω)
Let ω £ Ω, such that а;т(ш) converges to χ(ω) in X and υτη(ω)
converges to у (ω) in Υ.
By assumption (7.1), we know that for any neighborhood
U := VxW
4Strong convergence of a sequence in Lebesgue spaces Lp (1 < ρ < +oo) implies
that some subsequence converges almost everywhere. Let us consider indeed a
sequence of functions /„ converging strongly to a function / in Lp. We can
associate with it a subsequence fnK satisfying
||/»fc -/||lp < Tk\ ■■■ < nk < nk+1 < ...
Therefore, the series of integrals
°° Γ
Σ НЛчИ-/И||5-с^ < +~
fc=i ^
is convergent. The Monotone Convergence Theorem implies that the series
oo
ΣΐΙ/»*(ω)-/(ω)||^
fc=i
converges almost everywhere. For every ω where this series converges, we infer
that the general term converges to 0.

7.2. The Convergence Theorem
273
of (0,0) eX xY, there exists N such that for all q> N,
хд{ш) € χ(ω)+ν/2 к {хд{ш),уд{ш)) е Graph(Fg) +U/2
Hence
Уд(ш) € Fq(x{u>) + V)+W
Consequently,
Vm > N, ут{ш) e col \J{Fq{x{a>) + V) + W)\
\q>n )
Then its limit у (ω) belongs to these subsets for all N and all
neighborhoods of the origin V, W in X and Υ respectively, so that
yMe Π co[{j{Fq{x{u) + V) + W)\
N>0, V, W \q>N )
By assumption, there exist neighborhoods of the origin V and W
such that the subsets Fq(x(u) + V) + W are contained in a bounded
subset. Hence
2/H€ Π cd[\J(Fq(x^) + V/N)+W/N)\
N>0 \q>N /
Since the dimension of Υ is finite, Lemma 1.1.9 implies that у (ω)
belongs to the closed convex hull of the upper limit of the subsets
Fq(x(u) + V/N)+W/N
i.e., the closed convex hull of F^(x(uj)). □
The "continuous version" of the convergence theorem can be
stated for any Banach space in the following way:
Theorem 7.2.2 (Convergence Theorem II) Let X be a
topological vector space, Υ a Banach space and F be a nontrivial set-valued
map from X to Υ. We assume that F is upper semicontinuous on
its domain. - -

274
7 Graphical <fe Epigraphical Convergence
Let us consider measurable functions xm(·) and ym{·) from Ω to
X and Υ respectively, satisfying:
for almost all ω € Ω and for all neighborhoods Ы of 0 in the
product space X x Y, there exists Μ := Μ{ω,14) such that
Vm > M, (im(w),ym(w)) € Graph(_F)+W
If we assume that
i) xm{·) converges almost everywhere to a function x(-)
< и) Ут{·) € L1 (Ω; У, α) and converges weakly in L1 (Ω; У, a)
to a function у € ^(Ω;!'', α)
for almost all u> € Ω such that χ[ω) 6 Вот(^), ?/(α;) € ~cb~F(x(uj).)
Proof — The proof is analogous to that of the previous
theorem with Fn := F. With the same arguments, we arrive at the
conclusion that for almost all ω Ε Ω such that χ (ω) € Dom(F), all
neighborhoods V and W of the origin in X and Υ respectively,
у (ω) € co(F(x(u) + V) + Щ
Since F is upper semicontinuous at χ(ω), for every e > 0, there exists
a neighborhood V such that F(x(w) + V) С F(x(u>)) + eJ5. Taking
W = eB, we obtain
у(ш) € co{F(x(u;)) + 2eB
Hence, e > 0 being arbitrary, у(ш) belongs to coF(x(uj)). D
7.3 Epilimits
7.3.1 Definitions and Elementary Properties
Let us consider a metric space X and a sequence of extended
functions
Vn: X н-> R U {±oo}

7.3. Epilimits
275
whose domains
Dom(Fn) := {χ € X | Vn(x) φ ±οο}
are not empty.
Taking into account the order relation of R, as we did in
Chapter 6, when we introduced epiderivatives, we associate with each
extended function Vn two new set-valued maps Vnj and Vnj. defined
in the following way:
0 VnT :
<
it) VnJ. :
We see at once that
Graph(VnT) = Ep{Vn) L· Graph(Vnx) = Hp{Vn)
Using the concept of graphical upper and lower limits with these
associated set-valued maps, we come up with the concepts of epi
and hypo convergence, which are thus obtained by taking upper and
lower limits of their epigraphs and hypographs.
Definition 7.3.1 (Epilimits) Let us consider a metric space X
and a sequence of extended functions Vn : X >—> R U {±00} whose
domains are not empty. We shall say that
1. — the function lim^n^coVn whose epigraph is the lower
limit of the epigraphs of the functions Vn
SpQimfn-^ooVn) := Ыт.т.{п->со£р(Уп)
is the upper epilimit of the functions Vn
2. — the function Ivcain^ooV-n, whose epigraph is the upper
limit of the epigraphs of the functions Vn
Vn(x) + R+ if χ € Dom(14)
0 if Vn(x) = +00
R if Vn{x) = —00
Vn{x) - R+ if χ € Dom(14)
< 0 if Vn(x) = -00
R if Vn(x) = +00
£y(lim"p_ool4) := Limsupn_>co£p(Kl)

276 7 Graphical 8ε Epigraphical Convergence -
is the lower epilimit of the functions Vn
3. — the function lim|n_»ooV^ whose hypograph is the lower
limit of the hypographs of the functions Vn
Vn) := Liminfn-.ooWp(Vn)
is the lower hypo-limit of the functions Vn
4- — the function limi n->coVn whose hypograph is the upper
limit of the hypographs of the functions Vn
Hpilvailn-^oVn) := Limsupnl>coWp(14)
is the upper hypo-limit of the functions Vn-
If the upper and lower epilimits coincide at some point xq, then
we shall say that the common value
(limTn_0oKl)(a;o) := (Ит^г-юоКг) (ю) = \Jun\n->aoVn) (z0)
is the epilimit of the sequence of functions Vn at xq, and we define
the hypolimit lim|n_»ooVn at xq in the same way.
The terminology concerning the epilimits seems at odds with the .
choice of the limits; the upper epilimit is associated with the lower
limit5. However, they are consistent in the case of hypo-limits. This
is due to the analytical definitions of these epilimits, involving the
concepts of Γ-convergence and Urn sup inf introduced in
Definition 5.2.4:
Ит sup^j. из£у>^уф{х', у') := sup inf sup inf ф{х',у') ■
ε>0 V>0 χ'£Β(χ,η) У £В{у,е)
The concept of Urn inf sup is defined in a symmetric way, arid the
adaptation to sequences (or filters) is straightforward: For instance
lim sup^^ Ыу>^уфп{у') := sup inf sup inf фп(у')
ε>0 W>° n>N J/'eB(i/,e)
5This would never have happened had "Convex Analysis"; been "Concave
Analysis", and then, had maximization problems taken precedence over
minimization ones. Which is not the case, despite economists who prefer to maximize
profits, utilities and all these nice things.

7.3. EpUimits
277
Proposition 7.3.2 Let us consider a metric space X and a sequence
of extended functions Vn : X i-> R U {±00} whose domains are not
empty. We obtain the following formulas:
i) (lim^n^ooVn) (z0) = limsup^^ infz_I0 Vn{x)
ii) (lim^ool^J (x0) = liminfn-oo.i-ioKiiz)
Hi) (iwaln^ooVnj {xq) = liminfn-00 snpx_X0Vn(x)
^ iv) (limJn-ooVk) (zo) = limsupn_>OOi3._>3.0Fn(z)
Proof — We shall check these formulas for epigraphical
convergence only.
We set V^ := l\m^n-^c>oVn and show first that
4'
limsup inf Vn(x) < Vt(zo)
It is enough to consider the case when V^{xq) < +00. To compute
the value of V» at xq, we use the fact that for every λ > V}(xq),
there exist sequences of elements xn converging to xq and λπ to λ
such that λη > Yn{xn)·
Therefore, for all e > 0 and 77 > 0, there exists N such that, for
all η > iV, we have
and thus
inf Vn{x) < Vn{xn) < λη < λ + ε
\\Χ—Χ0\\<η
sup inf Vn(x) < A + e
n>N \\x-xo\\<v
from which we deduce that
I
I
I
VA > W(zd), limsup inf Vn{x) < A
Ί η—>oo x *XQ
and thus, .that
1: :_£ τ/ с„\ ^ τ,
limsup inf Vn{x) < VUxq)
n—юо x—+x0

278
7 Graphical <fe Epigraphical Convergence
2. — Conversely, to prove the other inequality, we have to
show that for any
λο > limsup inf Vn(x) φ +οο
η—>oo x ^o
the pair (xq, λο) belongs to the epigraph of VJ., i.e., to the lower limits
of the epigraphs of the functions Vn. But, by the very definition of
the infimum, we deduce that for all e > 0 and η > 0, there exist
N such that, for all η > Ν, we can find an element xn € Β(χο,η)
satisfying
Vn{xn) < inf Vn(x)+e < sup inf Vn(x)+e < λ0 + ε
||i-io||<7? n>N ||a:—aioll < V
By taking ε = η = 1/n and setting λπ := Xq + 1/п, we have proved
that xn converges to xq, λπ to λο and that Vn{xn) < \n for all n.
3. — We set vl· := limL^ool^ and prove next that
liminf VJx) < Vf(xQ)
n—>oo,i—>io '
It is obvious when vHxq) = +oo. Otherwise, let us estimate any
λ > Vf(x0). We know that for any ε > 0, η > 0 and N > 0, there
exist n' > N, (xn/, \n/) in the epigraph of Vn' satisfying
ll^n'-^oll < V & Vn'{xn') ^ λη/ < λ + ε
We deduce that
inf Vn{x) < λ + ε
п>АГ,||х-хо||<»?
Hence
liminf VJx) < λ
π—>οο,ι->xq
By letting λ —> νί(ΐο) we obtain
liminf VJx) < v!(xq)

7.3. EpUimits
279
4. — To show the opposite inequality
Vf(xo) < liminf Vn(x)
we begin by noting that it is always true when
liminf Vn(x) = +oo
n—>oo,x—>xo
We conclude by observing that the pair (xq, X\) where
λι > liminf Vn(x)
n—+ca,x^xo
belongs to the epigraph of Vj, because, by the very definition of the
Km inf, we can construct a subsequence of elements (again denoted)
(xn, \n) of the epigraphs of Vn converging to (xq, λι). □
It may be useful to store the following inequalities
i\n->ooVn < lm^n^oaVn < limj
i) limL_ooKi < limL^ool^ < limL^ooV^
(7.2)
ii) lim^n_ooKi < linixn-.ooVn < ^a\n-^ooVn
Remark — We have defined the concepts of epilimits from the limits
of the epigraphs. Conversely, we can recover the limits of subsets from the
epilimits of their indicators:
Proposition 7.3.3 Let us consider a metric space X and a sequence of
subsets Kn С X and their indicators ψκη· Let K^ and K^ denote the
upper and lower limits of the Kn's. Then
О Фк> = ^т^п-*ооФкп is the upper epilimit ofij)Kn
ii) φΚί = Уш\п-*°оФк„ is the lower epilimit of-ψκη
The proof is left as ал exercise. □
Prom the Inverse Stability Theorem 7.1.5, we can derive criteria which
imply some equalities in formulas (7.2.)

280 7 Graphical 8ε Epigrapbical Convergence
Proposition 7.3.4 Let us consider α metric space X and a sequence of
extended lower semicontinuous functions Vn: X *-^> RU{+oo} whose domains
are not empty. - - -'
Let xq € X and assume that there exist constants с > 0, N > 0 and
η > 0 such that, for alln> Ν, χ £ Β(χο,η), the domains of the contingent
epiderivatives D^Vn(x) are equal to the whole space and
sup |2?TV„(:c)(u)| < с (7.3)
Hl=i
Then - - .
n—ί-ехэ
or, equivalently,
limsup Vn(x) = limsup inf Vn(x)
n—+00,χ—*xq n—+oo x~*xo
Proof — Set V{ := lim^-ooVn. We apply Theorem 7.1.5 to the
set-valued maps Fn defined by
Fn{x) := VnT(a;)
It is easy to check that assumption (7.3) implies the stability assumption
of Theorem 7.1.5. This is straightforward when A„ = V(xn), since in this
case
))DFn(x,Xn))) := sup inf |μ| = sup |£>TV„(a;)(u)| < с
||u||=l μΕθΡη(χ,Χη)(η) N1=1
When A„ > V(xn), Proposition 6.1.4 implies that
Dom(DTVn(a;)) χ R С Graph(DFn(a;, λη))
so that
\\DFn(x,Xn))):= sup inf )μ\ = 0
||u||=l ^еВ^(1,Ап)(и)
Hence, there exists a constant с such that, for all n > Ν, χ £ Dom(Vn)
close to xq and A„ > Vn(x) close to V^(xq), we have
||№(a;,An)|| < с
Now, Theorem 7.1.5 implies that V^(xo), which by definition belongs to
the graphical lower limit of the Fn's, also belongs to the lower limit of the
V„T(a;) when п-юо and χ —* xq- Therefore:
limsup Vn(x) < V^(xQ)
n—юо,а:—Ho
The left hand side of this inequality is equal to V^(xo)· This, together with
inequalities (7.2), implies that V^(xq) = V^(xq). □

7.3. Epilimits
281
7.3.2 Convergence of Infima and Minimizers
We expect that the infimum of an epilimit is closely related to the
limit of the infima.
To prove this fact, we recall that an extended function V is said
to be inf-compact or lower semicompact if for any λ € R, the level
set
{x € X | V(x) < X} '
is relatively compact. A sequence of functions Vn is called
sequentially inf-compact if for any λ € R, every subsequence of elements
{xn> € X | Vn,{xn>) < X}
remains in a relatively compact set.
Proposition 7.3.5 (Limits of Infima) Let us consider a metric
space X and a sequence of extended functions Vn : Χ ι—» R U {+00}
whose domains are not empty. Then
limsup(inf Vn) < inf (Нт^-юсТ^.
If we assume that the functions Vn are sequentially inf-compact, then
inf (limL-^ooFn) < lim inf (inf Vn)
Consequently, if V is the epilimit of a sequence of sequentially inf-
compact functions Vn, then
i) inf У = lim^ootinf Vn)
ii) Limsup^oo-f?/ | Vn{y) = inf 14} С {z0 | V{x0) = inf V}
Proof
1. — Set
УТЬ := lun^n^ooVn
The first inequality holds true whenever V* = +00. Otherwise, let
λ > inf V}_h§ fixed and xq be chosen such that ^(zo) < λ. We know
that for all η > 0, there exists N > 0 such that
sup inf Vn(x) < X
n>NxeB{x0,v)

282
7 Graphical 8ε Epigraphical Convergence
Therefore,
sup (infl^i) < sup inf Vn(x) < λ
n>N n>N χ£Β{χ0,η)
Hence it is enough to let N go to oo and λ to V^(xq).
2. — Set
Taking a subsequence and keeping the same notations, we may
assume that
liminfiinf Vn) = lim (inf Vn)
n—>oo n—>oo
Consider first the case when
liminfiinfVn) > —oo
71—>CO
Let us choose for any η an element xn such that
Vn(xn) < inf!4 + l/n
They remain in a relatively compact subset since the functions Vn
are sequentially inf-compact. So a subsequence of xn does converge
to some xq. Therefore,
infV^ < vf(xo) = liminfn_»oo,i-.ioKi(a;)
<
< liminfTl_>ooHl(2;Tl) = liminfn^ooCmf Vn)
When
liminfiinf Vn) = —oo
n—>oo
we can choose xn € X in such a way that Vn(xn) —> —oo. Sequential
inf-compactness thus implies that a subsequence of xn converges to
some xq. Then, for all μ > 0, the pair (ζθ) —μ) belongs to the upper
limit of the epigraphs of the Ws, i.e.,
( lim^n_>00l/n)(a;o) = -oo
3. — Equality г) is a consequence of the first two inequalities.
Observe that the last statement holds true if the left-hand side of
inclusion ii) is empty. Otherwise, consider an element
xq € Limsupn_too{y | Vn{y) = infVn}

7.3. Epilimits
283
which is the limit of a subsequence of elements xnt such that Vni {xn') =
inf Vni. Then i) implies that
У\-(xo) < liminf Vni{xni) = inf У
so that νγ(ΐο) < inf У. Since V(xq) = Vf(xo) by assumption, we
have proved that xq minimizes V. Π ·
Unfortunately, there are counter-examples for the property that
the set of minimizers of the upper epilimit is the lower limit of the sets
of minimizers of the functions Vn. However, Theorem 7.1.5 provides
some results about the lower limits of level sets:
Proposition 7.3.6 Let us consider a metric space X and a sequence
of extended lower semicontinuous functions Vn : X н-> R U {±00}
whose domains are not empty.
Let xq belong to the domain of their upper epilimit and assume
that there exist N > 0 and constants с > 0, η > 0 such that, for all
n> Ν, χ € B(xq, 77) Π Dom(Fn),
(Ϊ) 3 u~ € cBx such that D^Vn{x){u^) = -1
(7.4)
ii) 3 и+ € cBx such that DfVn(x)(u+) = +1
Then there exists a constant I such that, for any sequence of elements
XQn converging to xq such thatVn(xQn) converges to (limi-n-^ooV^J {xq),
and for any \n converging to V^(xq), there exist xn € X satisfying
Vn(xn) < \n, Ha-n-zonll < l\\n-Vn{xQn)\
Proof — Set
Vj := Ип^п-юоКг
We apply Theorem 5.4.2 to the set-valued maps Fn(x) := Vn-\{x),
which are obviously consistent at (xq,V^(xq)) by definition of*Vp
SetA0 = FTb(z0).
Assumption (7.4) implies that the set-valued maps Fn are stable:
There exist constants η > 0 and с > 0 such that, for any
(ιη,μη) € GTaph(Fn)nB((x0,\o),v)

284
7 Graphical L· Epigrapbical Convergence
for any υ € R, there exists un € X such that
ϋ 6 DFn(xn, μη)(ηη) L· \\un\\ < c\v\
Indeed, if μη > Vn(xn), we can take un = 0. If μη = Vn(xn), we
take ν = vu% if ν > 0 and υ = —vu~ if ν < 0.
Therefore, Theorem 5.4.2 implies that for any λπ approximating
λο, we can find xn € F^r{\n) satisfying the required error estimates.
D
Remark — Observe that stability assumptions (7.4) imply
that for all η > N and all xn € Β(χο,η),
Xn)
ι ел
since the Fermat rule is violated. This excludes the case when the
level sets are set of minimizers. □
7.3.3 Variational Systems
Naturally, we can introduce the same definitions for "continuous"
parameters и GU, where U is another metric space.
In such a framework, we consider a family of extended functions
Vii £ U, V(u,-):X ^ Ru{±oo}
depending upon the parameter u, with which we associate the set-valued
maps
V(u,x)+n+ if (u,x) £ Dom(y)
i) V(it)T(a;) := { 0 if V{u,x) = +oo
R if V(u,x) = —oo
V(ii,a;)-R+ if (u,x) £ Dom(y)
гг) V(u)j.(a;) := { 0 if V(u,x) = -oo
R if V(u,x) = +oo
There is a subtle, but important, difference between the extended
function
V:(u,x) £ U χ X н-> RU{±oo}
for which the variables и and χ are on the same footing, and the set-valued
maps
V(-)T ·· и £ U — V(u)T

7.3. EpUimits
285
for which the variables play a different role: u, the role of a parameter and
x the role of the variable.
This type of situation occurs for instance in optimization where the
order relation on R involves the variable x, when we minimize the function
depending on the parameter и with respect to x.
To emphasize this difference, we borrow from Rockafellar and Wets
the following terminology: a set-valued map V(-)1- is called a variational
system.
Definition 7.3.7 (Variational system) Let us consider metric spaces X,
U and a variational systemV(-)^ defined on UxX. We shall say that V(·, ·)
(or V(-)f, to be precise), is
1. — upper epicontinuous at и if the set-valued map u' ~> V(u')j is
lower semicontinuous at и
2. — lower epicontinuous at и i/V(u)1- = limsup „/^„V^ii') τ
3. — epicontinuous at и if it is both lower and upper epicontinuous
at u.
The definition of upper and lower hypocontinuity are naturally
symmetric.
As in the discrete case, these concepts can be characterized in an
analytic way:
Proposition 7.3.8 Let us consider metric spaces X and U and a
variational system V(-)T defined on U x X. Then V is
• 1. — upper epicontinuous at и if and only if for any χ £ Oom.(V (и, ·))
V(и,χ) := limsup inf V(u',x')
2. — lower epicontinuous at и if and only if for any χ £ Ooxn{V(u, ·))
V(u,x) = lira inf V(u',x')
uf —*u,xf —*x
3. — epicontinuous at и if and only if for any χ £ Dom(y(u, ·))
V(u,x) = limsup inf V(u',x') = liminf V(u,x)
I .TV b.T ΊΙ· 17) T·' IT
11' —HL.X' —*X
Example — Definition 6;1.2-©f epiderivatives provides another
interpretation of the epiderivatives in terms of epilimits of the
difference quotients
V(x + hu)-V(x)
VhV(x) := и ^
h

286
7 Graphical <fe Epigraphies! Convergence
1. — the contingent epiderivative is the lower epilimit of the
difference quotients VhV(x) when h —> 0+
2. — the adjacent epiderivative is the upper epilimit of the"
difference quotients VhV(x) when h —> 0+
3. — the circatangent epiderivative is the upper epilimit of
the difference quotients
и ^ (V(x' + hu) - \')/h
when h —> 0+ and (x\ λ') G £p(V) converges to (x, V(x)). □
7.4 Epilimits of Sums and Composition
Products
This section is devoted to the upper and lower epilimits of the functions
Un := Vn + Wn о А
in terms of the upper and lower epilimits of the functions Vn and Wn.
Theorem 7.4.1 Let us consider Banach spaces X, Y, a continuous linear
operator A £ £(X, Y), two sequences of extended functions Vn and Wn
from X and Υ to R U {+00} respectively and define
Un := Vn + WnoA
1. — Then for every Xq £ X, the inequality
(limeTn_>00Vn)(a;o) + (limeTn_i00Wrn)(Aa;o) < (limeTn_>00[/n)(a;o) (7.5)
holds true whenever the left hand side is well defined.
2. — If in addition Vn and Wn are lower semicontinuous and the
following stability assumption holds true:
there exist constants с > 0, a £ [0,1[ and η > 0 such that, for all n,
i) \/x £ Όοτη(νη)Γ\Β{χ0,η), V у £ Dom(Wn)n Β(Αχ0,η)
BY с A(Oom(D\Vn{x))ncBx\ -Oom{D^Wn{y)) + aBY
<
") snPueOom(D)vn(x))\Di\Vn(x)(u)\/M ^ c
Hi) snpveOom{DiWn{y)) \DiWn(y){v)\/\\v\\ < с
(7.6)

7.4. Epilimits of Sums and Composition Product 287
then the upper epHim.it \т\^п^,хип satisfies:
n—+00 ип)Ы < (1Ц
τι—+oo
whenever the right hand side is well defined.
Consequently, if the sequences of functions Vn and Wn have epilimits
VT and WT at xQ and AxQ respectively and if V^(xQ) + W^Axq) is well
defined, then the functions Un have an epilimit U-\ at Xq and
Щ{х0) = V1(x0) + W1(Ax0)
Proof— For simplicity, we adopt the notations
Vf := lim^^Vn & V? := lim^^^Vn
for the functions Vn as well as for the functions Un and Wn-
Since inequality (7.5) holds true when one of the values Vf(x0) and
W^(Ax0) is equal to -00, or Ι/Ηχο) is equal to +00, we have to check the
formula when the two first values are larger than —00 and the last one
is smaller than +00. Then, by definition of the lim inf, there is a finite
number p, an integer N and a positive number η such that
Vn > N, \/x £ Β(χ0,η), Vn{x) > p, Wn(Ax) > ρ
By definition of the lower epilimit, the pair (хо,Щ(хо)) is the limit of a
subsequence (again denoted by) (arn,cn) satisfying
Vn{xn) + Wn{Axn) < cn
The two above inequalities imply that the sequences of real numbers an :=
Vn(xn) and bn := cn — an are bounded. Hence, subsequences (again
denoted) an and bn do converge to a and b satisfying
Uf(x0) = a + b, a > Vf(x0), b > W\{AxQ)
To prove the second statement, we begin by observing that if we set
' i) Kn := Sp(Vn) χ Sp{Wn) xR с XxRxFxRxR
< ii) G(x,a,y,b,c) := (Ax — y, a + b — c)
Hi) H(x,a,y,b,c) := (x,c)
then we can write
Sp(Un) = ff^nG-^O.O))

288 7 Graphical L·Epigrapb'ical Convergence
Therefore,„it is-sufficient to show that the lower limit of the subsets
Kn Π G-1(0,0)_ contains the intersection of the lower limit of the subsets
Kn with G~i(0,0). For that purpose, we shall use Theorem 3.4.5.
Let us consider any sequence of elements
(χοη,αοη) e £p(Vn), (y0n,bQn) £ Sp(Wn)
converging respectively to (xq, VHxq)) and (Axq, W*(Axq)) and let us set
con := aon + bon.
We observe that the elements (xon,aon)yon,bon)con) belong to Kn and
that
С(жо„,аоп,2/Оп)Ьоп,СОп) = {Αχαη — 2/Οτι,Ο) "
converges to (0,0).
We begin by checking that the assumptions (7.6) of our Theorem imply
the stability assumption (3.9) of Theorem 3.4.5, i.e., that there exists a
constant с > 0 such that, for all n,
V (x,a,y,b,c) ε Kn close to (xQ, V^(x0),yo,W\(y0),v)
for all (ζ, Α) ε X x R, there exist (u, μ, ν, у, δ) ε Τκ„ {х, а, У, Ь, с) and e ε Χ
satisfying
г) ζ = Аи — ν + е & Χ = μ + u — δ
, ii) ||e|| < α(|Μ| + |λ|)
^ ш) |Н| + |И| + и + и + |5|<с(|И| + |А|)
Assumptions (7.6) г) and гг) imply right away that there exist
u ε Oora{D\Vn{x)), ν ε Dom(DTWn(y)) & e ε Χ
satisfying
ζ = Au-v + e, ||e|| < a(\\z\\ + |λ|), ||U|| < φ||
Then
Ml < 1И1 + 1|е|| + ИИН| < 7(ΙΝΙ + λ)
for some positive 7 independent of (z, A).
Let us take now
μ := c||u||, и := φ\\, δ := c(||u|| + ||v||) - A
We deduce from (7.6) иг) that
(η,μ) ε Ep{DiVn){x), {υ,ν) ε Sp{D}Wn){y)

7.5. Conjugate Functions of Epilimits 289
and that
DiVn{x){u) + DiWn{y){v) < c(|H| + ||w||) = λ + δ
Consequently
\δ) < |A| + C(|H| + ||«||) < c'(\\z)\ + \X\)
The conclusion of Theorem 3.4.5 being available, we then know that there
exist elements (£П)аП)2/П)ЬП)Сп) 6 Кn satisfying ' *
G(xn,an,yn,bn,cn) = 0
and
||in-a;on|| + ||yn-2/On|| + |an-aon| + |bn-bon| + |c'n-con| <Ί))Λψθη-νοη)\
Hence lirrin-*^ \an — aon\ = lim,^,» \bn — bon\ = 0, so that wn and bn
converge to, Xf(xa) and ty*(xo) respectively. Furthermore, inequalities
Ul{xn) < Si +U imply tllat Ы(хо)'< Vhx'a) + W\Aa). Q
• 1 \i '· ■ ■ \ ι ' ' « ι : ,
7.5 Conjugate Functions of Epilimits
• ϊ i ι \ 1 i Ί
Let us consider a Banach space X, a sequence of nontrivial convex
I
lower semicontinuous functions
Vn:X i-> R U {+oo}
and their conjugate functions
vf.X* i-> Ru{+oo}
III * ' \
• '. ' ί . :
The purpose of this section is to characterize the upperj epilimit
of the! functions Vn. !
We can still define conjugate of an extended function
U:X *-+ RU{±oo}
by the formula
U*(p) = s\ip(<p,x> -U(x))
x'ex

290
7 Graphical <fe Epigrapbical Convergence
We observe that U* = +00 when U(x) = —00 for some χ G X and
that if U = +oo, its conjugate U* = —00.
When X* is supplied with the weak-* topology, we naturally
adapt the definition of lower epilimit of a sequence of extended
functions Wn : X* i-> R U {±oo} to the concept of sequentially weak
lower epilimit: The epigraph of σ — limS.n_,ooWn is .by definition the
sequentially weak upper limit of the epigraphs of the functions Wn.
Theorem 7.5.1 (Mosco) Consider a Banach space X and a
sequence of nontrivial convex lower semicontinuous functions
Vn: X н-> R U {+oo}
Assume that the upper epilimit of the functions Vn is not
trivial. Then it is equal to the conjugate of the sequentially weak lower
epilimit of the sequence of conjugate functions V£ :
lim^n_ooVn = (σ - ΙκΏ^η-χχ,ν*)
Proof — We set
Vf := lim'fn-.oo^n, W := a-lmi^^V*
Then for any ρ G X*, there exists a subsequence of elements (ρηι, ληι)
€ Sp{y*i) converging weakly to (p, W(p)).
We observe first that inequality
W*[x) < Vf(x)
always holds true: Indeed, it holds true when V^(x) = +00 or
W = +00, because, in this case, W* = —00. Assume now that
Vt (ж) < +oo and pick ρ such that W(p) < +00. Then there exist
sequences of elements xn and μη > Vn(xn) converging (strongly) to
χ and V}(x) respectively and subsequences pn> and λπ/ > V*i(pn')
converging weakly to ρ and W(p) respectively Therefore, by taking
the limit when n' —> 00, Fenchel inequalities
< Ρη',Χη' > < Vni{xni) + V*,(pnt) < μη' + Xn'
imply
<p,x> -W(p) < Vf(x)

7-5. Conjugate Functions of Epilimits
291
from which our claim follows.
If for some q € X*, W{q) = -oo, then for all χ € X, W*(x) =
+00. Consequently, in this case, inequality
Vf(x) < W*(x) (7.7)
is satisfied. It still holds true when V*(x) = —00.
It remains to prove that it is verified when W never takes the
value —00 and V^(x) > —00. To this end, we shall check that λ <
W*(x) for all λ < V^(x). Fix such (ж,λ). The latter statement
means that the pair (x,\) does not belong to the epigraph of V?,
which is, by definition, the lower limit of the epigraphs of functions
Vn. Therefore, there exist e > 0 and a subsequence of functions
(again denoted by) Vn such that
((x + eB)x(X + e[-ll+l]))neP(Vn) = 0
We infer from the Hahn-Banach Separation Theorem that there exist
elements (pn, —an) GPxR satisfying ||pn|| + \an\ = 1 and
V (χη,μη) € £p(Vn), (ρη,Χη) - θίημη < (pn,x) - 0ίη\ - ε (7.8)
This inequality implies that an > 0 because, if an < 0, it is enough
to take xn G Dom(14) and let
μ\ := Vn(xn) + к —> +oo with к
to derive the contradiction 00 < est.
We proceed now in three steps.
1. — Assume that lim infn-»00 απ is strictly positive. Then
there exist δ > 0 and a subsequence (again denoted by) an satisfying
an > δ. Therefore, we can divide inequality (7.8) by an > 0. Setting
qn ;= pn/an and taking the supremum over the xn G Oom.(Vn), we
deduce that,
V*(qn) < < <ln,x > -λ - ε/αη < < qn,x > -λ
On the other hand, ||^π|| < 1/5, so that there exists a subsequence
(again denoted) qn converging weakly to some cluster point q. Since
< qn, χ > -X converges to < q, χ > -λ, we infer that
W(q) < (q,x)-X

292
7 Graphical L· Epigraphical Convergence
and thus, that λ < W*(x).
2. — This happens at least when χ belongs to the domain
of Vp because, in this case, we claim that αη > ε/4ρ for η large
enough (where ρ := V^(x) — λ > 0.) Indeed, there exist sequences of
elements x$n and μη > V(xon) converging strongly to χ and V^(x)
respectively. Therefore, for η large enough, \\x — --xbnW < ε/2 and
μη < V^(x) + p. By taking such x0n and μη in inequality (7.8), we
obtain
0 < (pn, x ~ xon) + θίη(μη - λ) - ε
< ||ж - жоп|| + 071(^(0;) - λ + ρ) -ε < 2αηρ-ε/2
Therefore, inequality (7.7) holds true whenever V+(x) < +00.
Furthermore, we deduce from the first step that the domain of W is not
empty.
3. — Consider now the case when V+(x) = +00.
Either liminfπ_4θο απ > 0 and, by the first step, λ < W*(x),
or liminfn-xxian = 0, and a subsequence (again denoted by) an
converges to 0.
In this case, we take an element q in the domain of W, which is
not empty by the second step. Then there exist subsequences (again
denoted) qn and pn > V*(qn) which converge weakly to some q and
W(q) respectively. Next, we add FenchePs inequality
(ίη,Χη) < Vn(xn) + V*(qn)
to inequalities (7.8) multiplied by μ > 0 and we obtain
Vin€ Dom(14), <qn + μρ
< (1 + μαη)νη(χη) + Pn+ < μΡη, x > -μαπλ - με
Dividing by 1 + μαη and taking the supremum over the domain of
Vn, yields
y* f<ln + μΡη\ < pn+ < μρπ, x > -μαηλ - με
η\1 + μαη ) ~ 1 + μαη

7.5. Conjugate Functions of EpUimits
293
On the other hand, knowing that ||pn|| < 1, there exists still a
subsequence (again denoted by) pn converging weakly to some cluster
point p. The right hand side of the above inequality converges to
W(q)+ < μρ,χ > —με. Hence, by going to the limit, we infer that
ΪΥ(ς + μρ) < W(q) + < μρ,χ > -με
Therefore, taking into account that
<ς + μρ,χ> -Wiq + μρ) < W*(x)
we obtain
με - W(q) + < q,x > < W*(x)
Since e > 0, we deduce that W*(x) > λ by taking μ large enough.
Therefore, in both cases, λ < W*(x), so that, by letting λ —> +oo,
we infer that W*(x) = +oo. □
By taking the conjugates, we obtain the formula
(lim^ooVy* = (σ - lim^-^ooV^)
Applying it to the indicators Vn := φκη of closed convex subsets
Kn, the conjugate of which are the support functions of these subsets,
we obtain a formula for the support function of the lower limit of a
sequence of subsets Kn:
9
Corollary 7.5.2..Let X be α Banach space and Kn С X be a
sequence of nonempty closed convex subsets. Then the support
function of their lower limit К is the biconjugate of the sequentially weak
lower limit of the support functions σκη of the subsets Kn:
σ{Κ\·) = (a-lim»„.w7tfn (·))**
When the space X is reflexive, we can apply Theorem 7.5.1 to
the conjugate functions V£ to obtain the following consequence:
Theorem 7.5.3 Consider a reflexive Banach space X and a
sequence of nontrivial convex lower semicontinuous functions Vn '■ X >-^
Ru{+oo}.

294
7 Graphical L· Epigraphical Convergence
Assume that the upper epilimit of the conjugates V* is not trivial.
Then the conjugate of the sequentially weak lower epilimit of the
sequence of functions Vn is equal to the upper epilimit of the conjugates
y*.
(σ-ίπηξη-^ο^η)* = limfn-ooV^
Consequently, we deduce the following:
Corollary 7.5.4 Consider a reflexive Banach space X and a
sequence of nontrivial lower semicontinuous convex functions
Vn:X н-> R U {+oo}
Let us assume that its upper epilimit and the sequentially weak lower
epilimit do coincide with a nontrivial function V.
Then for any χ € Dom(V) and ρ G Dom(V*), there exist
sequences xn G X and pn £ X* converging to χ and ρ respectively and
satisfying
limsupVn(:cn) < V(x), \imsnpV*(pn) < V*(p)
n—>oo n—>oo
Applying Theorem 7.5.3 to the indicators Vn := φκη of subsets
Kn, we obtain a formula for the support function of the sequentially
weak upper limit of subsets:
Corollary 7.5.5 Let Kn be a sequence of nonempty closed convex
subsets of a reflexive Banach space X. Then the support function
of the sequentially weak upper limit of the subsets Kn is equal to the
upper epilimit of the support functions of the subsets Kn.
7.6 Graphical Convergence of Gradients
We study in this section the convergence properties of gradients and
generalized gradients of a sequence of functions Vn.

7.6. Graphical Convergence of Gradients
295
7.6.1 Convergence of Gradients of Smooth Functions
We begin with
Theorem 7.6.1 Let Ω be an open subset of a Banach space X.
Consider a sequence of Frechet differentiable functions Vn : Ω ι—» R
converging to a function V uniformly on bounded subsets σ/Ω. Assume
also that V is bounded from below on bounded subsets ο/Ω.
Then the set-valued map д-V : Ω ~> X* is contained in the
graphical lower limit of the gradients V^ of the functions Vn:
for any χ G Ω and any ρ G d-V(x), there exists a sequence xn
converging to χ such that V^(xn) converges to p.
If V is bounded from above on bounded subsets of Ω, the set-
valued map d+V is also contained in the graphical lower limit of the
gradients V^ of the functions Vn.
The same holds true if the functions Vn are locally Lipschitz and
Gateaux differentiable on Ω.
Proof— Let us fix e > 0 and χ G Ω. To say that ρ G d_V(x)
amounts to saying that there exists μ G]0, e[ such that
Vy € Β(χ,μ), V(x) - V(y)- <p,x-y> < ~\\x - y\\
We may assume that Β(χ,μ) С Ω. Then, the functions Vn
converge uniformly to V on the ball Β(χ,μ): For any δ > 0, there exists
Ns such that, for all η > N$ and for all у G Β(χ,μ), Vn(y) — V(y) <
δ/2. We take δ = με/A and fix η > Νβ.
We now apply Ekeland's Theorem to the continuous function
У ·-» Vn{y) - <p,y >
on the ball Β(χ,μ): There exists xn G Β(χ,μ) satisfying
i) Vn(xn)- < P> χη > +ε\\χ -χη\\ < Vn(x)- <p,x>
ii) Vy G Β(χ,μ),
Vn(xn)- <p,xn> < Vn(y)- <p,y> +e\\xn-y\\
We infer from the first inequality that
ε\\χ — xn\\ < δ + V(x) — V(xn)— < ρ,χ — xn > < δ + ~\\x — xn\\

296 7 Graphical L· Epigraphical Convergence
ι
from which we deduce that
' ; . · i
\\x< — Xn\\ < 2δ/ε = μ/2
) j Sipc.e \cjl belongs to the interior of B(x, μ) and minimizes on this
bail the function ; ;
ϊ '■ '; ; ί
Vn(y)~ < P, У> +ε\\χη - y\\
the Fermat Rule implies that
V и € X, 0 < (Vn(xn)-p,u) + e\\u\\ ■
: I \
which means that ||ν^(χπ) —ρ\\ < ε. Hence the pair (x,-p) is the limit
of a sequence of elements (xn, V^(xn)) of the graph of V^. Π
This statement is actually a consequence of a more precise
Theorem 7.6.2 Let us consider α Banach space X and a sequence
of extended lower semicontinuous functions Vn : X >—» R U {+00}
whose domains contain a ball B(x, a). Assume that the functions Vn
converge uniformly to an extended function V on this ball. Assume
also that V is bounded from below on B(x,a).
Choose any kind of δ-generalized gradients dsVn. Then for any
ρ € d-V{x), there exists a^sequence xn converging to χ and a
sequence of elements pn € dsVn(xn) converging to p.
; Proof— j We begin the proof as in Theorem 7.6.1, but this time
the Fermat Rule implies that
i /'·,..'
i€ Χ,.Οζ ΡΊνη(χη)(ΐί)- ,< p, и > +e\\u\\
Consequently/, the function] 14' is contingently' epidiffereniiabk a't«:,(.
rjLLBtnermbre, w;e deduce that for any convex epidenyatiye OsVh{xA) ■
[and 'for дШ ■ ' , >
j , : \ j - . i
Vu € X, 0 < δνη(χη)(ιή-<p,u>+e\\u\\
■it- i i
This can be written as
i
0 € dsVn(xn) -ρ + εΒ*
where. S* denotes the unit ball of -X*. This means that there exists
p\i^-BbYii{xn).such, that \\p —. pn\\ < e. Hence the pair (x,p) is the
limit of a sequence of elements (xn,Pn) of the graph of dsVn. □

7.6. Graphical Convergence of Gradients
297
7.6.2 Convergence of Subdifferentials of Convex
Functions
Theorem 7.6.3 Consider α reflexive Banach space X and a
sequence of nontrivial lower semicontinuous convex functions Vn: X *-^>
RU{+oo}. Let us assume that its upper epilimit and the sequentially
weak lower epilimit coincide with a nontrivial function V.
Then the subdifferential map dV is contained in the graphical
lower limit of the subdifferential maps dVn:
3V С Lim^oo^
More precisely, let ρ G dV(x). Consider sequences XQn and p$n
converging to χ and ρ respectively satisfying
limsupFn(a;on) < V(x), limsupV^POn) < V*(p)
П—+СО 71—>CO
(which exist by Corollary 7.5-4-) Then the error estimate
d((xon,Pon), Graph(<9Vn)) < yVn(x0n) + V*{p0n)- < p0n, x0n >
(7.9)
holds true.
, Proof — We introduce the "duality lack"
δη ■= Vn(x0n) + V*(pQn)- < P0n,XQn > > 0
which converges to 0, since by assumption
0 < limsupn_4005n
< < lim sup^oo Vn(xon) + limsup^,*, V*(p0n)- <p,x>
< V(x)+V*(p)- <p,x> = 0
If δη = 0, this means that p$n belongs to dVn{xQn). If not, we apply
Ekeland's Theorem to the lower semicontinuous function -
У ·-> Vn(y)- <p0n,y >

298 7 Graphical &ε Epigraphical Convergence
Since limsupj^oo V*(j>on) < +00, there exist constants с and N such
that
" Vn > TV, VJipon) < с
This implies that the functions у ι—> Vn(y) — < pon, у > are bounded
from below by — с
Hence, we can apply Ekeland's Variational Principle in the
generalized gradient form (Theorem 6.4.4): there exists a solution xn
satisfying
I
г) Vn(xn)- < ροη,Χη > +V%n\\
< Vn(xQn)- < pon, x0n >
(7.10)
^ it) 0 € dVn{xn) - p0n + л/δ^Β*
Inequality (7.10) гг) tells us that there exists pn in dVn(xn) satisfying
Upon ~Pn\\ < \β~η
On the other hand, inequality (7.10) г) yields
ll^On - xn\\ < -7= (Vn(xon) - Vn(xn)+ < pon, Xn> - < POn, XQn >)
VOn
Taking into account that
< Pon, xn> < Vn{xn) + V*(p0n)
we obtain
\\XQn ~ Xn\\ < -7= (Vn(x0n) + VniPOn)- < ΡΟη,ΧΟη >) = λβ~η
VOn
Hence inequality (7.9) is proved.
Since (ж, ρ) is the limit of (xon, pQn) by definition, we deduce that
it is also the limit of the sequence of elements (xn, pn) of the graph
of dVn. □
Since dV is a maximal monotone set-valued map when X is a
Hubert space, Proposition 7.1.7 implies:

7.6. Graphical Convergence of Gradients
299
Theorem 7.6.4 (Attouch) Let X be α Hilbert space. We supply X
with the norm topology and X* with the weak-* topology. Consider
a sequence of nontrivial lower semicontinuous convex functions Vn :
IhRU{+oo}.
Let us assume that its upper epilimit and the sequentially weak
lower epilimit do coincide with a nontrivial function denoted by V.
Then
dv = iW'n-oodVn = iJmin-oodVn
In particular, we deduce that under the assumptions of
Theorem 7.6.4, the graphical convergence of the subdifferentials dVn to
dV implies that for every χ £ X,
dV(x) = σ -LimsuO^^^dVnixn)
When the functions Vn are the indicators of nonempty closed
convex subsets Kn, we obtain the following theorem on the convergence
of the sequence of normal cones to these subsets:
Corollary 7.6.5 Let X be a Hilbert space and Kn С X be a sequence
of nonempty closed convex subsets. We supply X with the norm
topology and X* with the weak-* topology.
Assume that the lower limit К of the sequence {Kn)n coincides
with its sequentially weak upper limit. Then
NK(x) = tf-Lnnsupn^0OiXn^a.J\fan(a:n)
In order to state that some ρ 6 dV(x) belongs to the lower limit
of dVn{xn) for any sequence xn G Oom.(dVn) converging to ж, we need
some stability assumptions on the contingent second derivatives of
the functions Vn-
Naturally, the contingent Hessian d2V(x,p) := D(dV)(x,p) of У
at some point (ж, ρ) in the graph of dV is defined as the contingent
derivative of the set-valued map dV at {x,p).
Proposition 7.6.6 Let X be a Hilbert space. Consider a sequence
of nontrivial lower semicontinuous convex functions Vn : X н-» R U
{+00}.
Let us assume that their upper epilimit and their sequentially weak
lower epilimit coincide with a nontrivial function denoted by V.

300 7 Graphical L· Epigraphical Convergence
Let ρ belong to dV(x). We posit the following stability
assumption: There exist constants с > 0 and η > 0 such that
V (xn,Pn) € Graph(<9Vn) Π Β((χ,ρ),η), Oom(d2Vn(xn,Pn)) = X
< and
w \\d2Vn{Xn,Pn)\\ ■= SUpuejfinfiree2v'„(xBlp„)(«)lkll/ll«ll < C
Then ρ € Limb3£n^00iXn-^xdVn{xn).
Proof — We apply Theorem 5.4.2 to the set-valued maps Fn :=
dVn. U
7.7 Asymptotic Epiderivatives
We are now able to define asymptotic epiderivatives of a sequence of
functions Vn, by taking the asymptotic tangent cones to their epigraphs. "We
denote by
V\ := Нт^-ооК
their upper limit.
Definition 7.7.1 (Asymptotic Epiderivatives) Let us consider a
sequence of extended functions Vn :IhRU {+oo} whose domains are not
empty and an element xq in the domain of the upper epilimit of the
functions Vn (i.e., V^{xq) φ ±oo.) We shall say that the function C^{Vn}(xo)
defined by
C\{Vn}(x0)(u) : = Urn sup (Vn(x+hu') - X)/h
n—>oo,x—>xo,Vn(x)<X—*V^(x0),h—*0+
is the asymptotic circatangent epiderivative of the sequence of functions Vn
at xq in the direction u.
We see at once that the epigraph of C^{Vn}(xo) is the asymptotic
circatangent cone to the epigraphs of the functions Vn at (xo,V^(xq)), or,
equivalently, that C^{Vn}(xo) is the upper epilimit of the difference
quotients
u' i-+ (Vn(x+hu')-X)/h
when
n —> oo & (x,X,h) £ fp(K,) χ R+ converges to (xo,V%(xo),ty

7.7. Asymptotic Epiderivatives
301
We deduce that the asymptotic circatangent epiderivative is a positively
homogeneous, lower semicontinuous and convex function from X to R U
{±oq}.
We shall estimate the asymptotic circatangent epiderivative of a family
of functions Un :=Vn + Wn ° A.
Theorem 7.7.2 Consider two Banach spaces X and Y, a continuous
linear operator A £ £(X, Y) and two sequences of extended lower
semicontinuous functions Vn and Wn from X and Υ to R U {+00} respectively.
Assume thatx0 belongs to the domain ofV^ and Ax Q to the domain ofW\.
We posit the following stability assumption/ there exist constants с > 0,
α £ [0,1[ and η > 0 such that, for all n, (7.6) holds true.
Then, the asymptotic circatangent epiderivative of the sequence of
functions Un -.= Vn + Wn о A satisfies the estimate:
C\{Un}(xQ)(u) < &^{Vn}{x0){u) + C\{Wn}{AxQ){Au) (7.11)
whenever the right hand side is well defined.
Remark — Since the assumptions of this theorem are the same as
those of Theorem 7.4.1, we know that Щ(х0) < V^ (x0)+W^ (AxQ) whenever
the right hand side is well defined. □
Proof — We apply Theorem 3.4.5, since we have seen in the proof
of Theorem 7.4.1 that if we set
' i) Kn := £p{Vn) x Sp(Wn) xR с XxRx^xRxR
< ii) G(x, a, y, b, c) := (Ax — y, a + b - c)
Hi) H(x,a,y,b,c) := (x,c)
we can write
Ep(Un) = tf(JfnnG-i(o,0))
The stability assumption being the same as the one of Theorem 7.4.1, we
already know that they imply the stability assumptions of Theorem 3.4.5.
Hence, we deduce exactly as in the proof of Theorem 7.4.1 that
' C\, (x0, V\{x0), Ax0, W\(AxQ),Ifyxo)) П G~\Q,0)
<
(so, Vffro), AxQi W\(Ax0), υ\(χ0ή
It remains to show that this inclusion implies inequality (7.11.)

302 7 Graphical &ε Epigraphical Convergence
Inequality (7.11) being satisfied when one of the members of the right
hand side is +oo, we assume that both of them are different from +oo. Let
us consider
λ > C\{Vn}(x0)(u), μ > C\{Wn}(Ax0)(Au), ν := λ + μ
Hence the element (u, A, Au, μ, ν) belongs to
СЬкь (χ0,ν$(χο),Αχα,νή(Αχα),ΐή(χο)) nGT^O.O)
It then belongs to the asymptotic circatangent cone to Kn Л G~1(0,0).
Then, for all sequence hn > 0 converging to zero, there exist elements
(un, λ„, μη) converging to (u, λ, μ) such that, for all η > Ν,
(xn + hnu + Ηημη, αη + bn + Ηηνη)
belongs to Kn П G-\O, 0).
Therefore, the pairs (xn + hnun, an + bn + hn(A„ + μη)) belong to the
epigraph of Un. Since (iLn,Xn +μη) converges to (u, v), we deduce that
C\{Un}(x0)(u) < λ + μ
By letting A and μ converge to C^{Vn}(xo)(u) and C^{Wn}(Ax0)(Au)
respectively, we infer that
C\{Un}(x0)(u) < C\{Vn}(x0)(u) + C\{Wn}(Ax0)(Au) □

Chapter 8
Measurability and
Integration of Set»Valued
Maps
Introduction
We meet measurable maps whenever we deal with models of
systems having measurable data. Integrals of set-valued maps are
involved in many convexification (also called relaxation) problems,
since roughly speaking, the integral of a measurable set-valued map
is always convex.
Another important instance where measurable set-valued maps
do arise is related to linearization of differential inclusions along a
solution. Consider indeed a Lipschitz set-valued map F : Rn ~> Rn
and the differential inclusion
χ € Fix)
Let χ ·. [0,Г] н-> Rn be a solution to this differential inclusion, i.e.,
an absolutely continuous map satisfying x'{t) G F(x(t)) almost
everywhere in [0, T]. Linearization of this inclusion along x(·) leads to
the new differential inclusion:
w'(t) € DF(x(t),x'(t))(w(t)) almost everywhere in [0, T)
303

304 8- Measurability and Integration of Set-Valued Maps
where DF denotes the contingent derivative of F. Setting
G(t,w) := DF{x{t),x'{t)){w)
we obtain a set-valued map G merely measurable in time and Lips-
chitz in w (see Chapter 10.)
We investigate in this chapter some properties of measurable set-
valued maps with closed images, which we define in Section 1. Under
adequate assumptions, α set-valued map is measurable if and only if
its graph is measurable.
Measurable maps with nonempty closed images have measurable
selections. This result was born as a lemma by von Neumann in 1949
and has found different extensions and applications. It was given
in a stronger form by Kuratowski and Ryll-Nardzewski. Castaing
has proved that there exists a countable dense family of measurable
selections, a useful theorem since countability plays a major role· in
many problems of measurability.
These two statements constitute the core of what we call the
Characterization Theorem for measurable maps, that we prove in
Section 3.
But it is already used in Section 2 which is devoted to the
calculus of measurable maps. It is shown there that the main operations,
such as union, intersection, direct and inverse image, lower and upper
limits etc., do preserve measurability. The convex hull of a
measurable map is measurable, and measurable selections from the convex
hull have a Caratheodory representation. This is particularly useful
in control theory, when we have to deal with relaxed controls.
In Section 4 we investigate selections from limits of measurable
maps and in Section 5 tangent cones in Lebesgue spaces.
Section 6 is devoted to the concept of integral of set-valued maps
introduced by Aumann, who proved that the integral is closed and
convex when the set-valued map is measurable, integrably bounded
and takes closed values in a finite dimensional space, (He was
motivated at the time by problems of cooperative game theory and
mathematical economics where convexity properties are missing when the
set of players is finite. He then introduced the concept of continuum
of players and replaced the sum by the integral.)
Convexity of the integral is an immediate consequence of Lya-
punov's Convexity Theorem on convexity of the range of a nonatomic

8.0. Introduction
305
vector measure. Similar results concerning convexity of the integral
are still valid in the infinite dimensional case (the closure of the
integral is convex.) They are proved in Section 7.
The integral of a time-dependent set-valued map is also equal
to the integral of finite concatenations of extremal selections. This
result is sometimes referred as the bang-bang principle. The bang-
bang principle we state here is due to Olech. It says in essence, that
to integrate a map F : [a, b] ~> Rn, we can restrict the attention
to piecewise extremal trajectories with the number of switchings not
greater than η +1. This is the topic of Section 8, which does not use
the Lyapunov Theorem.
The last section deals with the extension to upper semicontinuous
maps of the Poincare Recurrence Theorem: Let X be a compact
metric space, V(X) denote the set of probability measures on X,
F : X ~> X be a closed set-valued map and μ G V(X) an invariant
measure of F. For any Borelian subset B, let
Boo := Π U F~n{B)
N>0 n>N
be the subset of points χ such that for all N, there exists η > N such
that F~n(x) Π Β φ 0. Then the measure of В Г) В^ is equal to the
measure of В.
The statement of this theorem is clear as soon as we have defined
the notion of an invariant measure of a set-valued map F:
Let В denote the σ-algebra of Borel subsets of X. We recall
that if F = f is single-valued, an invariant probability measure μ is
defined by:
VA&B, μ(Α)=μ(Γ1(Α))
When F is set-valued, we cannot extend this definition as it is
because А н-> μ(Ρ~ι(Α)) is no longer a measure. However, we shall
introduce the following definition: We say that μ is an invariant
measure of a closed set-valued map F : X ~> X if and only if
VAeB, μ{Α) < μ{Ρ~χ{Α)) (8.1)
Indeed we see that, for single-valued maps /, this definition coincides
with the classical one when we apply it to both A and its complement.
We shall prove the following result:

306 8- Measurability and Integration of Set-Valued Maps
Let X be α compact metric space and F : X ~> X be a closed set-
valued map with nonempty values. Then there exists an invariant
probability measure.
8.1 Measurable Set-Valued Maps
We define in this section measurable set-valued maps taking their
values in a complete separable metric space, (a Polish space1.) This
contains separable Banach spaces and, in particular, Lebesgue spaces
LP and Sobolev spaces Wm'p with 1 < ρ < oo.
Consider a set Ω and a family A of subsets A of Ω. Recall that
A is called a a—algebra if it verifies the following properties:
' i) 0 € A
<ii)A€A=^£l\A€A
Hi) An € A, n = 1, 2,... ==> Un>i Аг € -4
It is clear that i) and ii) imply that Ω € A and from й), га) follows
that the intersection of a countable family of elements of A is still
an element of A.
We call the pair (Ω, A) a measurable space and the elements of
A measurable sets.
If Ω is a topological space, then the smallest σ—algebra containing
all open sets is called the Borel σ—algebra. We denote it by β(Ω) or
simply by β, when it is clear from the context.
A map μ : A t—» R U {+00} is called a positive measure if for any
sequence of disjoint sets An £ A
μ(υη>ιΑη) = Υ^μ{Αη)
π>1
A positive measure is σ-finite if Ω is the union of a (countable)
sequence of measurable sets of finite measure. The σ—algebra A is
1 Polish space is the name given by Bourbaki to a topological space which is
metrizable in such a way that it becomes complete and separable. This space is
particularly often discussed in mathematics. Opial wrote: "...this touching tribute
to the Polish School of Mathematics testifies most expressively to its merits and
to its status in mathematics."

8.1. Measurable Set-Valued Maps
307
complete (or, to be more precise, μ—complete) if for every set A G A
satisfying μ{Α) = 0 any subset A\ с A is an element of A.
For instance for every open (or closed) subset Ω С Rn, the set
of all Lebesgue measurable subsets of Ω is complete (with respect to
the Lebesgue measure.) The Lebesgue measure is also σ—finite.
In summary, we shall say that the triple (Ω, Α, μ) is a complete
σ—finite measure space if μ is a positive σ—finite measure such that
A is μ—complete.
Consider a complete separable metric space X. A single-valued
/ : Ω ι—» X is said to be measurable (.Α-measurable, to be more
precise) if for every open subset О С X, f~1(0) G A (or equivalently
for every closed subset С С X, /-1(C) € A.) We say that a map
is simple if it takes only a finite number of values. A simple map is
measurable if and only if for every χ £ X, f~1 (x) G A.
The following equivalences provide in many instances convenient
tools to check measurability of a map:
' i) f is A — measurable
ii) f is the pointwise limit of simple measurable maps
in) f is the uniform limit of measurable maps
assuming a countable number of values
One immediately deduces that the pointwise limit of measurable
maps is measurable.
Measurable set-valued maps with closed images are defined in a
similar way·.
Definition 8.1.1 Consider a measurable space (Ω, A), a complete
separable metric space X and a set-valued map F : Ω ~> Χ with
closed images.
The map F is called measurable if the inverse image of each open
set is a measurable set: for every open subset О С Χ, we have
F~l{0) := {ω € tt\F(uj)nO φ 0} € A
Remark — The notion of measurability that we use here is sometimes
called in the literature weak measurability: a set-valued map is weakly
measurable if inverse images of open sets are measurable, in comparison to
strong measurability: inverse images of closed sets are measurable.
However in the framework we shall deal with (complete σ-finite measure spaces),

308 8- Measurability and Integration of Set-Valued Maps.
these two notions do coincide. That is why we simply use the word
"measurable." □
The domain of a measurable map is measurable as well as its ■
complement: {ω£Ω| F{uj) = 0}.
A measurable set-valued map has a measurable selection:
Definition 8.1.2 Let (Ω, Λ) be α measurable space and X a
complete separable metric space. Consider a set-valued map F : Ω ~> X.
A measurable map /:ΩηΙ satisfying
Vu>€ Ω, f{w) € F{lj)
is called a measurable selection of F.
Theorem 8.1.3 (Measurable Selection) LetX be a complete
separable metric space, (Ω, Λ) a measurable space, F a measurable set-
valued map from Ω to closed nonempty subsets of X. Then there
exists a measurable selection of F.
Proof — Fix a countable dense subset {xn}n>i of X. We
construct a sequence of measurable maps Д : Ω н-► X, к > 0 taking
values in {xn}n>i and converging uniformly to a selection / of F, so
that / is measurable. We proceed by induction to construct such a
sequence.
Denote by d the distance in X. For every ω G Ω, let η > 1 be the
о
smallest integer such that F(uj)r\ В (xn, 1) Φ 0· We set /ο(ω) = χη·
Then /ο(·) is measurable. Furthermore
\/ш&П, d{f0{u), F(u>)) < 1
Assume that we already constructed measurable maps
fk ■ Ω (->■ {xn}n>i> к = 0,...,m
satisfying
V0 < fc < m, \/ω € Ω, d(fk(u>),F(u>)) < ~ (8.2)
and
V0 < к < m-1, d(/feH,/fe+iH) < -lj (8.3)

8.1. Measurable Set-Valued Maps
309
For every n, define
Sn = {ω€Ω| fm{u) = xn }
The sets Sn are mutually disjoint and Un>i Sn = Ω. Furthermore
(8.2) implies that
V^ e 5n, F(lu) Π Β (ι»,2~m) ^ 0
Fix a; € Ω and let η be such that ω € 5η. Consider the smallest
integer г such that
FH П В (xn,2-m) Π 5 (xr> 2-(ш+1)) ^ 0
and set /m+i(o;) = жг. Then
d(/mH,/m+iH) < 2~m + 2-(ш+1) < 2~m+1
Moreover
d(/m+1 (07)^(0;)) < 2~(™+1)
Clearly, this defines a measurable map
fm+i ·' Ω ι-* {α;π}π>ι
and (8.2), (8.3) hold true with m replaced by m + 1.
From inequality (8.3) follows that for all ω G Ω, (fk(u))k>i is a
Cauchy sequence in a complete metric space X. Let f{uj) denote the
limit of /fe(o;). From (8.3) we deduce that /& converges uniformly to
/ and from (8.2) that
d(/H,F(u;)) = 0
Thus / is measurable and for every ω € Ω, /(ω) € ί^ω). □
The next theorem provides several useful characterizations of
measurability. For that purpose, let Λ ® В denote the σ—algebra
generated by products Ax B, where A £ Λ and В € В (В is the
Borel σ—algebra of a metric space X.)

310 8- Measurability ajad Integration of Set-Valued Maps
Theorem 8.1.4 (Characterization Theorem) Let (Ω, Α, μ) be а
complete σ—finite measure space, X a complete separable metric
space and F : Ω ~> X a set-valued map with поп empty closed
images. Then the following properties are equivalent:
i) F is measurable
ii) The graph of F belongs to A® B
Hi) F^iC) € A for every closed set С С I
< iv) F~1{B)eA for every Borel set В С X
ν) For all x£X the map d(x,F(·)) is measurable
vi) There exists a sequence of measurable selections (fn)n>i
of F such that V ω £ Ω, F(u) = Un>i /η(ω)
A countable family of selections satisfying the latter property is said
to be dense.
The equivalence of statements i) and ii) is fundamental in the
framework of our graphical approach, since it characterizes
measurable set-valued maps through the measurability of their graphs. Here
too, this happens to be quite important.
The equivalence of claims i) and vi), due to Castaing, is also very
useful, since countability is basic in many measurability questions.
We shall prove this theorem in the third section. But first we use
it to provide some properties of measurable set-valued maps.
8.2 Calculus of Measurable Maps
We develop in this section the calculus of measurable set-valued maps
in complete σ-finite measure spaces. However the reader should be
aware that some of the results below can be extended to more general
cases.
First, we note that semicontinuous maps are measurable:

8.2. Calculus of Measurable Maps
311
Proposition 8.2.1 (Continuity and Measurability) Consider а
metric space Ω and a complete σ-finite measure space (Ω,,Α,μ). We
assume that A contains all open subsets of Ω. Let X be a
complete separable metric space and F : Ω ~> X a set-valued map with
closed nonempty images. If F is upper semi-continuous (or lower
semi-continuous), then F is measurable.
Proof — If F is upper semicontinuous, then for every closed set
С С X, F~1{C) is closed. We deduce from Characterization
Theorem 8.1.4 that F is measurable. If F is lower semicontinuous, then
for every open set О С X, -F_1(C) is open. D
Theorem 8.2.2 (Convex Hull) Let(£l,A, μ) be a complete σ-finite
measure space, X a separable Banach space and F : Ω ~> Χ a set-
valued map with closed images. Then the map
Ω Э ω ~> coF(u>)
is measurable.
To prove this theorem we need the following well known result:
We recall that a map φ from Ω χ X to a metric space Υ is called
Caratheodory, if for every χ ζ. Χ, φ(·, χ) is measurable and for every
ω£ί], ¥>(ω, ■) is continuous.
Lemma 8.2.3 Consider two complete separable metric spaces X, Y,
a measurable space (Ω, Λ) and a Caratheodory map ψ : Ω χ X н-► Υ.
Then for every measurable f : Ω ι—» X, the map ω ι—» φ(ω,ί{ω)) is
measurable.
We provide its proof for convenience of the reader.
1
Proof — Since /(·) is measurable, there exists a sequence of
simple measurable maps fn : Ω (-► X converging pointwise to /.
Then ω ι-+ (p(u),fn(u>)) is measurable. Since φ is continuous with
respect to the second variable,
Va) £ Ω, \υοαοφ(ω,/η(ω)) = φ(ω,/(ω))
Hence φ(ω, /(ω)) is the pointwise limit of measurable maps and thus,
is measurable. D

312 8- Measurability and Integration of Set-Valued Maps
Proof of Theorem 8.2.2 — It is not restrictive to assume that
F.has; nonempty images. Characterization Theorem 8.1.4 implies
that there exists a dense sequence of measurable selections (fn)n>i
of F. Let Q+ denote all nonnegative rational numbers. Consider the
set
Λ := |(Ai>...,A„)JA1 € Q+> ]Γλ; = 1, η > ll
and the countable family of maps J2f=i Xifi, where (λι,..., λπ) G Λ.
By Lemma 8.2.3 these maps are measurable selections of coF. They
are also dense in coF. Characterization Theorem 8.1.4 ends the
proof. □
Theorem 8.2.4 (Union and Intersection) Let(Q,A, μ) be α
complete σ-finite measure space, X a complete separable metric space and
Fn : Ω ~> Χ set-valued maps with closed images. Then the maps
Ω Э ω ~> ω ~> Η(ω) := Q Fn(uj)
π>1 π>1
are measurable.
Proof — Fix an open set О С X. Then
G~\0) = {^Ω| |J(FnHnO)^0} = \JF-\0)
π>1 π>1
To prove the second statement observe that Η has closed images and
Graph(tf) = Γ) Graph (Fn)
71>1
By Characterization Theorem 8.1.4, Graph (Fn) belongs to Λ®Β
and so does the graph of H, which is measurable, thanks to the same
theorem. Π
Theorem 8.2.5 (Lower and Upper Limits) Let (Ω,Α,μ) be a
complete σ-finite measure space, X a complete separable metric space
and Fn : Ω ~> Χ, η > 1 set-valued maps with closed images. Then
the maps
Ω Э ω ~> LiminfTl_KX1i;iTl(a;) & Ω Э ω ~> Limsupj^oo-Fn^)

8.2. Calculus of Measurable Maps
313
are measurable.
Consequently, if for every ω € Ω, the images Fn(u) converge to
a subset F(uj), the set-valued map F (called the pointwise limit) is
measurable.
We need the following
Lemma 8.2.6 Consider a measurable space (Ω,Α), complete
separable metric spaces Χ, Υ and let g : Ω χ X н-► Υ be a Caratheodory
map. Then g is A® B-measurable.
Proof — Indeed it is enough to show that there exists a sequence
of maps gn : Ω χ Χ ι-> Υ measurable on A ® B, converging pointwise
to g. Consider a dense sequence xk € X, к > 1. Fix (ω, χ) € Ω χ Χ.
For every η > 1, let k be the smallest integer satisfying
о ( 1
X € В \Хк, -
\ η
and set дп(ш,х) = д(ш,Хк). Then gn converge pointwise to g.
Furthermore
Vy € Yk := В (zfc'-J \ U B \Xm'n)' 5n^'y) = ^(ω,ζ*;)
^ ' m<k ^ '
But Un>i ^ic = -^ and therefore for each η, ^π is A® B-measurable.
D
Proof of Theorem 8.2.5 — By Lemma 8.2.6 and
Theorem 8.1.4 the map (ω, χ) ι-> d(Fn(u), χ) is A ® 23(X)-measurable.
Thus for every ε > 0,
Graph(S(Fn(-),£)) := {(ω,χ) € Ω χ X\ d{Fn{u),x) < ε] С Л®В(.Х")
This and Characterization Theorem 8.1.4 yield that the map
ω ~> B(Fn(uj),e) is measurable
Then the result follows from Theorem 8.2.4 and equations
i) Liminfn^oo-FnM = Γ\^ιΌν>ιΓ\π>ν Β(Ρη{^)Λ)
w ii) Limsupn_too-Fn(^) = f|jv>i Un>jv Ρη{ω) □

314 8- Measurability and Integration of Set-Valued Maps
Definition 8.2.7 Consider metric spaces Χ, Υ and a measurable
space (Ω,Α).
A set-valued map G : Ω χ Χ ~> Υ with closed values is called
a Caratheodory map if for every χ £ X the map ω ~> G(u>, x) is
measurable and for every ω G Ω the map χ ~> G(u>, x) is continuous.
Theorem 8.2.8 (Direct Image) Let (Ω, Α, μ) be a complete
σ-finite measure space, X a complete separable metric space and F : Ω ~>
X a measurable set-valued map with closed images.
Consider a Caratheodory set-valued map G from Ω χ Χ to a
complete separable metric space Υ. Then, the map
Ω Э ω ~> (?(ω,ί(ω))
is measurable.
In particular for every measurable single-valued map ζ : Ω ι—»
X, the set-valued map ω ~> G(lj, ζ(ω)) is measurable and for every
Caratheodory single-valued map φ from Ω χ X to Υ, the set-valued
map
Ω Э ω ~> φ(ω, F(u))
is measurable.
Proof — It is not restrictive to assume that F has nonempty-
images. Prom Characterization Theorem 8.1.4, there exists a dense
sequence (fn)n>i of measurable selections of F.
We claim that the map ω ~> G{u>, fn{u)) is measurable. Indeed
consider a sequence of measurable simple maps fnk from Ω to X,
converging pointwise to fn when к —> oo. Then, since fnk are simple,
for every к the set-valued map ω ~» ΰ{ω, fnk{u)) is measurable. On
the other hand, since G(u>, ·) is continuous,
\f ω € Ω, Limfe^ooG^/nic^)) = G(w,/nM)
and from Theorem 8.2.5, we deduce that this limit is again a
measurable map.
On the other hand, since G is continuous with respect to the
second variable and (fn)n>i is dense, for every ω£ί]
G(u;,F(u;)) = \jG(u,Jn(u,))
71>1
Theorem 8.2.4 ends the proof. D

8.2. Calculus of Measurable Maps
315
Theorem 8.2.9 (Inverse Image) Consider α complete σ-finite
measure space (Ω,Α,μ), complete separable metric spaces X, Y,
measurable set-valued maps F : Ω ~> X, G : Ω ~> У with closed images.
Let g : Ωχ Χ ι-> У be a Caratheodory map. Then the set-valued map
Η defined by
Η (ω) := {ieF(w)| д(и,х) € G(w)}
is measurable. Consequently, if
VW€fi, g(Lj,F(tj)) П G(w) ^ 0
ΐ/ien i/iere eaasis α measurable selection f of F such that for every
ωξί], g(uj,f(uj)) belongs to G(uj).
Proof — Observe that the map Η has closed images. Define
the single-valued map φ: ΩχΧ\-+ΩχΥογ φ(ω, χ) = (ω, д(ш, х)).
Then, by Lemma 8.2.6,
VS € B(Y), {(ω,χ)\ g(u,x) € B}ei®B(I)
Observe that
Graph(tf) = Graph(F)n^1(Graph(G))
By Characterization Theorem 8.1.4,
Graph(F) e A®B{X) к Graph(G) e A®B{Y)
On the other hand for every Ay. В £ A® B(Y),
φ-^ΑχΒ) = {(ω,χ)\ д(ш,х) £ В} П{АхХ) £ А®В{Х)
Hence for every С С Α®Β{Υ), φ'1 {С) е А®В(Х). Thus Graph(tf)
belongs to A ®B(X) and from Characterization Theorem 8.1.4, we
deduce that Η is measurable. Finally the Measurable Selection
Theorem implies that it has a measurable selection, whenever Η has
nonempty images. □
As a consequence, we obtain the very useful

316 8- Measurability and Integration of Set-Valued Maps
Theorem 8.2.10 (Pilippov) Consider α complete σ-finite measure
space (Ω,Α,μ), complete separable metric spaces Χ, Υ and a
measurable set-valued map F : Ω ~> Χ with closed nonempty images. Let
g : Ω χ Χ ι—> Υ be a Caratheodory map. Then for every measurable
map h : Ω ι—» Υ satisfying
h{uj) G g(u>, -F(o;)) for almost all ω G Ω
there exists a measurable selection ${ω) G F(u) such that
h{uj) = д(ш, f(u>)) for almost all ω G Ω
Marginal functions and maps are measurable under the following
conditions:
Theorem 8.2.11 (Marginal Map) Consider a complete σ-finite
measure space (Ω,Α,μ), complete separable metric spaces X, Y, a
measurable set-valued map F : Ω ~> Χ with closed nonempty images
and a Caratheodory function f : Ω Χ Χ н-► R. Then the marginal
function ν : Ω ι—» R U {—oo} defined by
Vw£ii, ν(ω) := inf f(uj,x)
is measurable. Furthermore, the marginal map R defined by
Vwefi, R{u) := {x€F^)\f&,x)= inf }{ш,у)}
yeFH
is also measurable.
We recall the following:
Lemma 8.2.12 Consider a measurable space (Ω, Λ). Then for
every sequence of measurable real-valued functions fn : Ω >—» R, the
extended function
ω н-> f{u) := inf/n(u;) G RU{-oo}
has a measurable domain of definition and is measurable on it. A
similar statement holds true for s\iOn>i fn.

8.2. Calculus of Measurable Maps
317
Proof of Theorem 8.2.11 — Let us consider a dense family of
measurable selections (gn)n>i of the map F. Since the maps /(ω, ·)
are continuous, we infer that
Vw € Ω, ν{ω) = inf f{u>,gn(uj))
n>l
Since the maps ω ι-> f(uj,gn(uj)) are measurable by Lemma 8.2.3, we
deduce from Lemma 8.2.12 that ν is also measurable.
Theorem 8.2.9 on inverses of measurable maps imply that the
marginal map R is measurable, since
R{u) = {x € F{u;) \ί{ω,χ) = ν{ω)}
and because ν is measurable. Π
Corollary 8.2.13 Let us consider α complete σ-finite measure space
(Ω,Α,μ), a complete separable metric space X, a measurable set-
valued map F : Ω ~> Χ with closed images and measurable single-
valued maps / : Ω ь> Χ, ρ : Ω н-> R+. Then the following maps are
measurable:
1. — the map ω ~> Β(f (ω), ρ(ω))
2. — the distance function ПЭши <ί(/(α;), -F(o;))
3. — the projection map Ω Э ω ~> Πρ(ω)(/(α;)) defined by
• nFH(/H) := {x € FM| φ;,/Η) = d(f(u),F(u))}
Consequently if for every ω£Ω, Πρ(ω)(/(α;)) φ 0, ί/ien ί/iere exists
a measurable selection д(ш) G F(u) such that
d(f(u>),g(u>)) = d(f(u),F(u>))
Proof— Consider the function д(ш,х) = d(x,f(uj)). By Lemma
8.2.3 it is measurable in ω. Since it is also continuous in χ we may
use Theorem 8.2.9 with F = X, G{uS) = [0, ρ(ω)]. To prove the
second and third statements, we apply Theorem 8.2.11 to the function
(u,x)i-*d(x,f(uj)) □
Theorem 8.2.14 (Support Function) Let us consider a complete
σ-finite measure space (Ω, Λ, μ) and let X be a separable Banach

318 8- Measurability and Integration of Set-Valued Maps-
space, F : Ω, ~> Χ be a measurable set-valued map with nonempty
closed images. Then it has measurable support functions: for every
pel*,
the function ω ι-> a{F{uS),p) := sup < p, f > is measurable
/eF(«)
The converse statement holds true if the dual of X is separable and
images of F are convex and bounded.
Proof — Theorem 8.2.11 implies that the support functions
a(F(-),p) are measurable.
Assume next that for every ρ € X*, a(p,F(·)) is measurable,
that X* is separable and F has closed convex and bounded images.
To prove the last statement is enough to verify that for all x, the
map ω ι-> d(x, F(uj)) is measurable.
We first observe that since F{uj) is bounded, its support function
a(F(u>), ·) is continuous. Let pn e Χ*, η > 1 be a dense set of
points of the unit sphere of X*. By our assumption, for every η >
1, a(F(-),pn) is measurable. Fix χ £ X. Then, using that F{uj) is
closed and convex, we get
d(x,F{uj)) = d(0,F{uj)-x) = - inf Μΐ<ι a{F{u) - x,p)
= suPHpIL^iCCp-e) -σ(-ΡΉιίΟ) = supn>i((pn,a;) -a{F{u),pn))
Hence d(x,F(·)) is the supremum of measurable functions.
Consequently it is measurable. Π
The following result is very useful to study relaxation problems
in control theory.
Theorem 8.2.15 (Caratheodory Representation) Consider a
complete σ-finite measure space (Ω,Α,μ), a measurable set-valued
map G : Ω ~> Rn with nonempty closed images and a measurable
selection /(ω) G co(G(o;)). Then there exist measurable functions
\k ; Ω t—» R+ and measurable selections fk{u) € G(u>), к = 0, ...η
such that
/M = X>fe(u;)/fc(u;) & Σλ*Μ = l
fe=0 fe=0

8.3. Proof of the Characterization Theorem
319
Proof — Let Sn+1 denote the set of all points (λ0, ■··, λη) € R++1
such that 2fc=o λ& = 1. Consider the continuous map
h : R^+1 x (Rn)n+1 ^ Rn
defined by
η
h(Xo,...,Xn,xo,...,xn) = У^Л^ж^
fe=o
and the measurable set-valued map
F(u) = Sn+1 χ (G(u;))n+1
By the Caratheodory theorem, /(a;) € /i(-F(a;)). We conclude the
proof by applying Theorem 8.2.9 to this set-valued map F with
g(u>, x) = h(x) and the map G equal to /. Π
8.3 Proof of the Characterization Theorem
Characterization Theorem 8.1.4 is the consequence of both a
Theorem due to Castaing and a classical result on the projection of certain
classes of Borel subsets.
We show first that a map is measurable if and only if it has a
countable dense subset of measurable selections.
Theorem 8.3.1 (Castaing) Let (Ω,Α) be α measurable space, X
a complete separable metric space and F : Ω ~> Χ be a set-valued
map with поп empty closed images. Then the statements ί), υ), νϊ)
of Characterization Theorem 8.1.4 are equivalent.
Proof
1. — We begin by proving that г) =Ф г/г). Consider a dense
countable subset {χη}η>ι οι X. For all n, к > 1 define the set-valued
map Gnk : Ω ~> Χ by
Gn*M = ί Р{Ш)П °B {Xn' ^ if Ρ{ω) П °B {Xn'^ Φ 0
[ F{yS) otherwise

320 8- Measurability and Integration ofSet-Valued Maps
and set Fnk(u>) = Gnk{uj). Then Fnk has nonempty closed images
and is measurable. This follows from the very definition of Gnk and
equality'
for every open О С X, F^(0) = G~l(0)
Hence, by Theorem 8.1.3, Fnk has a measurable selection'/^. It
remains to verify that for every ω £ Ω, the values /nfc(^) are dense
in F(u).
Fix χ £ F(u>) and ε > 0. Let к > 1 be such that 1/fc < ε/2 and
η such that <ί(α;π, ж) < l/к. Hence
F(u>)nB(xn,l/k) φ 0
and /nfc(^) € B(xn,l/k). This yields
d{fnk(b>),x) < d(fnk{uj),xn) + d(xn,x) < ε
Since ε > 0 is arbitrary, the proof of this implication ensues.
2. — We prove now that vi) =*> v).
Let /π(·) be as in vi). Fix χ £ X and let d denote the metric
of X. Then for every n, the function ω t—» ci(a;, fn{u)) is measurable
(thanks to Lemma 8.2.3.) Consequently, the function
ω η-► (ίίχ, F(uj)) = inf d(x, fn(uj))
ri>l
is also measurable and v) follows.
3. — Finally assume that v) holds true. Then for every
χ £ X and r > 0 the set
{ω £ Ω\ d(x,F(u)) < r} = F'1 (в (s,r)J £ Λ.
Fix an open set О С X. Since X is separable, О is a countable union
о
of balls Β {χηιΤ·η). Hence
F~l{0) = Un^iF-1 (в (xn,rn)\ £ Λ
The proof is complete. D
To prove the rest of Characterization Theorem 8.1.4. we need a
very useful projection property enjoyed by complete measure spaces:

8.3- Proof of the Characterization Theorem
321
Theorem 8.3.2 (Measurable Projection) Let (Ω, Α, μ) be α
complete σ-finite measure space, X a complete separable metric space and
G € A®B(X). Then its projection is measurable:
7rn(G) := {ω£Ω\3χ£Χ, (ω,χ)€ΰ} € A
The proof can be found in [91] ?
Proof of Characterization Theorem 8.1.4 — We already
know г) Φ=Φ υ) <=$■ г/г). It is clear that iv) =Ф Hi).
To show that Hi) =*> г), fix an open subset О С X and define
the closed sets
Cn = {x € X\d(x,X\0) > l/n}
where d states for the distance. Then О = Un>i @η· Consequently
F{u>) Π Ο φ 0 if and only if for some η > 1, Ε(ω) Г\Спф%. This
yields
F-\0) = (J F~\Cn) e A
71>1
and ends the proof of г).
We show next that v) implies ii). Observe that
Graph (F) = {{ω,χ)€ΩχΧ \ d(x, F(u/)) = 0 }
By Lemma 8.2.6 the function (ω, χ) н-> d(x, -F(o;)) is A<g>B measurable
and therefore
{(ω,χ) £ Ω χ X I d(x, F(u)) = 0} e A® B
Thus Graph(F) belongs to A®B.
2It is quite tempting to think that the projection (and therefore any continuous
image) of a Borel set should be still a Borel set. This wishful thinking was made
once by Lebesgue. Suslin, who had discovered that error, could not believe that
the great Lebesgue would commit such a mistake. The discovery of Borel sets
whose continuous image is not Borel started the intensive study of analytic sets,
i.e., continuous images of Borel sets, which have many interesting properties.
As "tartes Tatin", "Rocquefort" and "Sauterne wine", this example shows how
frequently mistakes and errors can trigger unexpected advances in both everyday
life and sciences.

322 8- Measurability and Integration of Set-Valued Maps
We end the proof by assuming that ii) holds true and observing
that for every Borel set В £ В we have
F~\B) = 7rn(Graph(.F) П (Ω χ В))
But Graph(-F)n(fi χ Β) is an element of Л ® B. Therefore, using
Theorem 8.3.2, we deduce iv).
8.4 Limits of Measurable Maps and
Selections
Let (Ω, S, μ) be a complete σ—finite measure space and X be a
separable Banach space.
For a measurable map / : Ω ι-> X such that ||/(·)|| belongs to
L1(X; R, μ), denote by /Ω /άμ or /Ω /(ω)μ(άω) the integral of /.
Recall that two measurable single-valued maps are equal almost
everywhere if the set where they are different is of zero measure.
We denote by ΙΡ{£ΐ]Χ,μ), 1 < ρ < oo the Banach space of
(classes) of measurable maps /:ΩηΙ such that /Ω ||/||ρ <ίμ < oo.
Consider a sequence of measurable set-valued maps
Κη:Ω Э ω ~» Κη{ω) С X
We associate with it the subsets ICn С Ι?(Ω\Χ,μ) (1 < ρ < oo) of
selections, defined by
/Cn := {x{·) € Lp(£l;X, μ) | for almost all ω € Ω, χ (ω) € Κη(ω)}
The purpose of the next theorem is to compare the limits of the sets
JCn and the sets of selections x(-) of the limits of the sets Κη{ω).
Theorem 8.4.1 Let us assume that the set-valued maps Kn are
measurable, have closed images and that ω ι—» supra>1 d(0, Kn{u))
belongs to ^(Ω,-,Χ,μ), where 1 < ρ < oo. Then
{x(-) € ΙΡ{Ω\Χ,μ) | for almost alio;, χ(ω) € LiminfTl_tooi;irTl(a;)}
- С Liminfn_Kjo/Cn С Limsup^^/Cn С
{χ(·) € LP (Ω; Χ, μ) | for almost all ω, χ(ω) € Limsupj^oolfnC't')}

8.4. Limits of Measurable Maps and Selections 323
Consequently, if the subsets Κη{ω) have a limit Κ{ω) for almost all
ω € Ω, then the subsets KLn converge to the subset
Κ. := {x(·) € £Ρ(Ω;Χ,μ) | for almost alio;, χ(ω) € .Κ" (ω)}
Furthermore, if the dimension of X is finite and if LP (Ω; Χ, μ) is
supplied with the weak topology, then
σ - Limsupn_too/Cn С
{χ{·) € 1^(9,;Χ, μ) | for a.a. ω, χ(ω) G co(LimsupTl_tooi;!rTl(a;))}
Proof— Let χ(·) belong to the first subset. Then the functions
απ(·) defined by
αη{ω) := ά(χ(ω),Κη(ω))
are measurable and converge to 0 almost everywhere. Since
for almost all ω € Ω, αη(ω) < ||α:(ω)|| + sup<i(0, Κη(ω))
π>1
and since the right-hand side of this inequality belongs to 77(0; R, μ),
we deduce from Lebesgue's Theorem that the functions απ(·) do
converge to 0 in 77(0; R, μ). Let к G 77(0;R, μ) be a function with
strictly positive values. Corollary 8.2.13 allows us to choose a
measurable selection zn{·) of Kn(·) such that
\\χ(ω) — ζη(ω)\\ < αη(ω) + к(ш)/п
It belongs to LP(Ω; R, μ) since
for almost all ω € Ω,, \\ζη(ω)\\ < ||α:(ω)|| +αη(ω) + к(ш)/п
Therefore ζη(·) belongs to 1Сп and converges to x(-) in 77(Ω;Χ,μ),
i.e., x(-) does belong to the lower limit of the subsets K.n.
Let us choose some x(-) in the upper limit of the subsets K.n. Then
there exists a subsequence of elements ζηι(·) oiiCni converging to x(-)
in 77(Ω;Χ,μ). It has a subsequence converging almost everywhere
to x(-) and consequently, for almost all ω, χ (ω) belongs to the upper
limit of the subsets Κη{ω).
Finally, assume that x{·) -belongs to the sequentially weak
upper limit of the subsets /Cn. It is a weak limit in ΙΡ(Ω;Χ,μ) of a

324 8- Measurability and Integration of Set-Valued Maps
subsequence of functions zn{·) € lCn and also, thanks to Mazur's
Theorem, the strong limit of convex combinations υη(·) of elements
of the sequence zn{·). Then a subsequence (again denoted by) vn
converges almost everywhere to x{·). We conclude as in the proof of
Convergence Theorem 7.2.1 that
for almost all ω e Ω, χ{ω) G со {L\ms\XOn_^oaKn{u)) □
8.5 Tangent Cones in Lebesgue Spaces
Let (Ω, S, μ) be a complete σ—finite measure space and X be a
separable Banach space. Let us consider a measurable set-valued map
Κ:Ω Э ω ~> Κ (ω) С Χ
We associate with it the subset /С С 1^(0; Χ, μ) of selections defined
by
/C := { s(-) € Ι^Ω;*, μ) | for almost all ω e Ω, χ(ω) € АГ(а>) }
We shall characterize the contingent and adjacent tangent cones to
/C in terms of the tangent cones to the subsets Κ (ω).
Theorem 8.5.1 Let us assume that the set-valued map К is
measurable and has closed images. Then for every χ £ 1С, the set-valued
maps:
П Э ω ~> Τκ{ω){χ{ω)) к Ω Э ω ~> ТЬК{ш){х{ш))
are measurable. Furthermore
|t/(·) € £Ρ(Ω; Χ, μ) | for almost alia;, υ (ω) € Γ^(ω)(χ(ω))}
- С Г£(я(·)) С Гк(х(0)
С |г/(·) € υ^Ω,-,Χ,μ) | for almost alia;, υ(α>) € Τκ{ω){χ{ω))}
This theorem and its numerous applications motivate the
introduction of adjacent tangent cones and derivable sets.

8.5. Tangent Cones in Lebesgue Spaces
325
Proof— Fix χ £ 1С and ω ζ. Ω.
Let Q+ denote the set of all strictly positive rationale. It is
countable and from the very definition of the tangent cones, we can
write
г*м(.м) =п4 U *м-«сЛ
aeQ+ V^e]o,a]nQ+ /
and
n>0 \<*eQ+ ^e]o,a]nQ+ v " ri / j
We deduce the first assertion from Theorem 8.2.4.
To prove the second statement, consider υ(·) in the first subset.
We have to prove that when h > 0 goes to 0, there exist maps
Vh(·) € ΙΡ{Ω\Χ,μ) converging to υ(·) such that
for almost all ω € Ω, χ (ω) + hvh{u>) € Κ (ω)
Let us set:
ак(ш) := άΙυ(ω), —-—-—^j
Functions ah are measurable and converge to 0 almost everywhere
because for almost all ω G Ω, υ(ω) belongs to the adjacent cone to
Κ (ω) at χ (ω).
Since for almost all ω, α^(α;) < ||υ(ω)|| and υ € 77(0; R, μ), by
the Lebesgue dominated convergence theorem, the functions α/ι(·)
do converge to 0 in Ζ^Ω;!?., μ). Fix к G 1^(0; R, μ) with strictly
positive values. Corollary 8.2.13 yields that for some Zh € /C
Vwsfi, dft/H, ^H-gH^ < αΐι{ω) + ftjfc(a;)
We define now the maps υ/ι(·) by
Vh{bJ) ■= {zh{u)-x{uj))/h
They are measurable, satisfy
Vh{w)-v(u)\\ < ah{uj) + hk{uj)

326 8- Measurability and Integration of Set-Valued Maps
and thus, converge to v(-) in 1^(0; Χ, μ) since α^(·) converges to 0
in Ζ^(Ω;R, μ). We infer that υ(·) belongs to Г^(ж(·)) because
for almost all ω € Ω, ж (ω) + hvh(u) & Κ (ω)
Let us choose next some v(-) in the contingent cone to the subset /C.
Then there exist sequences hn > 0 and υπ(·) converging respectively
to 0 and to x(·) in LFifl; X) and satisfying
for almost all ω G Ω, ж(а>) + hnvn{u) & Κ(ω)
Then a subsequence (again denoted) υπ(·) converges almost
everywhere to υ(·) and consequently, for almost all ω, ν (ω) belongs to the
contingent cone to the subset Κ (ω) at χ(ω). Π
As an obvious consequence, we obtain
Corollary 8.5.2 Let us assume that the set-valued map К is
measurable and has closed images. Let x(·) G /C. If for almost every
ω£ί], the subsets Κ (ω) are derivable at χ(ω), so is 1С at χ and
Tk{x) = {«(■) € ΣΡ(μ·,Χ,μ) | for almost alia;, υ (ω) € Тщы)(х(ш))}
8.6 Integral of Set-Valued Maps
Let (Ω, Α, μ) be a complete σ—finite measure space and X a
separable Banach space supplied with the norm ||-||.
In this section we investigate some properties of the integral of a
set-valued map jP from Ω into closed nonempty subsets of X.
We denote by Τ the set of all integrable selections of F:
Τ = {/ € Ll{Q;X, μ) | /(ω) € F{ui) almost everywhere in Ω}
A set-valued map F : Ω ~> Χ is called integrably bounded if there
exists a nonnegative function к G £*(Ω; R, μ) such that
F{oj) С к(ш)В almost everywhere in Ω
In this case every measurable selection of F is an element of Τ thanks
to Lebesgue's Theorem.
Aumann did suggest definition of the integral of a set-valued map
in the following way:

8.6. Integral of Set-Valued Maps
327
Definition 8.6.1 The integral of F on Ω is the set of integrak of
integrable selections of F:
jf ity := {ja№\ /e^}
The integral is convex whenever F has convex images and we shall
prove later that its closure is still convex even when the images of F
are no longer convex.
It is also clear that
V λ € R, / ΧΡάμ = λ / Ράμ
Jo. Jo.
Proposition 8.6.2 Let us consider measurable, integrably bounded
set-valued maps F, Fi : Ω ~> X, i = 1,2 with nonempty closed
images and set G{u>) := Fi{uj) + F2(u>). Then
1. - Ι^σΐμ = /n Fim + /n F2dM
S- — Ιη^δΡάμ = со J^ Ράμ
3. - Vj, £ Γ, σ(/„%ρ) = /ησ(ίΉ,;ρ)μ(ώχ;)
^. — ///or some ж € /n -Fefyi. and ρ € -X"*
< ρ, ж > = <т ( / Ράμ, ρ 1
ΐ/ien /or every f £ Τ satisfying χ = /n /ίΖμ, эде obtain
for almost all ω € Ω, < ρ, /(ω) > = a(F(a;),p)
Proof— Theorems 8.2.8 and 8.2.2 imply that G and coF are
measurable. To prove the first equality it is enough to show that
/ βάμ С / Fia> + / F-ιάμ
JQ. JO. Jn
since the other inclusion is obvious.
Consider a measurable selection goiG and let (fin)n>i be a dense
sequence of measurable selections of Fi (which exist by
Characterization Theorem 8.1.4.) Then (/in + /2m)n,m>i is a dense sequence
of measurable selections of G. Set
Gnm(u>) = {/ϋ(ω) + /2jH | 1 < i < n,: 1 < j < m]

328 8- Measurability and Integration of Set-Valued Maps
Then
Vwefi, lim d{g{u),Gnm{u)) = 0
п-юо.т-юо
Thanks to Corollary 8.2.13, there exists a measurable selection gnm
of Gnm satisfying
HflH - flnmMH = d{g{u),Gnm{u))
Thus gnm(u) € -Fi(o;) + i^C^). Applying Theorem 8.2.9, we obtain
measurable selections fnm of i; such that f^m + f%m = gnm. This
yields
d[ / 9ηπι<1μ, / -Fidyu + / Ρζάμ) = 0
\^Ω ./Ω Jn J
By taking the limit when n, m —> oo, the first statement ensues.
It is clear that
со
/ Έάμ С / ΈδΡάμ
Jn Jo.
To prove the converse inclusion, fix к € L^fijR, μ) with strictly
positive values and a measurable selection g of coF. Recall that
Theorem 8.2.14 states that the support function
ω ι-> a(coF(u>),p) = a(F(u>),p)
is measurable and observe that for every ρ G X*
(p, / gdμ) < / σ(οδΡ(ω),ρ)μ(άω) = / σ(Ρ(ω),ρ)μ(άω)
Fix e > 0 and set ψ(χ) =< ρ, χ >. Theorem 8.2.9 on the
measurability of inverse images applied with
G(w) = W(F(u;),p)-ek(u>),a(F(u>),p)]
yields that there exists a measurable selection / G Τ such that
<P./M>> a{F{u),p)-ek{u)
Consequently
/ σ(Ρ(ω),ρ)μ(άω)< / (< ρ, f > +ε^άμ < σ ( / Fd^p j+e \\k\\Li
Jn Jci \Jn J

8.6. Integral of Set- Valued Maps
329
Since e > 0 and ρ € X* are arbitrary, using the separation theorem,
we end the proof of the second and third statements.
Fix x, f as in the claim 4. From the third statement
σ{Ρ{ω),ρ)μ{άω) = σ{ / Ράμ, ρ) = f < p,J > άμ
Consequently we deduce that
j (σ{Ρ{ω),ρ)-<ρ,~]{ω)>)μ{άω) = 0
But a(F(u>),p) >< p, f(u>) >. This achieves the proof. □
The famous Lyapunov's theorem on convexity of the range of a
vector measure implies that the integral of a set-valued map is convex
when X is a finite dimensional vector-space.
Recall that a set A £ Λ is called an atom (for the measure μ) if
μ(Α) > 0 and for every measurable subset Αι с Α, μ{Α{) is equal
to either 0 or μ(Α). A measure is nonatomic if Λ does not
contain atoms. Dirac and discrete measures are atomic. The Lebesgue
measure is nonatomic.
A point ζ of a convex set К is called extremal if there is no
x, у € К and 0 < λ < 1 such that ζ = λχ + (1 - X)y. We denote by
ext(K) the set of all extremal points of K.
When X = Rn, then the integral of any set-valued map is convex
(even when the values of F are not convex) and is closed when the
map is integrably bounded:
Theorem 8.6.3 (Convexity of the Integral) Let F : Ω ~> Rn
be a measurable set-valued map with nonempty closed images. If μ
is nonatomic, then
1. — The integral Jn Ράμ is convex and extremal points of
cd (Jn Ράμ) are contained in /n Ράμ.
2. — If a sequence Xk € Jn Ράμ, к = 1,2,... converges to an
extremal point χ of со (J^ Ράμ), then every sequence fj. G Τ such
that
fkάμ = xk

330 8- Measurability and Integration of Set-Valued Maps
converges in L\(Ω; Rn, μ) to some f G .F satisfying /n /<ίμ = ж.
/η particular-for all χ G ext (со J^ Ράμ) there exists a unique
f G Τ with χ = Jn /ίΖμ.
// гп addition F is integrably bounded, then the integral of F is
also compact.
When the dimension of the Banach space X is no longer finite,
the integral may no longer be convex, but its closure is convex:
Theorem 8.6.4 (Convexity of the Closure of Integral) LetX
be a separable Banach space and F : Ω ~»· X be a measurable set-
valued map with nonempty closed images. If μ is nonatomic, then
the closure of its integral is convex:
1. - !ηΡάμ = βδ(ΙηΡάμ).
2. — If χ G ext /n Ράμ, then f £ Τ satisfying /n }άμ = χ is
unique.
3. — If F is integrably bounded, then
Ι οδΡάμ = Ι Ράμ
Furthermore when X is reflexive and F has convex images and is
integrably bounded, then the integral Jn Ράμ is closed.
Remark — It is not difficult to check that the third statement
remains true if we assume that Τ Φ 0 and μ(Ω) < oo instead of the
integrable boundedness of F. Π
It is convenient to introduce the following
Definition 8.6.5 Consiaer a map F : Ω, ~»· X. We say that f G
Τ is an extremal selection of F if /n }άμ is an extremal point of
со (fn Ράμ). We denote by Te the set of extremal selections:
Te := { f G Τ | / ίάμ G ext (со ί Ράμ
I Jo. \ Jn
We deduce from Theorem 8.6.3 and the Caratheodory Theorem:

8.6. Integral of Set- Valued Maps
331
Corollary 8.6.6 Let F : Ω -^ Rn be α measurable integrably bounded
set-valued map with nonempty closed images. If μ is nonatomic, then
for every
χ € / Ράμ
there exist η + 1 extremal selections fk € Te and η + 1 measurable
sets Ah £ А, к = 0,..., η, such that
x = b\EXAjk)άμ
where хлк is the characteristic function of A^.
Proof— Fix x & fn Ράμ and let Xk > 0, Xk € ext (со /n Ράμ),
к = 0,..., η be such that
η η
fe=0 fe=0
which exist by the Caratheodory theorem. Theorem 8.6.3 implies the
existence of fk € Te such that /n fkd-l1 = £fc for every к = 0,..., п.
Then the set-valued map G(u) := {Λ(ω)}^=ο,ι,...,η has closed
nonempty images and is measurable. Since each Xk belongs to its
integral, which is convex by Theorem 8.6.3, so does x. Hence there
exists an integrable selection / := Sfe=oXAfc/fe of G such that χ =
Ιηίάμ- Π
Theorems 8.4.1 and 8.6.3 imply the set-valued version of the
Lebesgue Dominated Convergence Theorem:
Theorem 8.6.7 Let (Ω,,Α,μ) be a complete σ—finite measure space
ana X a separable Banach space. Consiaer measurable set-valued
maps
Kn : Ω Э ω ^ Κη{ω) с X
with closed images.
1. — If the function ω ι-> sup^j ά(0, Κη(ω)) is integrable,
then
I (Limmfn-too-Kn) άμ С Liminf^oo ( / Κηάμ

332 8- Measurability and Integration of Set-Valued Maps
2. —; -If the dimension of X is finite and ω ι-»· sup^j ||ΛΓπ(ω)||
is integrablej then ■ ·'■'-
Limsup^^ ( / Κηάμ)·<Ζ I (Limsu^^^Kn) άμ
\Jn J Jo.
Therefore, under the last asumptions, if the subsets Κη{ώ) have
a limit when η —> oo for almost every ω- G Ω, then
1лтп_юо ( / Κηάμ) = / (Limn_>oo-Kn) άμ
\Jn J JQ.
Proof — We begin by proving the first inclusion. An element
ν of the integral of the lower limit can be written ν = fn }άμ where
/ is the integrable selection of the lower limit of the maps Kn{·). By-
Theorem 8.4.1, it is the limit in Ll(Sl\ Χ, μ) of maps fn G JCn, whjch
are integrable selections of the set-valued maps Kn(·). Hence ν is the
limit of the integrals
vn = / ίηύμ € / Κηάμ
To prove the second inclusion, assume that
ν := Km / fn (ω)μ(άω) G Limsup^oo / Κη(ω)μ(άω)
where fnj G Knj. Since supn>0 ||ΛΓη(·)ΙΙ is integrable, the maps fnj(·)
are integrally bounded. By the Dunford-Pettis Theorem, a
subsequence (again denoted by) fnj(·) converges weakly to a map /(■),
which belongs to the sequentially weak upper limit of the subsets
K,n.
The second statement of Theorem 8.4.1 implies that
for almost all ω € Ω, /(ω) € со {Limsxxo^^^^K^u))
Therefore, by Theorem 8.6.3, we infer that
ν € / со (Limsupn_tooi;!rn(a;)) μ(άω) = / Ι,ϊπιδηΡη^,χ,Κη^μ^)
J9. Jn
This completes the proof of the second inclusion. □

8.7. Proofs of the Convexity of the Integral 333
8.7 Proofs of the Convexity of the Integral
We start with the more simple finite dimensional case.
8.7.1 Finite dimensional case
We assume here that X = R" and F, μ verify all the assumptions
of Theorem 8.6.3.
There exist many proofs of the first statement of Theorem 8.6.3.
We have chosen among them the one based on a lemma which gives
an estimate of the norm ||/ — g\\Li for f,g G Τ in terms of the
integrals of / and g. This lemma is intrinsically interesting since it
has found several applications in control theory, for instance in the
investigation of the Holder behavior of controls3.
We associate with any ρ G Rn and f,g G Τ
SP(f, g) ■= sup | / /άμ - у I У € / Ράμ к (ρ, J gdμ\ < (ρ, у) I
Lemma 8.7.1 Let f and g belong to Τ. Then for all ρ G R"
(ρ,/η9άμ) < (p>jafdv) => II/-Sb < 4n6p(/,fl) (8.4)
Furthermore if
(p,J ίάμ\ = <r(p,J Ράμ\
then
Vs e ^, ||/-fl||bi < (2n-i)6p(/,fl)
Proof — We first observe that
II /
II/-fllb ^ 2nsup / (f-g)dμ
apaWJa
AeAWJA
Indeed, if fy denote the components of /, let us define
Ai := {ω | /i(w) - 9ί(ω) > 0}
(8.5)
3See for instance [171].

334
8- Measurability and iniegraiioB of Set-Valued Maps
We observe that
ii/-sIIli < Σ / ΐΛ-Λμμ = Σ / (λ-λ)^+Σ А „ (si-/i)^
Then inequalities
/ (fi ~ 9ί)άμ < \\f (/ - 9)άμ
JAi \\JAi
II /
< sup / {f -ς)άμ
AeA\\JA
and
/ (βί-/ί)άμ < / (β-ί)άμ
Jn\Ai Jfl\Ai
II /
< sup / (/ - 9)άμ
imply estimate (8.5.)
It remains to show that for any A e A and any ρ G X* satisfying
(P. /n(/ ~ З)^) > 0, we have
ί(ί-9)άμ < 26p(f,g)
JA
(8.6)
in order to prove (8.4.) We note that we can write
/ (ί-9)άμ = / ίάμ-νΑ where yA := / £ίΖμ+ / ίάμ £ Ράμ
JA JD. JA Jn\A J9.
Therefore
/ /<*μ - 1/Λ + / ίάμ - yn\A = ίάμ- ^άμ (8.7)
Consequently assumption (ρ, /Ω(/ — 9)άμ) > 0 and equation
(ρ, Jn ίάμ - yA^i + (ρ, j^ ίάμ - yn\A^ = (ρ, j^f - g)d^j ^ (8.8)
imply that either < p, Jn ίάμ - yA > or < p, Jn ίάμ - yn\A > is not
larger than < ρ, /η(/ — 9)άμ >. Say for instance that
p, / ίάμ-уЛ < (ρ, ίάμ- 9άμ)
ι Λί / \ Jo. Jn /

8.7- Proofs of the Convexity of the Integral
335
Then
and therefore
p, j дdμ\ < (р,уА)
U ~ 9)άμ = / ίάμ -
JA II Λΐ
УА
< 5p(f,g)
(8.9)
(8.10)
Otherwise
P,j f^-yo.\Aj < (p,J ίΊμ- J 9άμ
and interchanging the roles of A and Ω,\Α, we obtain
/ (/ - 9)Ίμ = / ίάμ
On the other hand we always have
УП\А
< 5P(f,g)
\\ί(ί-9)άμ < 6p(f,g)
Equation (8.7) implies that
' \\JA(f - 9)άμ\\ = ΙΙ/η/φ-ϊ/ΑΐΙ
<
. < \\Ιηίάμ-Ιη9<Ιμ\\ + Ρηί<1μ-νίΐ\Αΐ < 26p(/,s)
In summary, we proved inequality (8.6) and thus, the first statement
of the lemma.
To prove the second one, let ρ be such that
p, J /άμ) = σ (ρ, Ι Ράμ
Changing coordinates, we may assume without any loss of generality
that ρ = (0,..., 0,1). Then Proposition 8.6.2 implies that
/πΜ - gn(ui) > 0

336 8- Measurability and Integration of Set-Valued Maps
almost everywhere in Ω. Hence in this case, proceeding as in the
proof of inequality (8.5) we get
' llZ-flb < 2(n-l)supAeU||JA(/-fl)dM|| + ||/n(/- 9)άμ\\
■
< (2n-l)supAe^||/A(/-fl)dM||
(8.11)
Fix a measurable А с Ω and define уд, ym a as before. Both terms
in the left hand side of (8.8) are nonnegative. Thus they are not
greater than (ρ,/Ω(/— 5)^)· Hence (8.9) holds true and (8.10)
follows. Consequently
sup
AeA
fu-9№
JO.
< W,g)
and from (8.11), the second statement ensues. D
Convexity Theorem 8.6.3 is the consequence of both the following
Theorem 8.7.2 and the Lyapunov Convexity Theorem 8.7.3 we recall
below.
We begin by proving
Theorem 8.7.2 Any extremal point of co(Jn Ράμ) belongs to Jn Έάμ.
Furthermore iffy £ J7, к = 1,2,... are such that /n fkd-μ converge
to an extremal point of со (fn Ράμ), then Д converges to a selection
/€f ίη^(Ω;Κπ,μ).
Proof— Set К :=сд (/Ω Ράμ) and let e € A" be an extremal point
of K. Observe that to prove this theorem it is enough to show that
every sequence /& G Τ such that
lim
к—юо
/ ίΗ<1μ =
Jo.
converges in ΐ}{Ω\ Rn, μ) to a selection / G Τ. We use Lemma 8.7.1
which allows derivation that it is a Cauchy sequence from the fact
that the sequence of their integrals is a Cauchy sequence.
Step 1. We claim that for every e > 0 therr ;> δ > 0 such that
V/i,/2€.F with J ϊιάμ
JO.
= 1,2

8.7. Proofs of the Convexity of the Integral 337
we have ||/i - f2\\Li < e.
Indeed it is enough to consider the case Κ φ {e}. Fix e > 0 and
pick 0 < Ύ] < e/8n such that
Q := ΚΠ(β + ηΣη~1) φ 0
where Σπ_1 denotes the unit sphere of Rn. Since Q is compact, so is
coQ С В(е, η) Π Κ. The point e being extremal, it does not belong
to coQ. Thus we can separate e from coQ: there exists ρ G R" with
||p|| = 1 such that
(j), e) > σς>(ρ) := max{ < ρ, χ > | χ € coQ }
Define the open halfspace
Я := {χ € X\ <p,x>> aQ{p) }
Then
Hn(K\B(e,V)) = 0 (8.12)
Indeed, if χ <Ξ Я П (К\В(е, ту)), then
У '■= е+ и η ц(^-е) = (1-й ^ η ) е+ „ V ..χ
\\х — е\\ \ \\х — е\\/ \\х — е\\
belongs to Q, so that < ρ, у >< σς>(;ρ). But
<p,y>> (l- V_ A aQ(p) + ^e,,o-Q(p) = <tq(p)
so that we get a contradiction.
Therefore Η ПК С B(e, η). Since e belongs to the interior of Я,
there exists δ > 0 such that J5(e, <5) С Я. Then
КПВ(е,6) С Н ПК С Β(β,η)
Consider /ι, /2 € ^ such that | fn ^άμ - e| < δ for i = 1,2. Then
both /n /jifyt belong to Я" П Я so that they satisfy
p, J ίίάμ\ > aQ{p)

338 8- Measurability and Integration of Set-Valued Maps
Changing notations if needed, we may assume that
P, J /гфЛ < (ρ,Ι ίιάμ)
Therefore
' { У € In Ράμ | (p, Jn /2άμ) <<p,y>}
<
Οίν^ΙηΡάμΙσοίρΥ <<p,y>} С КПН
By applying Lemma 8.7.1 with / = /i and д = /2 and using the
inclusion Κ Π Η с В(е,η), we deduce that
' ||/ι-ΛΙΙ* <
- 4nsup{||/n /ιίΖμ -y\\ | у e /n FdM, < ρ, /η /2^μ > < < ρ, у >}
k <4nsup{||/n/idM-y|| | у&НпК] < 4η(2τ7) < ε
and our claim follows.
Step 2. We next claim that the extremal point e of К belongs to
the closure of the integral of F:
Vt7>0, (j Fdfi\ ηΒ(β,η) φ 0
Indeed suppose for a moment that it does not hold true for some r\ >
0. Taking r] > 0 small enough, we may assume that the set Q defined
in Step 1 is nonempty. Let Η be as in Step 1. If J5(e, η) Π /η Ράμ = 0
we deduce from (8.12) that
Я П / Ράμ = 0
Jo.
and consequently ΗΠΚ = 0, because К is the convex hull of Jn Ράμ.
This contradicts the fact that e G Κ Π Η.
Step 3. Finally, we show that the extremal point e of К belongs
to the integral of F: Indeed, by Step 2, it belongs to the closure of
Jn Ράμ. Consider a sequence of elements Xk ■= Jn /ι^άμ of Jn Ράμ
converging to e.

8.7. Proofs of the Convexity of the Integral
339
By Step 1, (fk)k>i is a Cauchy sequence in Ζ^Ω; Κπ,μ). Thus
it converges to some / and has a subsequence (fk-)j>i converging
almost everywhere to /. The values of F being closed, we finally
obtain that for almost every ω G Ω, /(ω) G F(u>). Therefore / is an
integrable selection of F satisfying /n /άμ = e, so that e belongs to
the integral of F. □
We recall now Lyapunov's Convexity Theorem on the range of a
vector valued measure which we need to prove the convexity of the
integral in Theorem 8.6.3.
Theorem 8.7.3 (Lyapunov's Convexity Theorem) We assume
that μ is nonatomic and let f G Ι^(Ω; Rn, μ). Then the set
u{A) := { ί ίάμ )
is convex and compact.
The proof can be found in [45, Appendix C] and [8].
Proof of Theorem 8.6.3— Fix/jG.F, i = 1,2 and λ <Ξ [0, l].
We have to prove the existence of / £.!F satisfying
λ / №μ + (1 - λ) / /2άμ = ί ίάμ
Define the vector measure 1/:У1н Rn χ Rn by
v(A) = ЦА№, jAhdv) (8-13)
Then и is finite and nonatomic. Lyapunov's Theorem 8.7.3 implies
that the range of и is convex. Since
!/(0) = {0} & ι/(Ω) = (j^/ιΦ, jaf*dv)
there exists AeA satisfying u{A) = λι/(Ω), i.e.,
λ / /ίάμ = / /ίάμ, λ / /2άμ = / /2£?μ
Jn JA Jo. Ja

340 8- Measurability and Integration of Set-Valued Maps
Therefore the map / = xaIi + Χω\α/2 is an integrable selection of
F we were looking for.
Prom Theorem 8.7.2, we know" that the integral /n Ράμ contains
all extremal points of the convex set со$ς).Ράμ. This proves the
second part of the first statement. Theorem 8.7.2 implies also the
second assertion.
Finally, if F is integrably bounded, then the set со Jn Ράμ is
compact. Since /n Ράμ is convex and contains all the extremal points
of its closed convex hull, we deduce from the Caratheodory theorem
that
/ Ράμ = со ( / ΡάμJ = со ( j Ράμ)
The proof is completed. □
8.7.2 Infinite Dimensional Case
In this section we impose on F the assumptions of the infinite
dimensional Convexity Theorem 8.6.4.
To prove it, we need the following extension of Lyapunov's
theorem to the infinite dimensional case.
Theorem 8.7.4 Let Υ be α separable Banach space. Consider an
integrable map f G ^(Ω,-,Υ,μ) and define the map (vector measure)
ν : Α ι—» Υ by
v{A) = j /άμ
J A
If μ is nonatomic, then v(A) is convex and compact.
Proof— Let fn G ^(Ω,-,Υ,μ), η = 1,2,... be simple maps
converging to / in Ь1(0;У, μ). Define continuous linear operators
Γ, Tn : L°°(fi; R, μ) ^ Υ, η = 1, 2,... by
V5 € L°°(fi;R,^), Tn(g) = f 9/ηάμ, T(g) = f g/άμ
which satisfy
Urn ||Γη-Γ|| < lim ||/„ -/||Li = 0
П-ЮО 71—>CO "■"

8.7- Proofs of the Convexity of the Integral
341
Since fn is simple, the range of Tn is a finite dimensional space. Thus
Tn is compact4 and the limit Γ of the Tn's is compact as well.
To prove that u(A) is convex it is enough to check that for any
x,y £ v(A) and λ € [0,1], λχ + (l - \)y € v(A). Fix such x, y, λ
and let C,D £ Abe such that
χ = / /d/i, у = / ίάμ
Jc Jd
By the Lyapunov Convexity Theorem, for every η
{ Тп(ха) I A € Λ} is convex
Hence there exist АпёД η = 1,... such that
Vn > 1, ATn(Xc) + (1-λ)Γ„(χβ) = Tn(XAJ
Hence
f ИЛ>) -(λζ + (1-λ)ι/)|| < \\T(XAn) - Tn(XAn)\\
+ ||λΓη(χσ) + (l-\)Tn(XD) -(\T(Xc) + (l-\)T(xD))\\
k < ||ГП -ГЦ + λ ||Γ„ - ГЦ + (1 - λ) ||Γη - ГЦ =2 ||Γη - ГЦ
converges to 0 since ||Γπ — ГЦ converges to 0. □
The above infinite version of Lyapunov's Convexity Theorem
implies Theorem 8.6.4:
Proof of Theorem 8.6.4 — The proof is similar to the one of
Theorem 8.6.3. We have to show that for all /г € Τ, г = 1,2, ε > 0
and λ € [0,1], there exists / € Τ satisfying
|λ / /κΖμ + (1-Х) f /2άμ - J /J
< ε
Define the vector measure ν : A (-► Χ χ Χ by (8.13.) By Theorem
8.7.4 the closure of the range of и is convex. Since
!/(0) = {0} & !/(Ω) = (7/κΖμ, //2dM
4 A linear operator is compact if it sends the unit ball into a relatively compact
set. The set of compact operators is closed in the uniform operator topology ([148,
Lemma VI.5.3]) or [44, Chapter 10].

342 8- Measurability and Integration of Set-Valued Maps
.we have for some A £ Λ
λ / /ιάμ- / f-ίάμ + λ / /2£?μ - / /2Φ
Ι ./Ω ./Α II ./Ω JA
< ε
Setting / = хл/ι + Χω\α/2> we deduce that ^Ράμ is closed and
convex.
To prove the second assertion, fix an extremal point χ of /n Ράμ
and let /, д G .7-" be such that ж = /n /<ίμ = /n <7<ίμ. The space X
being separable, there exists a sequence of pn G Χ*, η > 1, which
separates points of X. Define
An = {ω € Ω Ι <ρη,/(ω) > > <pn,g(u>) >}
and set
/1 = XA„/ + ΧΩ\Αη5 & /2 = XA„5 + Xn\Anf
Then /1, /2 € J7. Observe that if μ(Αι) > 0, then
\Pn> / hdH > \Pn4 ^άμ)
and therefore /n /ι<ίμ ^ Jn /2^· On the other hand
- / ίιάμ + - / /2£?μ = ж
Since ж is extremal, we deduce that for every η, μ{Αη) = 0. Similarly
the set
A'n = {ω £ П\ <Pn,g&) > > <pn,/M >}
has zero measure. Since (pn)n>i separates points of X,
μ({ω € Ω |/(ω) ^ sM» < X>(An^«)) = 0
π>1
The third statement follows from the first one and Proposition 8.6.2.
To prove the last claim, assume that X is reflexive and that F is
integrably bounded and has nonempty closed convex images. Then
the set Τ is weakly compact in a{Ll(Ω; Χ, μ), L°°(fi,·Χ, μ)) and the
proof follows. □

8.8. The Bang-Bang Principle
343
8.8 The Bang-Bang Principle
When in addition Ω is ал interval [a, b] С R and μ is the Lebesgue
measure, then we have a Caratheodory type result for measurable
selections known under the name of the Bang-Bang Principle:
Recall that for a closed convex set К С Rn, a convex subset
Q С К is called an extremal face of К if for every x,y € К and
0< λ< 1
\x + (1 - X)y € Q =Ф ж, у € Q
Every extremal point is an extremal face. The dimension of the
extremal face Q is equal to the dimension of the affine space spanned
by Q. The dimension of an extremal point is zero.
Definition 8.8.1 Consider a map F : [a,b] ~»· Rn. A selection
f G Τ is called piecewise extremal if there is a partition
a = to < ti < ... < tk = b
such that Jt*+1 $άμ is an extremal point of со [Jt!+1 FduA.
In the next theorem we assume that Ω = [α, 6], Λ is the σ—algebra of
Lebesgue measurable subsets of [a, b] and μ is the Lebesgue measure.
Theorem 8.8.2 (Bang-Bang Principle) Consider a measurable,
integrably bounded set-valued map F : [a, b] ~»· Rn with nonempty
closed images. Then
1. — Ja Ράμ is convex and compact.
2. — If χ belongs to an extremal face of ^Ράμ of dimension
k, then there exists a piecewise extremal selection f G Τ such that
x = Ja }άμ and the number of subintervals on which f is extremal
can be made not greater than к + 1.
The above result in particular implies that for every igj' Ράμ,
there exist / G Τ and a partition of [a, b] in η + 1 subintervals
a = to < ... < tn < tn+i = b

344 8- Measurabihty and Integration of Set-Valued Maps
such that for every 0 < i < n, the restriction f\[tuti+1] is extremal for
the restriction of F to [t{, i;+i] an(^ x — L· f^f1·
Although the first assertion of the Bang-Bang Principle follows
from Theorem 8.6.3, we obtain it directly as a by-product of the
proof without any help of the Lyapunov Theorem. In this way the
proof given below is self-contained.
Proof of Theorem 8.8.2 — Since μ is the Lebesgue measure
we shall adopt the usual notation dt for άμ and μ(άω).
Observe that convexity and closedness of the integral Ja Fdt
follows from the fact that every extremal face of со Ja Fdt is a subset of
Ja Fdt. Thus our first statement would result from the second one.
To prove it, we proceed by induction with respect to the
dimension m of the face Q.
We already know, thanks to Theorem 8.7.2, that for every
measurable, integrably bounded set-valued map G : [a, c] ~»· BP with
nonempty closed images, every extremal point of со (/^ Gdt) belongs
to J£ Gdt (where с is any number larger than a and ρ > 1 is any
integer.)
Thus every extremal face of со (J^ Gdt) of dimension m = 0 is a
subset of J^Gdt.
Assume next that the above is valid for any such map G and
every extremal face of dimension smaller than k. Fix a measurable,
integrably bounded map G : [a, c] ~»· Rp with nonempty closed
images such that a nonempty set Q is a fc-dimensional extremal face of
со (J£ Gdt). Since Q is extremal, we obtain
Q Π ext fed J ΰάμ) = ext Q (8.14)
Denote by Q the set of all integrable selections of G. Pick an extremal
point e of Q and let ge € Q be an (extremal) selection of G such that
f£ge(t)dt = e. (It exists by (8.14) and Theorem 8.7.2.) Replacing
G by G — ge, we may assume that e = 0 and ge = 0. Let Ρ be the
vector space spanned by Q.
We claim that
V g € Q such that / g(t)dt € P, g(t) € Ρ a.e. in [a, c]
Ja .
(8.15)

8.8. The Bang-Bang Principle
345
Indeed it is enough to show that for every measurable set A, we
have JA g(t)dt € Ρ {Ρ is the intersection of a finite number of hyper-
planes.) Fix a measurable A and define 51 = xAg, g2 = (х[а,с]\д)з·
Then 51,52 € Q and
l£gi(t)dt + \fa92{t)dt = \£g{t)dt + ~o e Q
Hence, because Q is an extremal face,
/%ι(ΐ)ώ = f g(t)dt € Q С Р
Ja J A
This ends the proof of our claim.
Consider the set-valued map Gi(t) = G(t) Π Ρ with nonempty
closed images. For every τ G [a, c], set
π(τ) = / Gidi
Jo
Then
\/t < τ', ВД С π(τ') С Ρ
and the map τ ~»· ΈοΠ(τ) is continuous, because Gi is integrably
bounded. Fix жо € Q and let ί G [a, c] be such that xq € coH(t) and
for every ί < ί, жо £ WTZ{t).
■ Then жо is a boundary point of coTZ(t). Consequently it belongs
to an extremal face Q' of со (ja G\dt\ of dimension smaller than k.
Since со (/J Gdt) С со (J^ Gdt) and Q' с Q, (8.15) imply that
Q' is also an extremal face of со (Ja Gdt]. Thus there exist a = to <
ti < ... <tk = t and a selection g of G such that /^ t/(i)di = жо and
for all 0 < г < к - 1, j£+1 s(i)di is an extremal point of со /t*i+1 Gdi.
Set *
_ J 5(0 if ΐ€[α,ΐ]
90{l) ~ \ 0 otherwise
Then 50 € £ and f£go(i)dt = x0. Furthermore, since
) Γ Gdt = со I Gdt + со / Gdt
Ja Ja- Jt
CO

346 8- Measurability and Integration of Set-Valued Maps
Ji 9o(t)dt is an extremal point of cojf Gdt. □
Remark — Note that if xq is not an extremal point of Q, then
we have a choice of e in the proof above and, therefore, such xq can
be reached by, at least, two different piecewise extremal elements.
D
8.9 Invariant Measures L· Poincare's
Recurrence Theorem
8.9.1 Linear Extension of Set-Valued Maps
Let X and Υ be two compact metric spaces and B(X) and B{Y)
their Borel σ-algebras. We recall that the dual C*(X) of the space of
continuous functions is isomorphic to the space of Radon measures
on X and that a continuous single-valued /:1нУ can be extended
to a continuous linear operator Τ from C*(X) on X to C*(Y) by the
formula
V^C*(I), \/BGB(Y), Τ(μ)(Β) := μ(Γ\Β))
This fact can be extended to set-valued maps F : X ~»· Y. We
denote by V(X) CC*(X) the (weakly compact convex set) of
probability measures on X.
Definition 8.9.1 Let F : Χ ~»· Υ be α set-valued map. Denote by
Τ, the linear extension of F — the set-valued map from V{X) to
V(Y) defined in the following way:
ν G V(Y) belongs to Τ(μ) if and only if
VBeSQO, v(B) < μ{Ρ-\Β))
We extend it as a set-valued map from C*(X) to C*(Y) by setting
0 if /x is nonpositive,
Τ(μ) := I {0} if/A = 0
μ{Χ)Τ{μ/μ{Χ)) if μ is positive

8.9. Invariant Measures and the Poincare Recurrence Theorem 347
Proposition 8.9.2 Consider compact metric spaces Χ, Υ and a
closed set-valued map with nonempty values F : X ~»· У. Then Τ is
a closed convex process with nonempty values.
Furthermore, for any μ G V{X), ν belongs to Τ{μ) if and only if
for every open subset OcY, ν{Ό) < μ{Ρ"ι{0))
The closed convex process Τ is a (set-valued linear) extension of F in
the sense that for Dirac measures, 6y G Τ(δχ) if and only if у G F(x).
If G : Υ ~»· Ζ is a closed set-valued map with nonempty values
from Υ to a compact metric space Z, then the extension Η of the
product Η := GoF contains the product of the extensions: QoT С Ή,-
Proof — Let us consider a measure μ G V{X). The image Τ'(μ)
is not empty, Indeed, by the Measurable Selection Theorem 8.1.3, F,
being upper semicontinuous with closed images, is measurable, so
that there exists at least one measurable selection / of F. Define Vf
by the formula Vf{A) := μ($~~ι(Α))^ which is a probability measure.
Since f-\A) с F~l{A), we infer that μ{Γ\Α)) < μ^-^Α)) so
that Uf belongs to Τ(μ).
We prove now that Τ(μ) can be defined as the set of measures
ν € V{X) satisfying
for every open subset О с У, ν{Ό) < μ{Ρ"ι{Ό))
. We first extend this formula to compact subsets К с Υ. Since
the graph of F, and thus, of F~l, is closed and X is compact, F~l
is also upper semicontinuous. We then know that for any
neighborhood On D F~l{K), there exists an open neighborhood Mn D К
satisfying F~l{Mn) С Оп. Let us choose open subsets On such that
μ{Οη) \ μ{ρ-\Κ))
Hence inequalities
v{K) < v(Mn) < μ{ρ-\Μη)) < μ{Οη)
imply by going to the limit that v{K) < μ{Ρ~~ι{Κ)).
Take now any measurable subset В G B(Y). There exists a
sequence of compact subsets Kn с В such that
v[Kn) / v(B)

348 8- Measurability and Integration of Set-Valued Maps
Then inequalities
HKn) < μ(Ρ-\Κη)) < μ(ρ-1(Β))_
imply that v{B) < μ(Ρ~1(Β)).
Assume that у G F(x). Then 6y G Τ{δχ) since, for any open
subset О CY, 6y(0) < 5x{F~l{0)). This is obvious when у £ О.
If not, the left-hand side is equal to 1, and so is the right-hand side,
because
χ G F-\y) С F~\0)
Conversely, if у £ F(x), there exists an open subset О Э у such that
F(x)nO = 0, i.e., χ 0 F~l{0). ТЬеп^(б>) = 1ααάδχ ((F"1^))) =
0, so that δυ £ Τ{δχ).
Formula Q ο Τ С Ή. is obvious as well as convexity of the graph
oiT.
It remains to prove that Graph(^r) is closed when the spaces' of
Radon measures are supplied with the weak-* topology.
For that purpose, consider a sequence of measures (μη, νη) G
Graph(^") converging to (μ, ν) in the weak-* topologies of the duals
C*{X) and C*(Y) respectively.
It is sufficient to prove that the graph of the restriction of Τ to
V{X) is weakly closed. Indeed, when the measures μη and vn G
Τ(μη) are positive and converge weakly to μ and v, subsequences of
the probability measures ~μη := μη/μη(Χ) and
Vn ··= νη/μη{Χ) € Ρ(μη)
converge weakly to probability measures Д and V G Τ(β) respectively,
because the sets of probability measures are weakly compact and
the graph of the restriction of Τ to V{X) is assumed to be weakly
closed. Since the measures μη{Χ) converge to μ(Χ), we deduce that
μ = μ(Χ)μ~ and ν = μ{Χ)ν. Then ν = 0 when μ = 0 and otherwise,
ν = μ{Χ)ν G μ(Χ)Τ(μ/μ(Χ))
Hence we consider a sequence (μη, un) G V{X) χ V(Y) of the
graph of Τ converging to (μ, и). In order to prove that ν G Τ{μ), it
is enough to check that for any open subset Ό С X, inequality
u{0) < μ{Ρ-\θ))

8.9. Invariant Measures and the Poincare Recurrence Theorem 349
holds true thanks to the first part of the proposition.
Fix an open subset О С X. Since X is compact, О is the union
of an increasing sequence of open subsets Op С Ο, ρ > 1 such that
for every ρ > 1, Ov С О. Let us fix ρ and observe that F~1(Op)
is compact, as the image of a compact by the upper semicontinuous
set-valued map F-1.
Inequalities
Vn>l, i/„(Op) < μη(^-1(0Ρ)) < /^(V1^))
imply, thanks to a version of Alexandroff's Theorem recalled just
after the end of the proof, that
j v(Op) < liminfn-^ooi/nCOp)
\ < limsupn_wMn(i'-1(^)) < μ{ρ-\θρ)) < μ{Ρ-\Ό))
It remains to observe that v{0) = supp>1 v{Op) to conclude. D
We prove now the version of Alexandroff's Theorem (see [148,
Theorem 4.9.15]) we needed:
Theorem 8.9.3 (Alexandroff) Let X be α compact metric space.
Consider a sequence of Radon probability measures μη £ V(X)
converging weakly to μ. Then, for any open subset О С Х,
μ(0) < Hminf/infO)
η—► со
and for any closed subset К С X,
limsup^n(ii) < μ{Κ)
π-too
Proof — Let О С X be an open subset. Since a Radon measure μ
is regular, we can associate with any e > 0 a compact subset К С О
such that μ{Ό\Κ) < ε.
Let / G C(X) be a continuous function equal to 1 on the closed
subset К and to 0 on the complement of O. We observe that χκ <
f < Xo and thus, that
μ(0) < μ{Κ)+ε < ί ίάμ + ε

350 — 8- Measurability and Integration of Set-Valued Maps
Therefore, inequalities / /άμη < μη(0) imply that
μ(0) < [ fdu + ε < Uminf^n(0)+e
J η—►со
We conclude by letting e converge to 0.
The proof of the second inequality for compact subsets К is
analogous, since.for any e > 0, there exists an open О D К such that
μ{Ό\Κ) < ε. Define the continuous function / as above. Inequalities
μη{Κ) < J ίάμη imply that
limsup^n(UT) < ί ίάμ < μ{Ό) < μ(Κ) + ε □
п—юо J
Remark — We deduce that for any open subset О such that
μ(0) = μ(0), weak convergence of probability measures μη to μ
imply that μη{0) converges to μ(Ο). The converse is also true. (See
the proof of [148, Theorem 4.9.15]) for instance.) □
8.9.2 Invariant Measures
Taking X = Υ, we are able to derive the existence of invariant
measures by showing that the extension Τ has fixed points on V{X).
Theorem 8.9.4 Let X be α compact metric space and F:l-^1
be a closed set-valued map with nonempty values. Then there exists
an invariant probability measure μ, i.e., satisfying
VA£ B{X), μ{Α) < μ{Ρ-\Α)) (8.16)
Proof — Let Τ be the set-valued map from V{X) to itself
associating with any μ G V{X) the (nonempty) set of probability
measures ν G ^Ρ(μ). Then the above proposition states that Τ is a
map with closed graph and convex values from the convex compact
subset V{X) С С*(Х) to itself, and thus, upper semicontinuous with
convex compact values. Kakutani-Fan's Fixed-Point Theorem 3.2.35
5Recall that the proof we provided, based on the Ky Fan inequality, showed
that Ky Fan's theorem remains true in locally convex Hausdorff topological vector
spaces.

8.9. Invariant Measures ала the Poincare Recurrence Theorem 351
implies the existence of a fixed-point μ G V{X) of T, which is a
measure satisfying
VAeB(I), μ(Α) < μ(Ρ-\Α)) (8.17)
Therefore, μ is invariant by F. □
Naturally, we can derive most of the properties of invariant
measures enjoyed by single-valued maps. Let us show for instance that
Poincare's Recurrence Theorem holds true for closed set-valued
dynamical systems.
Theorem 8.9.5 (Poincare's Recurrence) Let X be α compact
metric space, F : X ~»· X be a closed set-valued map and μ G V(X)
an invariant measure of F. For any Borelian subset В С X, let
Soc := Π U Р~П(В)
Ν>0 η>Ν
be the subset of points χ such that for all N, there exists η > N such
that F~n(x) Π Β φ 0. Then the measure of В П Дх> is equal to the
measure of B.
Proof— The proof is a straightforward extension of the proof
in the single-valued case: We introduce the subsets
BN := (J F~n(B)
n>N
and we observe that В С Bq, that · · · С Дл/· С Β^-ι С · · · С В0
and that BN = F-N{B0).
Since μ is invariant, we deduce that
μ{Β) < μ{Ρ~\Β)) < ... < μ{Ρ~Ν{Β)) < ···
and thus, that
μ(Β0) < μ(Ρ-Ν(Β0)) = μ(ΒΝ) < μ(Β0)
Since the sequence of Β ν is not increasing and since μ(-Β/ν) = μ(-Βο),
we infer that μ(-Βοο) = μ(Βο)· Therefore,
μ{ΒΠΒ00) = μ(ΒηΒο) = μ{Β) D

Chapter 9
Selections and
Parametrization
Introduction
One of the first questions of set-valued analysis was how to relate
set-valued and single-valued maps, mainly in the hope of avoiding
the necessity of dealing with set-valued maps.
This can be done in two ways:
1. — Find a selection / : Jf н У of a set-valued map
F : X ~»· Y, i.e., a single-valued map / satisfying
Vi£X, f(x) € F(x)
2. — Find a parametrization (U, /), where U is a control or
parameter space and f : X xU ι-+ Υ is & single-valued map satisfying
Vx € X, F(x) = {f(x,u)}ueu
Naturally, we require that in both cases the selection / or the
parametrization (U, f) inherits the regularity properties of F, such
as continuity or Lipschitzianity.
Furthermore, keeping in mind applications to control theory, we
shall also study the case when both F and / are time-dependent in
a measurable way.
353

354
9 - Selections and Parametrization
We proved in the preceding chapter the existence of a measurable
selection of a measurable map. When we ask for more regularity of a
selection, we meet more technical difficulties. This is why continuous
(or Lipschitz) selections are associated with continuous (respectively
Lipschitz) set-valued maps with closed convex images.
These selections should be obtained in a constructive way
whenever this is possible.
The most natural constructive way amounts to using the minimal
selection, i.e., the selection / with minimal norm. Unfortunately, this
procedure does not preserve semicontinuity, but only continuity for
maps with closed convex images, as one can see in Section 3. Such
selection is also Caratheodory for Caratheodory maps with closed
convex images (see Section 5.)
We shall prove the existence of continuous selections of lower
semicontinuous maps with closed convex images (Michael's Theorem)
in Section 1, and the existence of approximate continuous selections
(CelUna's Theorem) of upper semicontinuous maps with closed
convex images in Section 2.
If the set-valued map is Lipschitz continuous, then, contrary to
an intuitive wish, it may happen that its minimal selection is not
Lipschitz.
There exist different ways to get Lipschitz selections for
Lipschitz set-valued maps with convex images. The barycentric selection,
which also comes easily to mind, is Lipschitz, but difficult to handle.
By constrast, the selection based on the Steiner point1 enjoys
attractive properties. For instance, it preserves measurability,
continuity and Lipschitzianity of set-valued maps.
We first single out a point of a nonempty convex compact set
'The Steiner point of a convex compact set, known for over a century, has
many important properties. It is defined for subsets of Euclidean space R".
Jacob Steiner (1796-1863) introduced this concept in 1840. From a peasant
family, he had only a village school education before coming late in life to
Mathematics. He had a strong preference for geometrical proofs over analytical ones.
Actually, he never provided any of them and succeeded however in keeping his
audience of Berliner Universitat awake without drawing any pictures. He was
overshadowed by Diersteweg, in Mors, who purposefully draws the curtains of
the class room to darken it before his lectures (...die Stube audi noch ausdriick-
lich verfinsterte!.) Certainly he intended to help his listeners to dream freely of
geometrical (with no doubts curvaceous) shapes...

9.1. Case of Lower Semicontinuous Maps
355
К С Rn, the Steiner point sn(K), and characterize it.
The Steiner selection sn(·) being Lipschitz in the Hausdorff
metric, the single-valued map χ н-> sn(F(x)) is a Lipschitz selection
whenever the Lipschitz set-valued map F : X ~»· Rn has convex
compact images.
.To be able to deal also with unbounded set-valued maps, we prove
in Section 4 an Intersection Lemma, which enables us to associate
with a closed convex set its convex compact subset in a Lipschitz
way. Hence we obtain in a constructive way Lipschitz selections for
Lipschitz set-valued maps with closed convex images.
The Steiner selection and the Intersection Lemma allow us to
find selections (Section 5) and parametrizations (Sections 6 and 7)
of continuous, Caratheodory and Lipschitz maps with closed convex
images, preserving regularity of the set-valued map under
consideration.
9.1 Case of lower semicontinuous maps
We shall prove the celebrated Michael's theorem stating that lower
semicontinuous convex-valued maps do have continuous selections.
Definition 9.1.1 Consider α set-valued map F : Χ ~»· Υ with
nonempty images. A single valued map f : X н-> У is called a selection
of F if for every χ G X, f(x) € F(x).
Theorem 9.1.2 (Michael's Theorem) Let F be a lower
semicontinuous set-valued map with closed convex values from a compact
metric space X to a Banach space Y. It does have a continuous
selection.
Corollary 9.1.3 Let F be as in Theorem 9.1.2. Consider a subset
К С X and a continuous single-valued map φ : Κ ι—» Υ such that
for every χ G Κ, ψ(χ) G F(x).
Then ψ can be extended to a continuous selection of F, i.e., F
has a continuous selection f (defined on the whole space X) such
that the restriction of f to К is equal to φ.
In particular this yields that for every у G F(x) there exists a
continuous selection f of F such that f(x) = y.

356
9- Selections and Parametrization
Proof — It is enough to consider the lower semicontinuous set-
valued map
G(x) := J ρ(χ) if x i K
\ {ψ{χ)} otherwise
and to apply Theorem 9.1.2. D
The proof of this Theorem requires the following Lemma.
Lemma 9.1.4 Let F be α lower semicontinuous convex-valued map
from a compact metric space X to a Banach space Υ. Then for each
ε > 0, there exists a continuous function f£ satisfying
Vxel, fe(x) € B(F(x),e)
Proof — Since F is lower semicontinuous, we can associate
with any χ £ К and any yx G F(x) an open neighborhood Ux such
that
Vx'£ Z4, B(yx,e)nF(x') φ 0
Since X is compact, it can be covered by η such neighborhoods UXi.
Let us consider a continuous partition of unity2 {ai}i=i,...,n associated
with this covering, i.e. ai : X i—> [0,1] are continuous functions such
that ai vanishes outside of UXi and
η
V χ G X, ^ai(x) = 1
i=l
Define /e by:
71
It is continuous since the functions ai are continuous. Let
I{x) := {i = 1,.. .,n\ αι{χ) > 0}'
It is not empty since Y2=x ai(x) = 1. When i € I{x), then χ belongs
to UXi and thus,
yXi€B(F(x),e)
2 See Chapter 3, Section 1.

9.1. Case of Lower Semicontinuous Maps
357
The set B(F(x),e) being convex, we infer that fe{x) belongs to
B(F(x),e.) □
We are now ready to prove the Michael Theorem.
Proof of Theorem 9.1.2 — We shall construct by induction
a sequence of continuous maps ип:ХиУ, n=l,... satisfying the
following properties:
' i) Vx € X, d{un{x),F(x)) < 2~n
(9.1)
^ и) Vx € X, ||un(aO-iin-i(aO|| < 2~n~1
For η = 1, we apply the lemma with e = 1/4. Assume that we
have constructed the maps un up to η and let us construct un+\. We
introduce for that purpose the set-valued map Fn+\ defined by
Vx€X, Fn+1{x) :=B{un{x),2-n)nF{x)
By (9.1) i), Fn+i(x) is convex and nonempty. The set-valued map
Fn+i is lower semicontinuous: Indeed, if xp converges to χ and if
у belongs to Fn+i(x), by the lower semicontinuity of F, there exist
elements yp G F(xp) converging to y. For ρ large enough, we see that
||уР-Ип(я:р)|| < 2~n
because и is continuous, and therefore, that yv belongs to Fn+i{xp).
This being established, we deduce from Lemma 9.1.4 the
existence of a continuous map un+i satisfying:
\/x € X, d{un+1{x),Fn+1{x)) < 2-(n+1)
which, together with the definition of Fn+i, implies (9.1) for η + 1.
We infer from inequalities (9.1) ii) that the sequence of functions
un is a Cauchy sequence in the Banach space of continuous maps
from the compact space X to the Banach space Y. They converge
uniformly to a continuous map и : Χ ι—> Υ. Inequalities (9.1) i)
imply that for all χ € X, u(x) belongs to F(x), for this subset is
closed by assumption. Hence и is the continuous selection we were
looking for.

358
9- Selections and Parametrization
9.2 Case of upper semicontinuous maps
Upper semicontinuous set-valued maps in general do not have
continuous selections even when their values are closed and convex. The
following example illustrates this issue:
Example Consider the upper semicontinuous set-valued map
F : R ~»· R defined by
F(x) := <
' { -1 } if χ < 0
[-1,1] if χ = 0
{ 1 } if χ > 0
Clearly F does not have any continuous selections defined on R. D
This is why it is more realistic to speak about approximate
selections. We state the following result due to A. Cellina:3
Theorem 9.2.1 Let F : Χ ~»· Υ be an upper semicontinuous map
from a compact metric space X to a Banach space Y. If the values
of F are nonempty and convex, then for every ε > 0, there exists a
locally Lipschitz single valued map fe : X н-> У such that
Graph(/e) С £(Graph(F),e)
and for every χ £ X, fe{x) belongs to the convex hull of the image
ofF.
Remark — This inclusion implies that the graphical upper limit
of the approximate selections is contained in F. □
Proof— Fix e > 0. Since F is upper semicontinuous, for every
χ £ X, there exists 0 < δχ < 2e such that
Vy e Ή(χ,δχ), F(y) С F{x) + E-B
3We prove here the approximate selection result only for a compact metric
space X, but the result is valid with a similar proof for any metric space X■ It
can be proved using paracompactness of metric spaces and the fact that a locally
Lipschitz partition of unity may be subordinated to any locally finite covering of
a metric space. Details can be found in [33, Section 1].

9.2. Case of Upper Semicontinuous Maps 359
Figure 9.1: Approximate Selection of ал Upper Semicontinuous Map
.
0
GraphCf^
—^
СгатЛС.Е}-—"
<- Graph(5(F,e))
Approximate Selection in a Neighborhood of Graph(F)
The family of balls {В {х,йх/4)}хех covers X. Since X is compact,
there exists a finite sequence / of indices such that the family of balls
{B (xiJXi/4)}iei covers X.
We set δ{ = δχ; and take a locally Lipschitz partition of unity {щ}
subordinated to this covering4. That is a,- : X ь-> [0,1] are locally
Lipschitz functions such that for all г 6 /, α,- vanishes outside of
B(xi,S{/4) and
\f χ £ X, Y^ai(x) = 1
iei
Let us associate with every г ζ I & point τχ £ F(B(xi,Si/4)) and
define the map /e : X ь-> Υ by
iei
Then /e is locally Lipschitz and for every χ £ X, fe(x) £ coF(X).
To show that /e is the desired approximation, fix χ £ X and
let I{x) denote the subset of all г £ I such that сц{х) ф 0. Thus
4See for instance [33, Theorem 1.1.2].

360
9- Selections and Parametriz&tion
χ € B{xi,8ijA) for every i €. I(x). Therefore for all i,j € I{x)
d(xi,Xj) < d(x,Xi) + d(x,Xj) < (<5i + <5j)/4_
Fix к € I(x) satisfying
6k = max δ{
i€l{x)
Then the latter inequality implies that
\fi€l{x), Xi£B(xk,6k/2)
Consequently
yi € F(B(xi,6i/A)) С F(B(xk,6k)) С F(xk) + |β
Since the right hand side of the above relation is convex, from the
definition of /e follows that
fe(x) € F(xk) + £-B
Pick zk G F{xk) satisfying ^\zk — fe{x)\\ < ε/2. Thus, by the choice
of δχ and zk,
d((x,fe(x)),(xk,zk)) < d(x,xk) + ||^-/е(ж)|| < — + - < e
which ends the proof. □
9.3 Minimal Selection
Consider a metric space X, a Banach space Υ and a set-valued map
F : X ~> Y.
We define the minimal map
m(F(x)) := \ и € F(x) I |Ы| = min \\y\\ \ (9.2)
j_ y€F{x) J
When У is a Hilbert space, or, more generally, a reflexive strictly
convex space5 and F has closed convex images, the minimal map is
single valued. In this case it is called the minimal selection.
5A Banach space Υ is called strictly convex if for all ж, у 6 Υ which are not
colinear, we have ||ж + 2/|| < ||ж|| + \\y\\.

9.3. Minimal Selection
361
A quite natural question arises:
What are continuity properties of the minimal selection'?
We prove below that if F is continuous and takes its values in a
compact subset of a strictly convex reflexive Banach space, then the
minimal selection is continuous.
The upper semicontinuity (or lower semicontinuity) of F, even
when it takes convex values in a compact subset, is not strong enough
to imply the continuity of the minimal selection, as the following
examples show:
Example — We consider the set-valued map F : R ~»· R
defined by
F(x)-= I {2} if x*°
*{h \ [1,2] if x = 0
It is upper semicontinuous with compact convex values and its
minimal selection is obviously not continuous at zero. Π
Example — We consider the set-valued map F : R ~»· R
defined by
F(x).= { [o,i] if χϊο
*{X)- \ {1} if x = 0
It is lower semicontinuous with compact convex values and its
minimal selection is not continuous at zero. □
First of all we investigate continuity properties of the function
d(0,F(-)).
Lemma 9.3.1 Let F be a set-valued map from a metric space X to
a Banach space Υ with nonempty images. Set
\\F(x)\\ = sup Иг/11
yeF(x)
1. — If F is lower semicontinuous (or upper
semicontinuous), then the function χ *—» d(0,F(x)) is upper semicontinuous
(respectively lower semicontinuous.)
2. — If F has bounded images and is lower semicontinuous
(or upper semicontinuous), then the function χ н-> ||-F(a:)|| is lower
semicontinuous (respectively upper semicontinuous.)

362
9— Selections and Par&metriz&tion
Consequently if F is continuous, then so is d(0,F(-)) and if in
addition F has bounded images, then ||-Ρ(·)|| *s also continuous.
Furthermore if F is Lipschitz with a Lipschitz constant c, then
d(0, F(·)) is c-Lipschitz. If moreover F has bounded images, then
\\F(-)\\ is ah о c-Lipschitz.
In the above c-Lipschitz means Lipschitz with the constant с
Proof — If F is lower semicontinuous^hen, by Corollary
1.4.17, the function χ ι-> d(0,F(x)) is upper semicontinuous and,
in the case when F has bounded images, the function χ ι-> ||F(a;)||
is lower semicontinuous by Theorem 1.4.16.
If F is upper semicontinuous, then for every ~x € Χ, ε > 0 there
exists a neighborhood λίε of χ such that for all χ € λίε, we have
F(x) С F(x) + eB. Hence
d(0,F(x) > d(0,F(x))-e
and therefore d(0,F(·)) is lower semicontinuous. On the other hand
if F(x) is bounded,
||F(x)|| < \\F(x)\\+e
This yields that ||-Р(-)|| is upper semicontinuous whenever images of
F are bounded.
The latter statement follows from the very definition of Lipschitz
set-valued maps. □
Proposition 9.3.2 Let F be a lower semicontinuous set-valued map
from a metric space X to a Banach space Υ whose graph is closed.
If the minimal map m(F(·)) has nonempty images, then its graph is
closed.
Consequently if Υ is strictly convex, reflexive, F has nonempty
closed convex images and the range of m(F(·)) is contained in a
compact subset ofY, then the minimal selection is continuous.
Corollary 9.3.3 Let F be a continuous set-valued map from a
metric space X to Rn with nonempty closed convex images. Then the
minimal selection is continuous.

9.3. Minimal Selection
363
Proof— Lemma 9.3.1 implies that ||d(0, F(-))\\ is continuous. Fix
x € X. Then for some δ > 0 the set
{\\m(F(y))\\}yeB(F(x),6)
is bounded. Proposition 9.3.2 implies that m(F(·)) is continuous at
x. The element χ € X being arbitrary, the proof follows. □
Proof of Proposition 9.3.2 — Indeed by Lemma 9.3.1 the
map
χ ^ d{0,F{x)) = \\m{F{x))\\
is upper semicontinuous. This yields that the set-valued map
χ ~> B(0,\\m(F(x))\\)
has closed graph. By our assumption the graph of F is closed. Thus
the intersection map
χ ~> F(x)nB(0,\\m(F(x))\\)
has closed graph.
On the other hand
m(F(x)) = F(x)nB(0,\\m(F(x))\\)
and the proof of the first statement ensues.
To prove the second statement it is enough to observe that a
single-valued map taking its values in a compact set is continuous if
and only if its graph is closed. □
When У is a Hubert space, then the compactness assumption in
the above proposition may be replaced by the upper hemicontinuity
of F:
Theorem 9.3.4 Consider α metric space X, a Hubert space Υ and a
lower semicontinuous set-valued map F : Χ ^λΥ with closed convex
values. If F is upper hemicontinuous, then the minimal selection is
continuous.

364
9 - Selections and Par&metriz&tion
Proof — We identify Υ with its dual... The projection of 0
onto the closed convex set F(x) is the element „
и := m(F(x)) € F(x)
such that
IMI2 + ^(-F(x),u) = sup <u-0,u-y>-< 0 (9.3)
ye-F(i)
(It is actually equal to 0.) Fix e > 0 and χ € X. Since F is upper
hemicontinuous there exists a neighborhood" V of χ such that for all
x' € V
' (-m(F(x')), m(F(x))) < a(-F(x'),m(F(x)))
<
w < a(-F(x),m(F(x))) + ε < -\\m(F(x))\\2 + ε
(thanks to (9.3).)
On the other hand, by Lemma 9.3.1, the function χ н-► ||m(-F(a;))||
is upper semicontinuous, since F is lower semicontinuous. Then there
exists another neighborhood W of χ such that
\/x' €W, \\m{F{x'))\\2 < \\m{F{x))\\2 + ε
These two inequalities imply that for all x' € V Π W
' ||m(iV)) - m(F(x))f
« = limbs'))!!2 + \\m(F(x))\\2 + 2 <-m(F(x')),m(F(x))>
k < 2\\m{F{x))\\2 + ε + 2(-||^(ζ)||2 + ε) < Зе
Therefore, the minimal selection is continuous. □
9.4 The Steiner Selection
To construct Lipschitz selections of set-valued maps and to
parametrize Caratheodory set-valued maps, we need a selection procedure
associating with a closed convex set К С Rn a point s(K) € К in

9.4.1. Steiaer Points of Сол vex Compact Sets
365
a regular enough way. We start this section with such a selection
procedure for convex compact sets K.
To be able to deal also with unbounded sets, we prove an
Intersection Lemma which enables us to associate with a closed convex set
К a convex compact subset P{K) С К in a Lipschitz way. The last
part of this section is devoted to Lipschitz selections of set-valued
maps with closed convex images.
9.4.1 Steiner Points of Convex Compact Sets
For a nonempty convex compact subset К of Rn, we define its Steiner
point or, for the sake of simplicity, Krummungsschwerpunkte, (also
called sometimes the curvature centroid) sn(K) by:
' for η = 1, Sl{K) = σ(Κ,+ΐ)/2 - σ{Κ,-ΐ)/2
<
for η > 2, sn{K) = η^η~ιρσ(Κ,ρ)ω(άρ)
where Σπ_1 denotes the unit sphere in Rn, σ(Κ, ·) is the support
function of Κ, ω is the measure on Σπ_1 proportional to the Lebesgue
measure and satisfying ω(Ση~1) = 1.
' Since σ{Κ,ρ) = σ(-Κ, -ρ), it follows that sn{K) = —sn(—K).
The support function being also additive with respect to K, the map
δπ(·) is linear:
For all convex compact K, L С Rn and all λ, μ € R,
<
δπ(λΛΓ + μ^) = Asn(ur)+^sn(L)
We also observe that if К is symmetric, i.e., К = —К, then
sn{K) = 0. \
We show below that sn is a selection in the sense that sn(K) € К
and that it is Lipschitz with respect to the Hausdorff distance:
We recall that the extended Hausdorff distance between two closed
subsets К-, L С Rn (which may be equal to +oo when К or L is

366
9 - Selections and Paxametrization
unbounded or empty) is denned by6
H(K,L) := max< sup d(x,L), sup d(x,K) >
l хек xeL J
The Steiner point can be alternatively defined by using the sub-
differentials of the support function of a compact convex subset K.
Recall that the subdifferential да(К,р) of the support function
σ(Κ, ■) is given by
θσ(Κ,ρ) = {χ € K\ <p,x>= σ(Κ,ρ)}
We denote by т(да(К,р)) the element of да(К,р) with the
minimal norm.
The function σ(Κ, ·) being continuous, Theorem 8.2.9 implies
that the subdifferential θσ(Κ, ·) is measurable.
Hence т(да(К,р)) being the projection of 0 onto δσ(Κ,ρ), the
single-valued map т(да(К, ·)) is also measurable (Corollary 8.2.13.)
Therefore, the formula stated in the following theorem has a meaning:
Theorem 9-4.1 Let /C denote the family of all nonempty convex
compact subsets o/Rn. Then
УК € /С, sn(K) = Щщ/Вп™(да(К,р))ар
where Vo\(Bn) is the measure of the η-dimensional unit ball Bn С
Rn.
Consequently,
\/K € /C, sn{K) € К
Furthermore sn(·) is Lipschitz with the constant n:
VK, L €/C, \\sn(K) - sn{L)\\ < nH(K,L)
6It can be verified easily that Ή is a distance. Furthermore, a set-valued map
with nonempty compact images is continuous if and only if it is continuous with
respect to the Hausdorff distance.

9.4.1. Sterner Points of Couvex Compact Sets 367
Proof— Fix if, LcRn.
— When η = 1, set
2/ = σ(Χ,+ΐ), ζ = -σ(Κ,-1)
Then y,z Ε Κ and
*i(*0 = \y+\z Ζ Κ
Furthermore
vohky /Bl m (δσ(χ> p)) *=5 (Γ **+£ **)=^+r
and
' ||Sl(X) - Sl(L)||
<
^ < 1\σ(Κ,+ΐ) - a(L,+l)\ + \\a(K,-l) - a(L,-l)\ < H{K,L)
— We assume next that η > 2. Consider the Moreau-Yosida
approximation σ\(Κ, ·) of the support function σ(Κ, ■) of K. Recall
that it is continuously differentiable. By integrating its gradient on
the unit ball Bn whose boundary is the unit sphere Σ71"1, Stoke's
formula implies that
/ Vax(K,p)dp = / σχ(Κ,ρ)ρμ(άρ)
where μ is the Lebesgue measure on the unit sphere Σπ_1.
We also recall that σ\{Κ, ·) satisfies
-\\K\\ < inf ΜΚ,ρ) < σχ(Κ,ρ) < σ(Κ,ρ) < \\K\\
(where ||ΛΓ|| := sup^g^ \\x\\) and converges pointwise to σ(Κ, ·) (see
Theorem 6.5.7.) Therefore
lim / ρσχ(Κ,ρ)μ(άρ) = Ι ρσ(Κ,ρ)μ(άρ)
λ->0+ J^n-i JT,n~i
On the other hand, by Theorems 3.5.9 and 6.5.7, Vax(K, ·) is the
Yosida approximation of the sub differential θσ(Κ, ·). Hence
Vax(K,p) € θσ(Κ,ρχ) с К

368 9 — Selections and Paxametrization
where p\ is the minimizer of the function
q н-> a(K,q) + —\\q-pf
and for any ρ € Bn, Va\(K,p) converges to т(да(К,р)). Therefore,
the maps Va\{K, ·) being measurable and bounded by \\K\\, we infer
that
Urn / Vax(K,p)dp = / τη{θσ{Κ,ρ)) dp
λ->Ό+ Jb-"- JBn
Dividing by Vol {Bn), and setting ω (dp) := μ(άρ)/ω(Ση~1), we obtain
Since ω(Ση~1) = nVo\(Bn), we derived the formula we were looking
for.
Proposition 8.6.2 and Theorem 8.6.3 yield that
/ К dp = Vo\{Bn)K
JB71
Consequently
™p=)/«. "■<*'<*■'»* ε й^)//* = K
To prove that sn(·) is n-Lipschitz observe that for every ρ € Σπ-1
VK, L € 1С, \a(K,p)-a{L,p)\ < H{K,L)
Hence
\\sn(K) - sn(L)\\ < η!^1\σ(Κ,ρ)-σ(£,ρ)\\\ρ\\ω(άρ)
w < пЩК,Ь)ш{Т,п~1) = nH{K,L)
The proof is complete. □

9.4.2. The Intersection Lemma
369
9.4.2 The Intersection Lemma
To obtain Lipschitz selections for unbounded closed convex sets we
need the following technical lemma.
Lemma 9.4.2 Let /C denote the family of all nonempty closed
convex sets in Rn. Then the map Ρ : Rn χ JC ^* 1С defined by
P(y,K) := КпВ(у, 2d(y,K))
is Lipschitz: for all K,L € /C and x, у € Rn
H(P(x,K),P(y,L)) < 5(H(K,L) + \\x - y\\)
Proof — First we show that our statement holds true with χ =
2/ = 0:
\/K,L € /C, H{P(0,K),P(0,L)) < 5H{K,L) (9.4)
Pick K,L in /C such that H{K,L) < +oo and set e := H(K,L). If
e = 0, then К = L and (9.4) holds true.
Suppose next that e > 0 and let у € P(0,L) be arbitrary but
fixed.
To prove inequality (9.4), we need to find a point χ in P(0,K)
such that
\\x-y\\ < 5e (9.5)
Observe first that
||y|| < 2d(0,L) < 2d(0,K) + 2e (9.6)
Let x\ € К be such that
||?/-a;i|| < e
If ||xi|| < 2d(0,K) then set χ := x\. Otherwise let X2 € К be such
"that
d{0,K) = \\x2\\
If x2 = 0 set ж = 0. Then (9.6) implies (9.5.)
It remains to consider the case
||xi|| > 2d{0,K), x2 φ 0

370
9 — Selections and Paxametrization
Figure 9.2: Illustration of the proof
We claim that there exists χ in the interval \x\, x%\ such that
||x|| = 2d(0,K)
Indeed define the continuous map ψ : [0,1] ι-> Rn by
VA € [0,1], ψ{\) = \\\Xl + (1-λ)χ2||
Since the image of φ is a connected set and
ψ{0) = d{0,K) L· φ(ΐ) > 2d{0,K)
there exists λ € ]0,l[ which satisfies φ(λ) = 2d(0,K). So setting
χ := \xi + (1 — X)x2, our claim follows. The set К being convex,
x€P{0,K).
We prove next that this χ is the point we are looking for.
Indeed, by the choice of x\,
||яг — £Ci|| > \\x — y\\ — \\y — xi\\ > \\x — y\\ — ε (9.7)
In the triangle with vertices at 0, x, X2 denote by α the angle at x

9.4.2. The Intersection Lemma
371
(see Figure 9.2.) Inequality ЦжгЦ < ||x|| yields that α € [0, f [ and
^ II^H 1 \/3
sine* < y-~ = -, cosa > — (9.8)
If α = 0 (that is always the case when η = 1), then
ll^ill = \\x\\ + \\x ~~ χι\\
and (9.7) yield
\\y\\ > H^ill-||y _ χι\\ > \\χ\\+\\χι — χ\\~ε > 2d(0,K)+\\x — y\\—2e
Thus we deduce from (9.6) that
Ik-2/11 < 4e
and derive (9.5.)
Assume next that α > 0 and consider the triangle with vertices
at 0, x, x\. Then the angle at χ is equal to π — α > ξ. Let ζ €]0, χι[
be such that ζ — χ ± x. Then
||z|| > ||x|| = 2d(0,K) (9.9)
In the triangle formed by points ζ,χ,χι, the angle at χ is equal to
π — α—| = | — α. Therefore (9.8) implies
V3
2Г — χι Ν > \\x — xi||sm ( — — a J = \\x — xi\\ cosa > \\x — хл\\ —
J-ll — II -Ml V П /II -Ml — II J-ll q
(I-)-
and from (9.9) follows that
л/3
\\XU = \\z\\ + \\z ~ XU ^ 2d(0,.K") + —- Цж — xi\\
This and (9.7) imply
л/Я
112/11 > Iki||-||y-si|| > 2d{0,K) + ~{\\x-y\\-e) - ε
Therefore, thanks to (9.6), we derive
л/3
2 Mi1-У|| -e)-e < 2e

372
9 — Selections and Parametriz&tion
and thus (9.5.) So (9.4) is proved.
Pick next any x, у ε Χ and observe that
H(P(x,K),P(x,Ljf^ff(P(0, Κ - x),P(0,L - x))
Hence, using (9.4), we deduce
' H(P(x,K\P(y,L))
< H(P(x,K),P(x,L)) + H(P(x,L),P(y,L))
< m(K-x,L-x) + H(B(x,2d(x,L)),B(y,2d(y,L)))
k < 5H{K,L) + \\x-y\\ + 2\\x-y\\ = 5H(K,L) + 3||s-y||
The proof is complete. □
9.4.3 Lipschitz Selections of Lipschitz Maps
An immediate application of the above construction is the existence
of a Lipschitz selection for a Lipschitz set-valued map:
Theorem 9.4.3 Consider α Lipschitz set-valued map F from a
metric space to nonempty closed convex subsets ofRn. Then F has a
Lipschitz selection f.
Remark — In the next section, we prove a stronger result, which
in particular implies that for any у € F(x) there exists a Lipschitz
selection f οι F such that /(ж) = у. D
Proof — We first observe that a map F : X ~»· Rn is Lipschitz
if and only if there exists c> 0 such that
Vs, у € X, 7i{F{x),F{y)) < c\\x-y\\
Define the set-valued map G : X -^ Rn with nonempty compact
convex images by
Vz € X, G(x) = F(x)nB(0,2\\m(F(x))\\)

9.5. Caratheodory Selections 373
By Lemma 9.4.2 it is Lipschitz. Consider the single-valued map
f(x) = sn(G(x))
where sn(·) is the Steiner selection. The proof follows by the
application of Theorem 9.4.1. □
Remark — Observe that if the set-valued map F is merely
continuous in the Hausdorff metric·.
Ve>0, 3 <5 > 0, Vie Β(χ, δ), H(F(x), F(x)) < ε
then the function / defined in the above proof is a continuous
selection of F. □
9.5 Selections of Caratheodory maps
Prom now on a measurable map always means a Lebesgue measurable
map. Let X be a metric space and а < b be real numbers. Consider
a set-valued map
F : [a, b] χ Χ ~> Rn
Definition 9.5.1 For every t € [a, b] let C(t) С X be a given set.
1. — The map F is called Caratheodory on {C(t)}te[au if
V χ € X, F(-, x) is measurable
<
Vt € [a, b], F(t,·) is continuous on C(t)
2. — The map F is measurable/Lipschitz on {C(£)}ieraw if
for every t € [a, 6], there exists k(t) > 0 such that
V χ € X, F(-tx) is measurable
<
Vt € [a, b], F{t,·) is к(t)- Lipschitz on C(t)
where k(t)—Lipschitz means Lipschitz with the constant k(t).

374
9- Selections and Paxametrization
Let a single valued map ζ : [α, b] н-► X and a measurable single valued
map w : [a, b] i-> Rn be such that-
w(t) € F(t, z(t)) almost everywhere in [a, b]
We refine here the selection result proved in the preceding section:
Namely we show the existence of a measurable/Lipschitz selection
f(t, x) € F(t, x) such that
for almost every t € [a, b], w(t) = f(t,z(t))
whenever F is measurable/Lipschitz.
Results of this type are sometimes needed in control theory when
we investigate the local behavior of solutions for nonsmooth systems.
Theorem 9.5.2 (Caratheodory Selection) Let X be a metric
space, F : [a, b] x X ~»· Rn be a set-valued map with nonempty
closed convex images, ζ : [a, b] ι—» X be a single valued map and
w : [a, b] ι—» Rn be measurable.
If F is Caratheodory on {C(£)}ie[a,6] and
w(t) € F(t,z(t)) almost everywhere in [a, b]
then there exists a Caratheodory selection f of F on {C(i)}ie[a,6]
such that, for almost every t € [a,b], w(t) = f(t,z(t)).
Proof — Define the map / by
V(i,s) € [a,b]xX, f(t,x) = UF{ttX)(w(t))
where Π^(ν) is the projection map from Corollary 8.2.13, i.e., for
every (t,x), HF^tiX^(w(t)) is an element of F(t,x) such that
d(w(t),F(t,x)) = ||nF(ti!B)(ii;(i))-ii;(0
Corollary 8.2.13 implies that for every χ € X, the map f(-,x) is
measurable. It is also clear, that w(t) = f(t, z(t)) almost everywhere
in [a, b]. The last statement follows from Corollary 9.3.3. □
If F(t,·) is Lipschitz, then the selection / may be chosen to be
Lipschitz with respect to x:

9.5. Caratheodory Selections
375
Theorem 9.5.3 (Measurable/Lipschitz Selection) Under all
assumptions of Theorem 9.5.2 assume that the set-valued map F is
measurable/Lipschitz on {С^)}щад and let k(t), t € [a, b] denote
the corresponding Lipschitz constants.
Then there exist с > 0 and a measurable/Lipschitz selection f of
F on {C(£)}te[a,b] such that f(t,·) is ck(t)—Lipschitz on C(t) and
for almost every t € [a, b], w(t) = f(t,z(t))
Furthermore с is independent of F.
Proof — Let Ρ be the set-valued map defined in Lemma 9.4.2
and sn be the Steiner selection.
For every t € [a, b] and χ € Rn set
f(t,x) = sn(P(w(t),F(t,x)))
Since w(t) € F(t,z(t)), by the very definition of the map P, we get
P(w(t),F(t,z(t))) = w(t) almost everywhere in [a, b]
Thus f(t,z(t)) = w(t) almost everywhere.
Theorem 9.4.1 implies that f{t,x) € F(t,x). We deduce from
Lemma 9.4.2 and Theorem 9.4.1 that for every t € [a,b], the map
f(t; ·) is Lipschitz on C{t) with Lipschitz constant 5nk(t).
It remains to show that / is measurable with respect to t. Fix
χ € X. We first show that the map
t ^ Φ(ί,χ) := P(w(t), F{t,x)) = F{t,x)nB(w(t), 2d(w(t),F(t,x)))
is measurable.
Indeed, by Theorem 8.2.4, the intersection of two measurable
maps is measurable. On the other hand F(-,x) is measurable by
assumption and
t ~> B(w{t),2d(w(t),F(t,x)))
is measurable by Corollary 8.2.13.
Hence Theorem 8.2.14 implies that for every ρ € Sn~1, the
support function σ(Φ(·,χ),ρ) is measurable. Thus, by the definition of

376
9- Selections and Parametrization
Steiner point, f(-,x) is measurable when η = 1. When η > 2, it is
enough to observe that the map ρ ι-> ρσ(Φ(ί, χ), ρ) is continuous and
therefore the map
t ^ / ρσ(Φ(ί,χ),ρ)ω(άρ)
Js*-1
is measurable. □
9.6 Caratheodory Parametrization
We prove in this section that a Caratheodory set-valued map F with
closed convex images admits a Caratheodory parametrization.
Consider a metric space X, reals а < b and a set-valued map
F : [a, b] χ Χ ~> Rn
Definition 9.6.1 Let U be a metric space and C{t) С X, t € [a, b]
be given subsets of X. We say that a single-valued map
f : [a,b] xX x U ■-» Rn
is a Caratheodory parametrization of F on {C(£)}ie[a,6] if
' i) V(i,s) € [a,b]xX, F(t,x)=f(t,x,U)
ii) V (x,u) € Χ χ U, f(-,x,u) is measurable
<
Hi) V (t,u) € [a, b] χ U, f(t,-,u) is continuous on C{t)
iv) V (t,x) € [a, b]xX, f(t,x,·) is continuous
We recall that \\F(t, x)\\ is defined by
\\F{t,x)\\ := max ||у||
y£F(t,x)
Let В denote the closed unit ball in Rn.
Theorem 9.6.2 (Caratheodory Parametrization) Consider a
metric space X and a set-valued map F : [a,b] χ Χ ~> Rn with
nonempty convex compact images.

9.6. Car&theodory Paxametrization
377
IfF is Caratheodory on {С(Ь)}щащ, then there exists a Carathe-
odory parametrization f of F on {С^)}щац such that U = В and
for all t € [a,b], χ € X, и,υ € В we have
||/(i,s,u)-/(i,s,t/)|| < c\\F(t,x)\\\\u-v\\
where с is independent of F.
Furthermore if F is continuous, so is f.
Finally, if images of F are merely closed (instead of compact),
then the same statement holds true with U = Rn and \\F(t,x)\\
replaced by 1.
Proof— To simplify the notations, we set Mx(t) := ||.F(i,a;)||.
Let (t, x, u) € [a, b] χ Χ χ Rn. Consider the closed ball G(t, x, u) of
radius 2d(Mx(t)u,F(t,x)) and center Mx(t)u, i.e.,
G{t,x,u) = B(Mx(t)u,2d{Mx(t)u, F(t,x)))
We claim that G is measurable with.respect to t.
Indeed, fix χ € X, и € Rn. Prom Theorem 8.2.11 it follows that
Mx(·) — ||.Р(-,а;)|| is measurable and from Corollary 8.2.13 that the
function d(Mx(-)u,F(-,x)) is measurable. Thus, using again
Corollary 8.2.13, we deduce that G(-,x, u) is measurable.
Let Ρ be the map defined in Lemma 9.4.2 and set
$(t,x,u) := P(Mx{t)u,F{t,x)) = G(t,x,u)riF(t,x) С F(t,x)
Since G(-,x, u) is measurable, Theorem 8.2.4 and our assumptions
on F imply that <&(·,£, u) is measurable.
Define the single-valued map / from [a,b] χ Χ χ Rn into Rn by
V {t,x,u) € [a,6] x-X" x Rn, f(t,x,u) := sn($(t,x,u))
where sn is the Steiner selection.
Since Φ is measurable with respect to t, using the definition of
sn, we deduce that / is also measurable with respect to t.
Lipschitz properties of sn and Ρ proved in Theorem 9.4.1 and
Lemma 9.4.2 imply:
' \\f(t,x,u)-f(t,y,v)\ < nHmt,x,u),$(t,y,v))
(9.10)
< 5n{H{F(t,x),F{t,y)) + \\Mx(t)u - My{t)v\\)

378
9- Selections and Paxametrization
Since F has compact images, F(t,·) is continuous on C(t) in the
Hausdorff metric. This and (9.10) imply that for every и € Rn,
/(·, ·, u) is Caratheodory on {C(i)}ie[aib].
In order to prove that / is a parametrization, it remains to check
г) of Definition 9.6.1 with U = B. For this aim fix t € [a, b], χ € X
and у € F(t,x). Setting
u ._ ί y/Mx(t) if Mx(t) φ о
1 0 otherwise
we derive
и € В, Mx(t)u = у, $(t,x,u) = у
Hence
f{t,x,u) - sn($(t,x,u)) = у
This means that F(t,x) с f(t,x,B). The opposite inclusion
resulting from Theorem 9.4.1, we have proved г).
Assume next that F is continuous. Then, by Lemma 9.3.1, so
is ||-F(·, ·)ΙΙ· Thus the set-valued map G with convex compact
images is continuous. Consequently its graph, Graph (G), is closed and
therefore the intersection map Φ has closed graph. Since for all
(t, x), F(t, x) с Mx(t)B, we deduce that Φ is upper semicontinuous.
We claim that Φ is actually continuous. Indeed in order to prove
this claim, it is enough to show that it is lower semicontinuous. Fix
(t, x, u) and an open set О С Rn such that
Φ(ί,ι,η)ηΟ φ 0
Let z\ € Φ(ί, χ, и) П О. From the very definition of G we deduce
that there exists
z2 € F(t,x)nlat(G(t,x,u))
Thus the interval
}zi,z2] С Int(G(i>s>u))ni,(i>s)
Consequently we can find an element ζ € Rn satisfying
ζ € Int(G!(i>s,u))ni,(i>s)nO

9.7. Measurable/Lipschitz paxametrization
379
Hence for some e > 0,
Β(ζ,ε) С G(i,i,ti)nO
This and continuity of G imply that for every (t',x',u') sufficiently
close to (t,x, u),
Β(ζ,ε/2) С G{t',x',u')
On the other hand, F being continuous, for all (t1, x') near (t, x),
B{z,e/2)nF(t\x') φ 0
Thus Φ is lower semicontinuous, and thus, is continuous.
Since Φ has compact images, it is also continuous in the Haus-
dorff metric. Prom the very definition of sn, we deduce that / is
continuous.
To prove the last statement, it is enough to apply the same proof
replacing everywhere Mx(t) by 1. □
9.7 Measurable/Lipschitz Parametrization
In this section we show that the Caratheodory parametrization
defined in the preceding section is measurable/Lipschitz on {C{t)}t&[a,b}
whenever F is measurable/Lipschitz on {C(t)}teia,b]·
Theorem 9.7.1 (Parametrization of Unbounded Maps)
Consider a metric space X and a set-valued map F : [a, b] x X ~> Rn
with nonempty closed convex images.
Assume that F is measurable/Lipschitz on {C{t)}te[a,b] and ^
k(t), t € [a, b] denote the corresponding Lipschitz constants.
Then there exists a Caratheodory parametrization f of F with
U = Rn on {C(£)}ie[a,6] such that:
' V(i,u) € [a,6]xRn, f{t,-,u) is ck(t) - Lipschitz on C(i)
<
V (t,x) € [a,b]xX, f(t,x,·) is с - Lipschitz on Rn
where с is independent of F.
Furthermore if F is continuous, so is f.

380
9- Selections and Parametrization
Proof— It is enough to apply the same proof as the one of
Theorem 9.6.2 replacing everywhere Mx{t) by 1 and to observe that (9.10)
implies that for all (t,u) € [а,Ь] х Rn, f{t,-,u) is 5nfc(£)-Lipschitz
on C(i). □
Theorem 9.7.2 (Parametrization of Bounded Maps) We posit
the assumptions of Theorem 9.7.1 and we assume that the values of
F are compact.
Then there exists a Caratheodory parametrization f of F on the
family of sets {C(t)}te[aq with U = В such that:
i) V (t, u) € [a, b] x B, f(t, ·, u) is ck(t) — Lipschitz on C{t)
< ii) Vi € [a,b], \/x € X, \fu,v € В
\\f(t,x,u)-f(t,x,v)\\ < cllFfoaOllllu-t/H
where с is independent of F.
Furthermore if F is continuous, so is f.
Proof— We consider the same parametrization as in the proof of
Theorem 9.6.2. We already know that / is a Caratheodory
parametrization of F on {C(i)}ie[ai6].
Then (9.10) and Lemma 9.3.1 imply that for all (t, u) € [a, b] x B,
\\f(t,x,u)- f(t,y,u)\
< < 5n(H(F(t,x),F(t,y)) + \\Mx(t)u - My(t)u\\)
< 5n (k(t) \\x - y\\ + k(t)\\x-y\\)
Consequently, f(t,-,u) is 10nfc(i)-Lipschitz on C(t). □
When we allow the set of parameters to range in a Banach space,
then actually the parametrization may be taken linear with respect
to the parameter.
Theorem 9.7.3 (Linear Parametrization) If all the assumptions
of Theorem 9.7.1 are satisfied and X is compact, then there exists a
map
ψ : Χ χ C{X;Rn) н-> Rn

9.7. Measurable/Lipschitz paxametrization 381
and a measurable set-valued map U : [a,b] ~> C(X;~Rn) with closed
convex images such that :
' i) V(i,s) € [a,b]xX, F{t,x) = <p(x,U(t))
ii) ψ{χ,·) is linear nonexpansive
<
Hi) Vt € [a,b], Vu € U(t), Vx,y € X,
\\tp(x,u) - tp(y,u)\\ < ck(t)d(x,y)
where с is independent of F.
If moreover F has compact images, then the values of U are
compact.
Proof — Let / be a paxametrization map as in the conclusion
of Theorem 9.7.1. Then
V (t,u) € [a,b] χ Rn, f(t,-,u) is ck(t) - Lipschitz
Hence for all t € [a,b], {f(t, -,и)}иец is a family of equicontinuous
maps.
Define the map U with closed convex values by :
Vi € [a,b], U(t) := co({f(t,;u)}ueRn) С С(Х;Кп)
If F has bounded images, then Ascoli's Theorem yields that for all
t, U(t) is compact.
To show that it is measurable, consider a dense sequence (?χ^)^>ι
in Rn. Then for every к the map
t -> f(t,-,uk) € C(X;Rn)
is measurable. Set
V(t) = \Jf(t,;Uk)
k>l
Theorem 8.2.4 implies that V is measurable and Theorem 8.2.2 yields
that U is measurable.
Consider the linear map
ψ : Χ χ C{X;Rn) ■-» Rn, <p(x,u) := u{x)

382
9- Selections and Parametrization
Clearly it is nonexpansive.
Then equality г) holds true by the very definition of ψ and U(t).
Notice that иг) is verified for every и € V(t). Hence, by definition
of φ, it is verified for every и € coV(t). The set U(t) being equal to
the closure of coV{t) the proof ensues. Π

Chapter 10
Differential Inclusions
Introduction
This chapter is just ал introductory survey of differential
inclusions in finite dimensional vector-spaces. Many results below are
stated without proofs. Their proofs and further developments can
be found in [33], [50], [195].
Differential inclusions
x'{t) € F{t,x{t)) (10.1)
where F is a set-valued map from R χ Χ to a finite dimensional
vector-space X have been studied since the thirties. Their
consideration was initiated by Zaremba1 in 1934 and Marchaud in 1938.
These investigators were mostly interested in the existence results2
and also examined some qualitative properties of solutions sets.
We have first to agree on what we shall call a solution to such a
differential inclusion.
1who kept close relations with Painleve and other French mathematicians after
his study in Paris.
2While Zaremba investigated the paratingent solutions, Marchaud was mainly
concerned with the contingent ones, i.e., the derivatives were taken in the
contingent and paratingent sense. Since 1961, when Wazewski showed that one can
use more "classical" solutions than the contingent and paratingent ones, solutions
have been understood in the Caratheodory sense, i.e., absolutely continuous
functions verifying (10.1) almost everywhere.
383

384
10- Differential Inclusions
In the case of differential equations, there is no ambiguity since
the derivative x'(·) of a solution x(-) to the differential equation
x'(t) = f(t,x(t))
inherits the regularity properties of the map / and of the function
x(-). This is no longer the case with differential inclusions and is one
of the reasons why their study brings more difficulties than that of
the ordinary differential equations.
Solutions to differential inclusion (10.1) are understood in the
Caratheodory sense, i.e., absolutely continuous functions verifying
(10. l) almost everywhere.
Interest in differential inclusions (10.1) was revived in the early
sixties, when mathematicians became attracted by a new domain:
control theory3. Filippov in 1959 and Wazewski in 1961 proved that
under very mild assumptions the control system
x' = f(t,x,u(t)), u(t) € U is measurable (10.2)
may be reduced to the differential inclusion (10.1.)
This considerably simplified the study of the closure of solutions
to (10.2) and led to the celebrated Filippov-Wazewski relaxation
theorem, which we state in Section 4.
But differential inclusions also encompass much more
sophisticated control systems:
1. — Closed loop control systems
x'{t) = f(t,x{t),u(t)), u(t) € U(t,x(t)) (10.3)
2. — Implicit control systems
f(t,x(t),x'{t),u{t)) = 0, u{t) € U(t,x(t)) (10.4)
3. — Systems with uncertainties
x'{t) € f{t,x(t),u(t)) + e(t,x)B, u(t) € U{t,x(t)) (Ϊ0.5)
3Set-valued maps have been carefully — maybe unconsciously — hidden in
control theory, despite the early contributions of Wazewski and Filippov. They
were thought of as being useless. This is often the justification of something new
one is reluctant to learn... "Everybody wants to teach, nobody wants to learn..."
bitterly complained Abel.

10.0. Introduction
385
where e(t,x) is a function incorporating the disturbances of the
model.
Setting
F(t,x) := f(t,x,U(t,x))
in the first case,
F(t,x) := {υ\ Ο € /(ΐ,χ,ί/,Ε^ΐ,α:))}
in the second one and
F(t,x) := /(ί)χ(0,ί/(ί)χ))+ε(ί)χ)β
in the third one, we replace the above control systems by differential
inclusions of the type (10.1.)
We shall survey three main classes of differential inclusions.
First, the case when the right-hand side is upper semicontinuous
with closed convex values: in this instance, we can extend Peano's
Theorem on the existence of solutions to differential equations with
continuous right-hand side and show that the set of solutions depends
upper semicontinuously upon the initial state. This is the topic of
Section 1.
In Section 4, .we trade convexity of the images with more
regularity on F, which is assumed to be Lipschitz with respect to the
state and measurable with respect to the time. Filippov's Theorem
provides not only the extension of the Cauchy-Lipschitz Theorem,
but also very useful estimates ά la Gronwall, implying in particular
the Lipschitz dependence on the initial state. Under this condition,
we can regard Fasa kind of infinitesimal generator of a semigroup
of set-valued maps, which are the reachable maps.
Furthermore, it allows us to estimate the contingent, adjacent and
circatangent derivatives of the solution map S := <S[o,:r] associated
with the differential inclusion
for almost all t € [Ο,Γ], x'{t) € F(t,x(t))
where F:[0,T]xX^X.
We shall express these estimates in terms of the solution maps of
adequate linearizations of differential inclusion of the form
w'(t) € F'{t,x(t),x'{t)){w{t))

386
10- Differential Inclusions
where for almost all t, F'(t, x, у) (u) denotes one of the (contingent,
adjacent or circatangent) derivatives of the set-valued map F(t, ·) at
a point (ж, у) of its graph (the set-valued map F is regarded as a
family of set-valued maps χ ~> F(t,x) and the derivatives are taken
with respect to the state variable only.)
These linearized differential inclusions can be called the
variational equations, since they extend (in various ways) the classical
variational equations of ordinary differential equations.
The Relaxation Theorem, stated in Section 4, connects these two
classes of differential inclusions: when F is Lipschitz and compact-
valued, the set of solutions to differential inclusion (10.1) is dense in
the set of solutions to the relaxed or convexified differential inclusion
x'(t) € coF(t,x(t))
We also mention the third case when
F(t,x) := -A{x)
where Л is a maximal monotone map. (This covers gradient
inclusions of the form
for almost all t € [Ο,Γ], x'{t) € -dV(x(t))
where the "potential" У is a lower semicontinuous convex function,
describing the continuous version of the steepest descent method.)
Monotonicity implies the uniqueness of the solutions. Existence
is guaranteed if A is maximal monotone, so that A can be regarded
(and is regarded) as the infinitesimal generator of a semigroup of
(nonlinear) operators. This is elucidated in Section 3.
We shall also devote Section 1 to the Viability Theorem and
Section 2 to the description of some of its applications.
If if is a closed subset of the domain of F (assumed to be
independent of the time for simplicity), we say that a solution x(-) to a
differential inclusion is viable if x(t) remains in К for all t.
The purpose of the Viability Theorem is to characterize the
viability property: for any initial state xq € K, there exists at least one
viable solution.

10.1. The Viability Theorem
387
When F is upper semicontinuous with closed convex images, К
enjoys the viability property if and only if К is a viability domain of
F, i.e., if and only if
Vz € K, F{x)nTK{x) φ 0
When F = f is single-valued, this theorem has been proved by
Nagumo in 1942 and rediscovered 14 times since4. The above set-
valued version has been proved by G. Haddad in 1981.
One can also show that if К is not a viability domain of F, it
still contains the largest closed viability domain, called the viability
kernel. This result provides a quite useful tool.
The second section is devoted to applications of the Viability
Theorem. For instance, differential inequalities of the type
Vi € [0.Г], V(x(t)) < w(t)
can be regarded as a viability constraint of the form
Vi € [Ο,Γ], (x(t),w(t)) € EV{V)
so that the Lyapunov method can be adapted to differential
inclusions and to functions V which are merely lower semicontinuous.
This allows us to deduce the existence of the smallest lower
semicontinuous Lyapunov function larger than or equal to a given function.
10.1 The Viability Theorem
In all this section Χ, Υ denote finite dimensional vector-spaces,
except an explicit mention to the contrary.
We describe the (non deterministic) dynamics of the system by
a set-valued map F from the state space X to itself.
Consider initial value problems (or Cauchy problems) associated
with differential inclusion
for almost alii € [Ο,Γ], x'(t) € F(x(t)) (10.6)
satisfying the initial condition x(0) = xq.
4This does not imply necessarily that it is true.

388
10- Differential Inclusions
10.1.1 Solutions to Differential Inclusions
We have first to agree on what we shall call a solution to such a
differential inclusion.
Denote by L1(0,oo·, X, e~btdt) the space of integrable functions
for the measure e~btdt (weighted Lebesgue spaces) and by
' i) W := W0 := π^^,Τ-,Χ) if Γ .< +oo
<
w ii) W-ь := W^1(0,oo;X,e-btdt) if Γ = +oo
(for some b > 0) the weighted Sobolev spaces defined by
' W := {s(-) € Ρ{0,Τ·Χ) | s'(.) € Р{0,Т;Х)}
<
W-b ■= {x(·) € L^O.oo;X, e"btdt) \ χ'(·) € ^(0,οο;Χ, e'^dt)}
We shall supply them with the topology for which a sequence xn{·)
converges to x(·) if and only if
г) χη(·) converges uniformly to χ(·)
(on compact intervals if Τ = +oo)
<
ii) x'n(·) converges weakly to x'(·) in L1 (0, T; X)
or in ^(Ο,οο-,Χ,ε^άί) if Γ = +oo
Definition 10.1.1 (Viability and Invariance Properties) Let
К be a subset ofOom(F).
— A function x(-) from [Ο,Γ] to X is called viable in К if
Vi € [Ο,Γ], x(t) € К
— We shall say that К enjoys the local viability property
or controlled invariance (for the set-valued map F) if for any
initial state xq in K, there exist Г > 0 and a solution on [Ο,Γ] to
differential inclusion (10.6) starting at xq and viable in K.
It enjoys the global viability property (or, simply, the viability
property^ if we can take Γ = oo.
— We shall say that a subset К С Dom(F) is a viability
domain of F if and only if
\/x € K, F(x)ilTK(x) φ 0 (Ю.7)

10.1. The Viability Theorem
389
Figure 10.1: Example of a Map with Nonconvex Values
1
i
i
о
There is no solution starting at 0
Since the contingent cone to a singleton is obviously reduced to
0, we observe that a singleton {x} is a viability domain if and only
if ж is an equilibrium of F, i.e., a stationary solution to the inclusion
0 € F{x).
In other words, the equilibria of a set-valued map provide the
first examples of viability domains, actually, the minimal viability
domains.
Open subsets of the domain of F are viability domains of F, since
contingent cones to open subsets are equal to the whole space.
10.1.2 Statements of the Viability Theorems
Viability theorems hold true for the class of nontrivial upper semi-
continuous set-valued maps with nonempty compact convex images.
We observe that the only truly restrictive condition is the convexity
of the images of these set-valued maps, since the continuity
requirements are minimal.
But we cannot dispense with it, as the following counter example
shows:
Let us consider X := R, К := [—1, +l] and the set-valued map

390
10- Differential Inclusions
F: K^R defined by
F(x) :=
{-1} if χ > 0
{-1,1} if χ = 0
{+1} if ж < 0
Obviously, no solution to the differential inclusion
x\t) € F(x(t))
can start from 0, since 0 is not an equilibrium of this set-valued map!
Theorem 10.1.2 (Haddad's Theorem) Let us assume that
i) F : X ~> X is upper semicontinuous
ii) the images of F are convex and compact
Hi) К is locally compact
Then К enjoys the heal viability property if and only if К is a
viability domain of F.
In other words, К is a viability domain if and only if for any
initial state xq in AT, there exist Τ > 0 and a solution on [0, Γ] to
differential inclusion (10.6) starting at xq and viable in K.
Since open subsets of finite dimensional vector spaces are locally
compact viability domains, we obtain the extension of Peano's
Theorem to differential inclusions:
Theorem 10.1.3 Let Ω be an open subset of a finite dimensional
vector space X and F : Ω ~> X be an upper semicontinuous set-
valued map with convex compact images.
Then, for all xq € Ω, there exists Γ > 0 such that differential
inclusion (10.6) has a solution on the interval [0, T) starting at xq-
The interesting case from the global point of view is the one
when the viability subset is closed. In this case, we derive from
Theorem 10.1.2 a more precise statement.
Theorem 10.1.4 (Local Viability Theorem) Let us consider an
upper semicontinuous set-valued map F : X ~> X with compact
convex images and a closed subset К С Dom(F).

10.1. The Viability Theorem
391
Then К is α viability domain if and only if it enjoys the local
viability property.
Actually, for any xq € К there exist a positive Τ and a solution
on [О, Г] to differential inclusion (10.6) starting at xq and viable in
К such that either Τ = oo or Τ < oo and lim supt_>T„ ||a;(i)|| = oo.
Further adequate information —a priori estimates on the growth
of F — allow exclusion of the case when limsupi_>r_ ||a;(i)|| = oo.
This is the case for instance when F is bounded on K, and, in
particular, when К is bounded. More generally, we can take Τ — oo
when F enjoys linear growth:
Definition 10.1.5 We say that a set-valued map F : X ~> X has a
linear growth if there exists с > 0 such that
Vi € K, \\F{x)\\ := sup ||v|| < c(||s||+l)
veF(z)
We shall call Peano maps nontrivial upper semicontinuous set-valued
maps with nonempty compact convex images and with linear growth,
or equivalently, nontrivial closed set-valued maps with convex values
and linear growth.
Theorem 10.1.6 (Viability Theorem) Let us consider a Peano
map F from X to X and a closed subset К С Dom(-F).
If К is a viability domain, then for x0 € K, there exists a
solution on [0, oo[ to differential inclusion (10.6) starting at xq, which is
viable in К and belongs to the space И^1'1(0, oo; X, e~btdi) for some
b>0.
Let us consider now a sequence of closed viability domains of a
set-valued map F. We would like to address the following stability
question:
Is the upper limit of these closed viability domains still a closed
viability domain"?
Theorem 10.1.7 (Stability of Viability Domains) Let us
consider a Peano map F : X ~> X. Then the upper limit of a sequence
of closed viability domains of F is also a closed viability domain of
F.

392
10- Differential Inclusions
10.1.3 Viability Kernels
Definition 10.1.8 Denote by S(xq) the (possibly empty) set of
solutions to differential inclusion (10.6) on [0,oo[ starting at xq € X.
We call the set-valued map
Dom(F) э χ н-> S(x)
the solution map of F (or of differential inclusion (10.6).)
Theorem 10.1.9 (Continuity of the Solution Map) Let us
assume that F : X ~> X is a Peano map. The solution map S is upper
semicontinuous with compact images from its domain to the space
C(0, oo\X).
Definition 10.1.10 (Viability Kernel) Let К be a subset of the
domain of a set-valued map F : X ~> X. We shall say that the
largest closed viability domain contained in К (which may be empty)
is the viability kernel of K.
Theorem 10.1.11 Let us consider a Peano map F : X ~> X and
a closed subset К С Dom(.F). Then the viability kernel of К exists
(possibly empty) and is the subset of initial states such that at least
one solution starting from them is viable in K.
The viability kernels may inherit properties of both F and K.
For instance, if the graph of F and the subset К are convex, so is
the viability kernel of K. If F is a closed convex process (i.e., its
graph is a closed convex cone) and if К is a closed convex cone, the
viability kernel is a closed convex cone.
In general, viability kernels are not necessarily connected.
The viability kernel contains for instance all the limit sets L(x)
of the solutions x(-) to differential inclusion (10.6) defined by
L(x(-)) := f|d(x([r,oo[))
T>0
Indeed, they are closed viability domains:

10.1. The Viability Theorem
393
Theorem 10.1.12 (Limit Sets are Viability Domains) Let us
consider a Peano map F : X ~> X. Then the limit sets of the
solutions to differential inclusion (10.6) are closed viability domains.
In particular, the limit of a solution x(-) to differential inclusion
(10.6) when t —> +oo, when it exists, is an equilibrium of F.
The trajectories of periodic solutions to the differential inclusion
(10.6) are ako closed viability domains.
10.1.4 Viability and Equilibria
Equilibrium Theorem 3.2.1 states that any compact convex viability
domain of a Peano map contains an equilibrium. When К is a
compact viability domain, the convexity of the image F(K) implies also
the existence of a viable equilibrium·.
Theorem 10.1.13 Let F be a Peano map. If К С Oom(F) is a
compact viability domain and if F(K) convex, then there exists an
equilibrium of F in K.
Proof — Assume that there is no equilibrium. This means
that 0 does not belong to the closed convex subset F(K), so that the
Separation Theorem implies the existence of some ρ G X* and e > 0
such that
sup <v,-p>= a(F(K),-p) < -ε
veF(K)
Let us take any solution x(-) to differential inclusion (10.6) viable in
K, which exists by the Viability Theorem. We deduce that
Vi > 0, (-p,x'(t)) < -ε
so that, integrating from 0 to t, we infer that
Vt > 0, et < <p,x(t)-x(0)>
But К being bounded, we thus derive a contradiction. Π
With the same kind of techniques, we derive the following
criterion on the existence of an equilibrium:

394
— 10- Differentia,! Inclusions
Theorem 10.1.14 Let us assume that F is upper hemicontinuous
with closed convex images and that К С Dom(F) is compact. If there
exists a solution x(-) of (10.6) viable in К such that
then there exists an equilibrium of F in K.
inf - / \\χ'(τ)\\άτ = 0
i>0
Proof — Let us assume that therels ho viable equilibrium,
i.e., that for any χ G К, О does not belong to F(x). Since these
subsets are closed and convex, the Separation Theorem implies that
there exists ρ in the unit sphere of X and ev > 0 such that
<r{F{x),-p) < -ep
In other words, we can cover the compact subset К by the subsets
Vp := {x € K\ a{F{x),-p) < -ep}
when ρ ranges over Σ. Since they are open, thanks to the upper
hemicontinuity of F, К can be covered by q open subsets VPi. Set
e := min eVi > 0
j = l,...,q ^
Consider now any solution to differential inclusion (10.6) viable
in K. Hence, for any t > 0, x(t) belongs to some VPj, so that
-||s'(i)|| < (-Pj,x'(t)) < a(F(x(t)),-Pj) < -ε
and thus, by integrating from 0 to t, we have proved that there exists
e > 0 such that for all t > 0,
e < -tj\\x'{r)\\dT
a contradiction of the assumption of the theorem. □
10.2 Applications of the Viability Theorem
In this section, we shall posit the linear growth conditions which
guarantee the existence of solutions to the differential inclusions on
the half-line [0, oo[ (Γ = +oo.)

10.2. Applications of the Viability Theorem
395
10.2.1 Linear Differential Inclusions
We now take for the right-hand side of F a closed convex process
and for viability domain a closed convex cone K. We recall that by-
Lemma 4.2.5,
Vi e K, TK{x) = K + Rx
Corollary 10.2.1 Let F : X ~> X be α closed convex process and
К С X be a closed convex cone. We posit the following assumptions:
{ i) V χ € K, R(x) := F(x) Π Τκ{χ) φ 0
) ii) the norm ||Д|| of R is finite
Then, for any initial state xq G K, there exists a solution x{·) to the
"linear differential inclusion"
for almost all ί > 0, x'(t) € F(x(t)) (10.8)
starting at xq and viable in the cone K.
10.2.2 Lyapunov E4inctions
We consider differential inclusion (10.6) and a time-dependent
function w(·) defined as a solution to the differential equation
w'(t) = -ip{w{t)) (10.9)
where ψ : R+ ·-» R is a given continuous function with linear growth.
Our problem is to characterize functions enjoying the ψ -Lyapunov
property, i.e., nonnegative extended functions V : Χ ι—► R+ U {+00}
(such that Dom(V) с Dom(F)) satisfying
Vt > 0, V(x(t)) < w{t), to(0) = V(x0) (10.10)
along at least one solution x(-) to differential inclusion (10.6) starting
at xq where w is a solution to differential equation (10.9.) Since
this condition amounts to saying that the epigraph of V enjoys the
viability property for the differential inclusion
(x?(t),w'(t)) € G{x(t),w(t)) where G(x,w) ·.= F(x) χ {-ip(w)}
we can apply the Viability Theorem on the epigraph of V.

396
10- Differentia,! Inclusions
This allows us to use lower semicontinuous instead of differen-
tiable functions among the candidates to satisfy this Lyapunov
property.
We have then to translate the fact that Sp(V) is a viability
domain of G, and for that purpose, to use Proposition 6.1.4 which
states that the contingent cones to the epigraph are the epigraphs of
the contingent epiderivatives: This leads us to extend the concept of
Lyapunov functions to lower semicontinuous functions
V:X ^ RU{+oo}
We recall that V is said to be contingently epidifferentiable at χ if
its contingent epiderivative never takes the value — oo (see
Definition 6.1.2.)
Definition 10.2.2 (Lyapunov Function) Let
V : X н-> R+ U {+oo}
be a contingently epidifferentiable function.
We shall say that V is a Lyapunov function of F associated with
φ if and only if V is a solution to the contingent Hamilton-Jacobi
inequalities
Vx 6 Dom(F), inf DfV(x)(v) + 4>(V(x)) < 0 (10.11)
v£F(x)
Theorem 10.2.3 Let V : Χ i-> R+ U {+00} be a contingently
epidifferentiable lower semicontinuous function and F : X ~> X be a
Peano map. Then V is a Lyapunov function of F associated with
φ(·) if and only if for any xq G Dom(V), there exist solutions x(-) to
(10.6) starting at xq and w(-) to (10.9) satisfying property (10.10.)
Example W-Monotone Set-Valued Maps
Let 1У:Хн R+ и {+оо} be a nonnegative extended function.
We say that a set-valued map F is И^-monotone (with respect to ψ)
if for every x, у £ X, we have
\/u€ F(x), υ e F(y), D^W{x - y)(v - u) + φ{λ¥{χ -y))<0
(10.12)
By setting V{x) — W(x — x) we obtain the following consequence
of Theorem 10.2.3:

10.2. Applications of tie Viability Theorem
397
Corollary 10.2.4 Let W : X i-> R+ U {+00} be a contingently
epidifferentiable, lower semicontinuous extended function and F :
X ~> X be a Peano map such that —F is W-monotone with respect
to ψ.
Let χ be an equilibrium of F (i.e., 0 G F(1e)). Then, for any
xo € Oom(F), there exist solutions x(-) and w(-) to (10.6), (10.9)
satisfying
s(0) = s0, ω(0) = W(xq-x) and Vi > 0, W(x(t)-x) < w(t)
In particular, for W(z) := |||.г||2, we find the usual concept of
monotonicity (with respect to φ):
V x,y £ X,u (Ε F(x),v G F(y), (u-v,x-y) > φ (-\\x - y\\2 J D
The function U : X ·-» R+ U {+00} being given, we can construct
the smallest lower semicontinuous Lyapunov function larger than or
equal to U, i.e., the smallest nonnegative lower semicontinuous
solution Ό ψ to the contingent Hamilton-Jacobi inequalities (10.11) larger
than or equal to U. Its epigraph is the viability kernel of the epigraph
of U:
Theorem 10.2.5 Let us consider a Peano map F : X ~> X, a
continuous function ψ : R+ >—» R with linear growth and
U:X^ R+U{+oo}
such that Dom(!7) С Dom(-F).
Then there exists a smallest nonnegative lower semicontinuous
solution
ΙΙφ : Dom(.F) 1-»· RU {+00}
to the contingent Hamilton-Jacobi inequalities (10.11) larger than
or equal to U (which can be the constant +00), which enjoys the
property:
V so € Dom(!7¥,) there exist solutions to (10.6), (10.9) satisfying
<
^ s(0) = s0, U(xo) = w(0), Vi>0, U(x{t)) < υφ(χ(ί)) < w{t)

398
10- Differential Inclusions
Let us single out the following consequence when we take for
function U := ам the distance to a closed set Μ с Χ:
Corollary 10.2.6 Let us consider а Реапо тар F : Χ ~> X. Then
for all a > 0, there exists a smallest lower semicontinuous function
dMa--X ·-> RU{+oo}
larger than or equal to ам such that
V xo € Oom(d,Ma), there exists a solution x(-) to (10.6) such that
x(0) = so, Vi > 0, dM(x(t)) < dMa(xQ)e~at
Therefore, we can regard the subsets Oom(dMa) as the basins of
exponential attraction of M.
10.2.3 Tracking a Differential Inclusion
Let us consider a differential inclusion (10.6) where F : X ~> X is a
Peano map and an observation map Η : X ~> У from X to another
finite dimensional vector-space Y.
We shall in some sense "project" the differential inclusion (10.6)
to a differential inclusion on the observation space Υ described by a
set-valued map G
for almost all t > 0, y'(t) e G(y(t)) (10.13)
in order to "track" (or "filter") a solution x(-) to differential inclusion
(10.6) in the following sense:
V xq € Dom(-F) and yo € H(xq), there exist
< solutions x(-) & y(-) to (10.6), (10.13) such that
x(0) = xo, 2/(0) = г/о and V t > 0, y(t) e H{x(t))
(10.14)
This property may be called the tracking property.
Proposition 10.2.7 Let us consider a closed set-valued map Η from
XtoY and Peano maps F : X ~> X and G :Y ~> У'. Then tracking
property (10.14) holds true if and only if for every у G lm(H), we
have
Vx € H-\y), F(x)nDH(x,y)~1(G(y)) φ 0 (10.15)

10.3. Nonlinear Semi-Groups
399
It follows obviously from the Viability Theorem 10.1.6, because
the above condition amounts to saying that
V(x,y) € Graph(tf), (F(x)xG(y))nTGmmH)(x,y) φ 0
i.e., that the graph of Η is a viability domain5 of the set-valued map
FxG.
10.3 Nonlinear Semi-Groups
We provide now an existence and uniqueness theorem for differential
inclusions the right-hand side of which is minus a maximal monotone
set-valued map:
Theorem 10.3.1 (Crandall-Pazy) Let A be α maximal monotone
set-valued map from a Hilbert space X to X. Consider the initial
value problem for the differential inclusion
x'{t) € -A(x(t)), s(0) = xo
where the initial state Xq is given in Oom(A). Then it has one unique
solution x(-) defined on [0, oo[.
Furthermore if m(A(x)) denotes the element of A(x) with the
minimal norm, then x(-) is actually the solution to the differential
equation
for almost all t > 0, x'(t) = -m(A(x(t)))
(called the slow solution). Moreover, the function t —> ||a;'(i)|| is
nonincreasing.
Let x(-) := S(xq) and y(-) := S(yo) be the solutions starting at
xq and ?/o· Then
V* > o, ||s(0 - y(0ll < lko-2/o||
Finally,
ί Vi > 0, x'{t) = Yunh^Q+(x(t-\-h)-x{t))/h
у and x'(-) is continuous from the right
In this case, the solution map S is single-valued and is called in
the literature the nonlinear semigroup of the monotone map A.
5 When G = 0, i/o = 0 and Η is single-valued, the viability kernel of ff-1(0)
is closely related to the "zero dynamics" introduced by Byrnes and Isidori.

400
10- Differential Inclusions
10.4 Filippov's Theorem
Filippov's theorem for differential inclusions is as important as the
Gronwall lemma for ordinary differential equations.
Consider a finite dimensional space X, reals а < b and let F be
a set-valued map from [a, b] χ Χ into closed nonempty subsets of X.
We associate with it the differential inclusion
for almost all t € [a, b], x'(t) £ F(t,x(t)) (10.16)
Consider the solution map of (10.16), associating with each initial
point xq G X the subset
S[o,b](xo) :== {x I x is a solution to (10.16) on [a, 6], x(a) = xq}
of solutions to differential inclusion (10.16) on [a, b].
Let у G W1,1(a, b;X) be a given absolutely continuous map.
Filippov's theorem provides an estimation of the distance
between у and the set S[ajb](xo) С W1'1(a,b;X) under the following
assumptions on F:
i) V χ € X, F(-,x) is measurable
it) 3 β > 0, к G ^(α, 6; R+) such that for a.a.
(10.17)
ΐ € [a, 6], F(t, ·) is fc(i)-Lipschitz on y(t) + /35
иг) The map ti-^d(y'(t),F(t,y(t))) is integrable
where В denotes the closed unit ball in X.
Remark — Observe that under the above assumptions, the set-
valued map t ~> F(t, y(t)) is measurable: Indeed, let yn be a sequence
of measurable simple maps converging pointwise to y. Since the map
t ^ F(t,yn(t))
is measurable, Theorem 8.2.5 and assumptions (10.17) imply the
measurability of F(-, ?/(·))·

10.4. Filippov's Theorem
401
Therefore, Corollary 8.2.13 implies that the function
t -> d(y'(t),F(t,y(t)))
is always measurable. □
Theorem 10.4.1 (Filippov) Let us consider an absolutely
continuous function у : [α, 6] н-► Χ, δ > 0 and assume that conditions
(10.17) hold true. Set
7(0 = d{y'{t),F{t,y{t))), m(t) = exp (£ k(s)di
and
η(ί) = m(t) (s+f j{s)ds\
Ifv(b) < β, then for every xq € B(y(a), δ), there exists χ G 5[aib](a;o)
such that
Vie [a, b], И0-у(011 < v(t)
and
\W(t) - y'(t)\\ < k{t)v(t) + j(t) a.e. in [a, b]
Corollary 10.4.2 (Lipschitz Dependence) Let у £ S[aib](yo) and
assume that F, у satisfy assumptions (10.17) i) and ii).
'Then there exists I > 0 such that for all xq € X satisfying
m(b) \\xq — 2/o|| < β we have
dwi,i (y, S[ajb](xo)) < l\\xo-
2/o I
We state next a "nonlinear version" of Aumann's Theorem on
the convexity of the integral of a set-valued map.
This theorem compares solutions to (10.16) and to the convexified
(relaxed) differential inclusion:
for almost all t £ [a, 6], x'{t) G со F(t,x(t)) (10.18)
Theorem 10.4.3 (Filippov-Wazewski) Let у be a solution to the
relaxed inclusion (10.18) on [a,b]. Assume that F and у satisfy
assumptions (10.17) and that the map t ~> F(t,y(t)) is integrably
bounded on [a,b]. Let η(·) be defined as in Theorem 10.4-1-

402
10- Differential Inclusions
If 77(6) < β, then for every δ > 0 there exists α solution χ to
(10.16) on [a,b] satisfying
s(0) = ?/(0) & ||s-y||c < 5
The next theorem is a simple consequence of Theorem 10.4.3 and the
Convergence Theorem 7.2.1.
Theorem 10.4.4 (Relaxation Theorem) Let F :[a,b]x X *-* X
be a set-valued map with closed nonempty images. Assume that F
is measurable with respect to t and that there exists к G L1(a, b; R+)
such that for almost every t G [a, b],
F(t,-) is /c(i)-Lipschitz & V χ e X, F(t,x) С k(t)B
Then the solutions to (10.16) are dense in the set of solutions to
the relaxed inclusion (10.18) for the metric of uniform convergence.
Corollary 10.4.5 Let S?°bAxo) denote the set of solutions to (10.18)
on [a, b] with x(a) = xq. We posit all assumptions of Theorem 10.4-4·
Then for every xq £ X, the closure of S[atb](xo) in the metric of
uniform convergence is equal to <Su°u(£o)·
Filippov's theorem allows us to investigate infinitesimal
generators of reachable maps of the nonlinear dynamical system (10.16.)
For all a < t < Τ < b and xq € X, define
R(T,t)x0 = {x(T)\ χ € S[tiT\(xo)}
It is called the reachable set of (10.16) (or of F) from (t,xo) at time
T.
Reachable sets enjoy the semigroup property
Vi > s, R(T,s)x0 = R(T,t)R(t,s)x0
When F is sufficiently regular, the set coF(t, xq) is the infinitesimal
generator of the semigroup R(-, t)xo in the sense that the "differential
quotients" (R(t + h, t)xo — xo)/h converge to coF(t, xq).

10.5. Derivatives of the Solution Map
403
For that purpose, we posit the following assumptions:
i) V (t, x) € [a, b] χ X, F(t, x) is nonempty and compact
ii) V χ € X, F(-,x) is continuous on [a, 6]
<
ш) V (ίο, so) € [a, b] χ Χ, there exists a neighborhood
N с[а,Ъ]хХ, L>0 such that V {t, x), (t, у) е Л/",
F(i,s) С ^,у) + Ь||х-у||Б
(10.19)
Theorem 10.4.6 Assume that conditions (10.19) hold true. Then
for every (ίο,^ο) € [a,b[xX, for all (t,x) near (ίο, so) and all small
h>0
R(t + h,t)x = χ + /ι coF(i0,so) + o(t,x,h) (10.20)
гу/iere
Ит оЦМ = о .
(ί,ι)—>(ίθ,ΐο). Ί—>0+ Д
Remark — Equation (10.20) has to be understood in the
following way
R(t + h,t)x С x + hcoF(t0,x0) + \\o(t,x,h)\\B
and
x + hcoF(tQ,xQ) С R{t + h,t)x+\\o(t,x,h)\\B □
10.5 Derivatives of the Solution Map
We now provide estimates of the contingent, adjacent and circatan-
gent derivatives of the solution map S := <S[o,T] associated with the
differential inclusion
for almost all t e [0,Г], x'(t) e F(t,x(t)) (10.21)
where -F: [0,Γ] xl^l

404
10- Differential Inclusions
In this section the set-valued map F is regarded as a family of set-
valued maps χ ~> F(t, x) and the derivatives are taken with respect
to the state variable only.
Let ж be a solution of the differential inclusion (10.21.) We
assume that F satisfies the following assumptions:
г) V χ G X, F(-, x) is measurable
ii) V t e [0, Г], VieX, F(t, x) is a closed set
(10.22)
Hi) 3 β > 0, k(-) € 1^(0, Γ) such that for almost all
t e [0, Γ], F(t, ·) is k(t) - Lipschitz on x(t) + βΒ
Consider the adjacent variational inclusion, which is the
"linearized" along the trajectory χ inclusion
( w'(t) € L»bF(i,x(i),z'(i))K*)) a.e. in [0,Γ]
Ί w(0) = и
where и G X. In Theorems 10.5.1, 10.5.2 below we consider the
solution map S as the set-valued map from Rn to the Sobolev space
W^CO.TjRT1).
Theorem 10.5.1 (Adjacent Variational Inclusion) If the
assumptions (10.22) hold true, then for all и G X, every solution w G
W1,l{Q,T;X) to the linearized inclusion (10.23) satisfies
w e DbS(x(0),x)(u)
In other words,
{w{-)\ w'(t) € DbF{t,x{t),x'{t))(w(t)), w{0) = u}c DbS{x{0),x)(u)
Proof— Let hn > 0 be a sequence converging to 0. Then, by
the very definition of the adjacent derivative and Lipschitz continuity
of F(t, ·), for almost all t e [0, Г],
Ш. d Ut), *(«,»w+*.»(«»-*(«η _ 0 (10.24)

10.5. Derivatives of the Solution Map
405
Moreover, since
x'{i) G F{t,x{t)) a.e. in [Ο,Γ]
by (10.22), for all sufficiently large η and almost all t G [Ο,Γ]
d(x'(t) + hnw'{t),F(t,x(t) + hnw(t))) < hn(\\w'{t)\\ + k{t)\\w{t)\\)
This, (10.24) and the Lebesgue dominated convergence theorem yield
L
■T
d(x'{t) + hnw'(t),F(t,x(t) + hnw{t)))dt = o{hn) (10.25)
where limn_>oo o{hn)/hn = 0. By the Filippov Theorem and by
(10.25) there exist Μ > 0 and solutions yn G S(x(0) + hnu)
satisfying for all large η
Ьп-х1 - Κ-ω'\\ί1{0ιΤ,χ) < Mo{hn)
Since
(yn{0)-x{0))/hn = и = w(0)
this implies that
lim VJLZ^ = w mC(0,T;X); lim Vn~~X = w' ϊηΐΛθ,Γ;Χ)
тг-юо hn 71-4CO hn
Hence
lim / зто)+к„)-шл=0
я-юо \ hn J
Since ?x and ги are arbitrary the proof is complete. □
Consider next the circatangent variational inclusion, which is the
linearization involving circatangent derivatives:
ί w'(t) G CF(t,x(t)>x'(t))(w(t)) a.e. in [Ο,Γ]
1 ги(0) = u ^ ' '
where и £ X.
Theorem 10.5.2 (Circatangent Variational Inclusion) Let us
assume that conditions (10.22) hold true. Then for all и G X, every
solution w G ^■'^(Ο,Γ-,.Χ') to the linearized inclusion (10.26) satisfies
w£ С5(ж(0),ж)(?х).

406
10- Differential Inclusions
In other words,
{w(-) | w'{t) € CF(t,x(t),rf(t))(w(t)),'w(0) = u} C CS{x{0),x)(u)-
Proof — Consider a sequence xn of solutions to (10.21)
converging to ж in W1,1(0, T; X) and let hn —> 0+. Then there exists
a subsequence Xj .— Хщ
such that
lim x'-(t) = ж'(£) a.e. in [0,T] " (10.27)
i—»oo
Set Xj = /in.. Then, by the definition of circatangent derivative and
by (10.27), for almost all t e [0, Γ]
to лU),^wHy-tf»-*^ _ 0
Moreover, using the fact that x'j(t) G F(t,Xj(t)) a.e. in [0, T],'we
obtain that for almost all t € [0, Г] and all large j
d (x'jit) + Xjw'itlFfaxjtt) + Xjw(t)Yj < \j (\\w'{t)\\ + k(t) \\w(t)\\)
This, (10.28) and the Lebesgue dominated convergence theorem yield
( d(a;^(i) + AJV(i),F(i,a;i(i) + Ajii;(i)))di = o{Xj) (10.29)
where limj_>coo(Aj)/Aj = 0. By the Filippov Theorem and (10.29),
there exist Μ > 0 and solutions yj G S(xj(0) + Xju) satisfying
IIyj - x'j - Xjw'\\ < Mo{hj)
Since
(yj(0)-Xj(0))/Xj = и = ги(0)
this implies that
Pi ~ ai - ... ;„ ,νη т. vv „„ ?ir^i _
Hence
lim Οΐ—Ξί = ω inc(0,T;X); Urn L^ = ω' ίηΐ/(0,Τ;Χ)
J—>CO Aj j—>CO Xj
lim d L S(*№ + hnju)-Xj\ = o 30)
J-»00 V bnj /

10.5. Derivatives of the Solution Map
407
Therefore we have proved that for every sequence of solutions xn to
(10.21) converging to χ and every sequence hn —> 0+, there exists a
subsequence which satisfies (10.30.) This yields that for
every sequence of solutions xn converging to χ and hn —> 0+
v , / S(xn(0) + hnu) - xn\
,hm α w, —-—i = 0
™->°° V. hn J
Since и and w are arbitrary the proof is complete. □
We consider now the contingent variational inclusion
\w'{t) e coDF{t,x{t),x>{t)){w{t)) a.e. in [0,T\ (1Q 31)
1 w(0) = и V ■ )
Theorem 10.5.3 (Contingent Variational Inclusion) Consider
the solution map S as a set-valued map from Rn to И^1,оо(0, Г; Rn)
supplied with the weak-* topology and let x(-) be a solution to
differential inclusion (10.21) starting at xq.
Then the contingent derivative DS(xo,x(·)) of the solution map is
contained in the solution map of the contingent variational inclusion
(10.31), in the sense that
DS{xq,x{-)){u) С {w{-)\ w'{t) e WDF(t,-x(t),s?(t))(w(t)),w(p) = u}
Proof — Fix a direction u£R" and let w(-) be an element
of DS(xo,x(-))(u). By definition of the contingent derivative, there
exist sequences of elements hn —> 0+, Un —> и and wn(·) —> w(-) in
the weak-* topology of W1'00^, Г; Rn) and c> 0 satisfying
' 0 IKCOII < c a-e- in [°>r]
ii) x'(t) + hnw'n(t) e F(t,x(t) + hnwn{t)) a.e. (10.32)
Hi) wn(0) = un
Hence
i) wn(·) converges pointwise to w(-)
(10.33)
ii) w'n{·) converges weakly in Χ^Ο,Γ-,Κ71) to w'(-)

408
10- Differential Inclusions
By Mazur's Theorem and (10.33) и), a sequence of convex
combinations
CO
vm{t) := J2 a^w'pit)
p=m
converges strongly to w'(·) in I'1(0, Г; X), where a^ > 0,
CO
p=m
and for every m, a^ ^ 0 only for a finite number of p.
Therefore a subsequence (again denoted) vm{-) converges to w'(-)
almost everywhere. By (10.32) i), ii) for all ρ and almost all ΐ € [0, T\
w'p(t) e —{F(t,x(t)+hpwp(t))-xl{t))ncB
tip
Let t € [0,Γ] be a point where wm(i) converges to w'(t) and x'(i) g
-F(i, x(t)). Fix an integer η > 1 and e > 0. By (10.33) г), there exists
m such that hp < 1/n and ||гир(г) — w(t)\\ < 1/n for all ρ > т.
Then, by setting
$(y,h) := ±{F(t,x(t) + hy)-x'(t))ncB
we obtain that
/
vm(t) & со [J Ф(у,/1)
\ье]о,$[,у&оЮ+±в
and therefore, by letting m go to oo, that
/
w'(t) £ со (J Ф(у,/1)
^е]о,^],уешМ+^в
Since the subsets Ф(у, /г) are contained in the ball of radius c, we
infer that u/(i) belongs to the closed convex hull of the upper limit
by Lemma 1.1.9. Hence
/
w
''(0 e» Π
ε>0, η > 1
(J Ф(у,/0 + еЯ
\he]o,±],vew(t)+±B

10.5. Derivatives of the Solution Map
409
We observe now that
Π U $(y,h)+eB \cDF(t,x(t)^(t))(w(t))
ε>0;π>1 ^e]0,i],yew(i)+iB J
to conclude that w(-) is a solution to the contingent differential
inclusion
w'{t) e coDF(t,x(t),x'(t))(w(t)) a.e. in [Ο,Γ]
w(0) = и
Since w G DS(xo,x(-))(u) is arbitrary we end the proof. Π

411
Bibliographical Comments
Chapter 1
We already have mentioned that the concepts of upper and lower
limits of sets are due to Painleve in 1902 [318] as it is reported by
his student Zoretti in [467].
The compactness result (Theorem 1.1.7) was obtained by Zarankie-
wicz [462]. Many other proofs were provided since. The Duality
Theorem 1.1.8 was proved first by Walkup & Wets [427].
Some basic calculus of upper and lower limits of sets and set-
valued maps appeared before and after the second World War in the
two volumes of Kuratowski's basic monograph [255]. The continuous
version of Proposition 1.2.6 on the lower limits (or lower semiconti-
nuity) of intersections of convex-valued maps can be found in [35] for
instance.
The square product of set-valued maps was introduced and used
in>[40] for studying local observability of control systems under
uncertainty.
Upper and lower semicontinuous maps were introduced by Bouli-
gand [80] and Kuratowski [254] and [255]. Berge's-book [66] played
an important role in disseminating these results among
mathematical economists, as one can see by reading Debreu's basic monograph
[132]. The convergence issues were then reexamined by Choquet in
[100] in a more general case. Pseudo-Lipschitz set-valued maps have
been introduced in [47,49] in the framework of the Inverse Function
Theorem in finite-dimensional spaces, and later, in [39] for Banach
spaces, and have also been studied by Rockafellar in [370].
The Genericity Theorem is due to Kuratowski [254,255]. It was
extended later by Choquet [100] and Shi Shuzhong [388].
- The Maximum Theorem appeared in Berge's monograph [66].
The lower semicontinuity criterion of an infinite intersection of
lower semicontinuous set-valued maps was proved in the context of
differential games [51]. - -

412
Bibliographical
Chapter 2
Closed convex processes and their transposes have been
introduced by Rockafellar [356] and further studied in [358,360]. Robinson-
Ursecu's Theorem has been proved in [347,348,349,417] and the
Uniform Boundedness and the Crossed Convergence Theorem in [42].
The Closed Image Theorem is taken from [35]. Closed convex
process are related to Ioffe's fans ([233,238]) for which results of linear
functional analysis can also be extended.
Linear processes and their invariance properties were studied in
[197].
Chapter 3
The literature on nonlinear analysis is too broad to give any fair
account in this short review.
Ky Fan's Inequality has been proved in [160] and the proof we
gave is derived from Browder in finite dimensional vector-spaces and
from [45,35] in the general case, where extensions are provided and
further bibliographical comments can be found. The Equilibrium
Theorem was derived from Ky Fan's papers [159,161] in [45]. See
also recent contributions of Simons [395]. One can find in [35] further
developments on the existence of solutions to inclusions. Kakutani's
Theorem appeared in [243] and the set-valued version of the Leray-
Schauder Theorem in [45].
Ekeland's variational principle has been proved in [154] and used
very often since then. For more recent views, see [35, Chapter 5],
[156] and their bibliography. The proof given here is due to Siegel
[393].
One can find in [2] the history of the concept of derivative of maps
between Banach spaces, from Peano, Volterra, Gateaux and Frechet
to the eve of the nonsmooth era. We mention only the following
[323,201,199,200].
The Inverse Function Theorem for surjective single-valued maps
in Banach spaces is due to Graves [205,206]. See also the earlier
paper [220] in collaboration with Hildenbrandt. Ljusternik proved in
[273] that the tangent space to /_1(0) is the kernel of f'(x) when
the Frechet derivative /' is continuous and surjective (See [272] for
instance.) This has been extended by Leach in [265] to strong differ-

Comments
413
entiable maps.
The extension of this theorem to problems with constraints and
set-valued maps in finite dimensional spaces appeared in [266,46,47,
49] and [35, Chapter 7]. The use of the localization property of
Ekeland's Variational Principle is due to Lebourg [266]. It has been
adapted to the case of infinite dimensional spaces in [39,178,182,189,
194].
This theorem has been extended to the case when the surjectivity
of the first derivative is lacking, and also, when the set-valued map
is defined on metric spaces, thanks to the concept of high order
variations introduced in [177,182,194,189]. Such results were motivated
by control theory, in particular by local controllability of control
systems in [180,190]. See [194] and the forthcoming monograph [195]
for an exhaustive presentation of the theory and other applications.
Monotone maps have been introduced by Zarantonello [463]. For
a comprehensive account of monotone maps, see for instance the
books of Brezis [83], J.-L. Lions [270] and Browder [84]. Maximal
monotone maps were characterized by Minty [290] and the fact that
the subdifferential is maximal monotone is due to Rockafellar [354].
See for instance [35, Chapter 6] for developments of this important
domain of nonlinear analysis and further bibliography.
The existence of eigenvectors of a closed convex process on cones
with compact soles appeared in [31] for characterizing controllability
of'closed convex processes. Existence of positive eigenvectors and
extension of the Perron-Frobenius Theorem appeared in [43] and can
be-found in [35, Chapter 3].
Chapter '4
It is impossible to give an exhaustive account of the many papers
in which the various concepts of tangent cones appeared in the
literature. They have been introduced again and again in a manifold of
contexts, many of them unknown to the authors.
The need to introduce contingent and paratingent directions was
felt by Bouligand in the thirties in order to differentiate non differen-
tiable functions (see [78,80,79].) This was taken up by Zaremba [465]
and Marchaud [280] to define solutions to differential inclusions. See
also the paper [100] by Choquet.
The contingent cone was used under the names of tangent cone
and sequential tangent cone in optimization and control theory in

414
BibJiograpbicai
the fifties (see for instance the books by Neustadt [300] and Hestenes
[218]) overshadowed during the sixties when convex analysis was
expending, and was resurrected in this field as a parent pauvre of the
tangent cone introduced by Clarke in the middle of the seventies in
his thesis [106]. Clarke and Rockafellar's papers extended the main
results of convex analysis to the case of locally Lipschitz functions
and lower semicontinuous functions respectively. See also Clarke's
book [113].
Meanwhile, Russian and Eastern European mathematicians did
use many of these cones. Let us mention the books by Dubovitskij
& Miljutin [147], by Ioffe & Tikhomirov [229] and by Pchenitchny
[321] for instance, just to cite a few.
This period saw an explosion of notions of tangent cones. We
chose to emphasize only three of them, the contingent cone since it
appears that it is the one which is at the origin of most theorems,
the adjacent cone which is needed in Lebesgue and Sobolev spac'es
(see [172,173,183,181,186,196]), and the tangent cone introduced by
Clarke, because of its regularity and convexity properties.
The fact that it coincides with the contingent cone for sleek
subsets was first proved in [34] in finite dimensional vector-spaces. The
proof that the lower limit of the contingent cones is contained in
the Clarke tangent cone is due to Cornet in [118,117] in the case of
Hilbert spaces (see also [325]) and the extension to any Banach space
is due to Treiman [411]. We provided here a different proof based on
Ekeland's Variational Principle. See also [72] on this topic.
The characterization of the polar of a viability domain of a closed
convex process appeared in [31,32] to study the duality relations
between controllability and observability of closed convex processes.
The calculus of tangent cones to intersections and inverse images
appeared in [47] (finite dimensional case, generalizing formulas due
to Rockafellar in [361] under stronger transversality conditions) and
[39] (Banach spaces.) Theorem 4.3.9 was shown to us by Rockafellar.
A discussion of results on upper limits of normal cones, which
we deduce by polarity from the relations between the lower limit
of the contingent cones and the Clarke tangent cones can also be
found in Rockafellar and Wets' book [355]. These authors also proved
that the polar cones to the contingent cones coincide with the sets
of proximal normals in finite dimensional vector-spaces. See also
[294,232,233,237] and [365,72,70,104] among other contributions to
these issues, which were not treated in this book.

Comments
415
Convex kernels (or asymptotic cones) of adjacent (or
intermediate) cones have been used in [172,173,181] in the study of optimal
control of differential inclusions because the use of the mere convexity
of Clarke tangent cones required too strong properties in the infinite
dimensional spaces of solutions.
The results on paratingent cones and hypertangent cones are
taken from Shi Shuzhong's papers [391,392]. Ioffe's menagerie
appeared in [236]. See also [122], [283] among other contributions.
The results presented in Sections 4.6 and 4.7 appear here for the
first time.
Chapter 5
There were many definitions of pointwise derivatives of set-valued
maps proposed in the literature. Let us mention only Banks- Jacobs'
[58], Martelli-Vignoli's [282], DeBlasi's [125] and Petcherskaja's [326]
to name a few.
The concepts of graphical derivatives and their use in the
extension of the Inverse Function Theorem have been initiated in [46] for
the contingent derivative and in [47,49] for the circatangent
derivatives. Adjacent derivatives have been introduced and used in control
theory in [172,173,183,181,186,196]. See also Polovinkin [332] who
also used the graphical approach to define derivatives of set-valued
maps, as well as [378,373]. Proposition 5.2.6 was proved in [181].
Circatangent derivatives of the subdifferential of a convex
function, regarded as generalized Hessians, have been introduced in [47,
49] to study the stability of solutions to convex minimization
problems. More generally, derivatives of maximal monotone maps have
been extensively studied by Rockafellar in [371].
The idea to use Ekeland's Variational Principle for obtaining local
surjectivity criteria for single-valued maps is due to Ekeland and was
taken up in [46] in the case of set-valued maps by involving the
surjectivity of the contingent derivative. This result was improved
by Lebourg in [266], who used the localization property of Ekeland's
Variational Principle. This appoach has been further exploited in
[47,49,39,177,178,182] to provide more and more general versions of
this useful theorem. Analogous approaches have been used by Ioffe,
in [238] for instance.
For nondifferentiable single valued maps, an inverse function
theorem using generalized Jacobian can be found injthe papers [109,113]

416
Bibliographical
of Clarke. See also the contributions [434] of Warga, [214] of Halkin,
[302], and [232,237,238] of Ioffe, who uses the concept of fans, and
the references of these papers to Russian literature.
The inverse mapping theorem has been extended to the case when
the set-valued map is defined on a metric space, thanks to the concept
of high order variations introduced in [182,178,189].
Local inverse univocity and injectivity of set-valued maps has
been proved in [39,38,40] and used in the study of local observability
of control systems and differential inclusions. The localization results
are new.
The new field of "Qualitative Physics" (see for instance [67,130]),
an active branch of Artificial Intelligence, as well as of mathematical
economics (see [259,260]), provide many new motivations for using
the differential calculus of set-valued maps and set-valued analysis.
The results on qualitative analysis appear here for the first time.
Chapter 6
The characterization of the sub differential of a convex function
through the normal cone to its epigraph goes back to Moreau and
Rockafellar when they introduced the concept of subdifferential.
Clarke used this characterization in [106] to define the generalized
gradient of any lower semicontinuous function through the normal cone
to the epigraph. The dual relation between subdifferential of locally
Lipschitz functions and the epiderivatives (regarded as the support
function of the subdifferential) was established by Clarke.
Rockafellar extended it to lower semicontinuous functions in [362,363,
364,366]. Contingent epiderivatives have been introduced in [46] in
the study of Lyapunov functions for differential inclusions, where
they play a natural and crucial role. See also [33, Chapter 6] and"
[50]. They are also fundamentally involved in the study of Hamilton-
Jacobi equations and closely related to viscosity solutions: see [179,
185,188]. The proof of Theorem 6.1.6 is derived from an idea of
Quincampoix.
The epidifferential calculus has been developed in [39]. It has
been established by Clarke [113] in the case of locally Lipschitz
functions and by Rockafellar [362,363,364,366] under stronger transver-
sality conditions than the ones used in [39].
Relations between the subdifferential and the local subdifferential
used by Crandall, Evans and P.-L. Lions were elucidated in [191]. Lo-

Comments
417
cal sub and super differential were extensively exploited for studying
viscosity solutions of Hamilton-Jacobi equations: see [119,120] and
the bibliography of P.-L. Lions' book [271].
Other generalizations of the gradients or Jacobians using
approximations procedures of functions or maps have been proposed. Let
us mention the Warga's derivate containers [432], Halkin's [215] and
[174] among other contributions using this approach.
Demianov and Rubinov and their collaborators introduced pairs
of generalized gradients to break down the lack of symmetry due
to the epigraphical approach. See [140,141] for instance, and their
bibliography.
We refer to the books of Rockafellar [358], Moreau [296], Ekeland-
Temam [149], [35, Chapter 4], and the forthcoming book [355] by
Rockafellar and Wets for an account of convex analysis. For a short
introduction to convex analysis, one can also consult [48].
Results of Section 6.5 appear here for the first time.
Many suggestions have been made to define second-order
approximations of functions. The one we propose fits in the unified approach
of this book. Applied to the epigraph of a function, it yields Rock-
afellar's extension (parabolic derivatives) given in [377,369] of
analogous concept introduced by Ben-Tal and Zowe in [64]. The formula
on the second order contingent epiderivative of the Moreau-Yosida
approximation of a lower semicontinuous function is new.
Another concept of generalized Hessian was introduced in [47,
49] by taking circatangent derivatives of subdifferentials of convex
functions.
In the forthcoming book [355] by Rockafellar and Wets,
generalized Hessians are defined as со-differentials' of the generalized
gradients. See also the contributions of Hiriart-Urruty on this topic.
Chapter 7
This chapter is just a very short introduction to epigraphical (and
graphical) convergence.
The first results on this subject are due to Wijsman [454,454] and
Walkup & Wets [427] in the convex case.
It has been extended to infinite dimensional spaces and applied
to the approximation of variational inequalities by Mosco [297], and
then extended by Joly [242] and R. Robert [345].

418
Bibliographical
Extension to the non convex case has known a considerable
expansion since the resurrection of this topic with the concepts of Γ
and G-convergence by de Giorgi and his collaborators (see for
instance [127,126,128,128]) in the framework of calculus of variations,
with the concepts of epiconvergence by R. Wets and his
collaborators (see [447,379,380,381,382,448,450,451,146,25]) in the framework
of convergence of random sets and functions and with the papers of
Attouch motivated by the theory of homogenization in [27,28] and
above all in his book [30]. We refer to this book and the forthcoming
monograph by Rockafellar and Wets [355] for further references and
an exhaustive presentation of this topic, for which this chapter can
be used as an introduction.
The results on graphical convergence of maximal monotone maps
are due to Attouch [29] and the relations between lower graphical and
pointwise convergence appeared in [42].
The Convergence Theorem has been used by several authors
studying differential inclusions with upper semicontinuous convex-
valued right-hand side (see [33, Chapter 1] for instance.) We state
here a more general version of this theorem.
The results on the limits of infima are taken from Wets [449].
The theorem on conjugate functions of epilimits was proved by
Wijsman [453,454] in the finite dimensional case and by Mosco [297],
Joly [242] and R. Robert [345] in the infinite dimensional case.
The theorem on the lower limits of gradients of differentiable
functions converging uniformly is new. It extends to the infinite-
dimensional case a theorem of M. Crandall, C. Evans & P.-L. Lions
[119] which allows one to study the stability of viscosity solutions
to Hamilton-Jacobi equations. See P.-L. Lions' book [271] for
further comments on sub differentials of viscosity solutions to Hamilton-
Jacobi equations. Their relations with generalized gradients have
been investigated in [188,89].
The theorem on convergence of subdifferentials of convex
functions is due to Attouch [28,26]. See also his monograph [30].
Chapter 8
The measurable selection theorem, which, as many other
important mathematical innovations of this century, is due to J. von
Neumann. This result was given in a stronger form by Kuratowski and
Ryll-Nardzewski [253]. In [92], Castainghas proved that measurable

Comments
419
selections are dense, the result which is often referred to now as the
Castaing representation theorem.
The interested reader can learn more about the history of this
subject from Wagner's survey [426] of measurable selection theorems
and also from Ioffe's survey [231] of Soviet literature on this subject.
The book [91] by Castaing L· Valadier remains the reference
for measurable set-valued maps. In particular, the proofs of
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Chapter 9
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288] and the Approximate Selection Theorem of an upper semicon-
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The idea to use the Steiner points of the images of a set-valued
map as selection and parametrization procedures is due to Lojasiewicz

420
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Index
5-generalized gradient 249
σ—algebra 306
σ-finite measure 306
adjacent cone 126
adjacent derivative 189
adjacent epiderivative 237
approximate selection 358
asymptotic cone 159
asymptotic epiderivative 300
atom 329
Attouch's theorem 299
Baire's theorem 44, 58
ball 17
biorthogonal subspace 63
bipolar cone 63
bipolar set 63
bipolar theorem for closed convex
processes 70
bipolar theorem 64
bitranspose 67
Borel σ-algebra 306
bounded family of processes 61
bounded-valued map 36
Caratheodory map 311
Caratheodory parametrization 376
Caratheodory selection theorem 374
Castaing's theorem 319
Cellina's theorem 358
characterization of monotone maps
107
Choquet's theorem 163
circatangent cone to a sequence of
sets 166
circatangent cone 127
circatangent derivative 189
circatangent epiderivative 237
Clarke normal cone 157
Clarke tangent cone 127
closed convex process 56, 314
closed convex set-valued map 34,
373
closed graph criterion 73
closed graph theorem 60
closed range theorem 64
closed set-valued map 34, 41
closed-valued map 36, 56
coerciveness 88
compact-valued map 36
complementary convex 164
complete σ- finite measure space
310
complete σ-algebra 307
complete measure space 307
composition product 37
concave function 222
cone with compact sole 114
conjugate function 256
consistent sequence of functions
96
consistent 203
constrained inverse function
theorem 96, 207
constraint qualification condition
140
contingent cone 32, 121
contingent derivative 181
457

458
contingent epiderivative 225
contingent Hessian 299
contingent hypoderivative 226
contingently epidifferentiable 225
continuous partition of unity 81,
386
continuous selection 355
continuous set-valued map 40, 141
convergence of gradients 295
convergence of subdifferentials 297
convergence theorem 271
convex epiderivative 241
convex function 222
convex hull of an upper limit 26,
273
convex kernel 159
convex set-valued map 34, 56
convex-valued map 36
core 36
correspondence 33
cow 119
criterion of pseudo-Lipschitzeanity
101
crossed convergence 61
curvaceous shape 354
cut set-valued map 43
dense family of selections 310
derivable cone 127
derivable set-valued map 182
derivable set 127
differential geometer 119
differential quotient 94
Dini directional derivative 94
directed set 19
distance to a set 17
domain 34
dual norm 20
duality map 125
duality pairing 19
duality relation 258
Dubovitskij-Miljutin cone 125
eigenvalue of a convex process 114
eigenvector of a convex process 114
Ekeland's rule 91, 233
epicontinuous 285
epiderivative 225
epigraph 222
epilimit 275
episleek function 225
equilibrium map 207
equilibrium theorem 84, 276
extension of a set-valued map 36,
236
extremal element 329
extremal face 88, 343
extremal selection 330
Fenchel's inequality 256
Fermat's rule 232
fixed point theorem for inward or
outward maps 87, 250
Frechet differentiable single-valued
map 94, 250
Gateaux differentiable single-valued
map 94
generalized gradient (Clarke) 249
generalized sequence 19
generic continuity 44
generic property 44
God (spherical) 119
gradient 94
graph of the transpose of a
process 67
graph of a map 34
graphical convergence of monotone
maps 270
graphical convergence 267
graphical upper/lower limit of set-
valued maps 267
Graves' theorem 95
Hausdorff distance 365
higher order adjacent subset 171
higher order circatangent subset
171
higher order derivatives 216

459
higher order contingent subset 171
higher order epiderivatives 260
hypertangent cone 164
hypograph 222
hypolimit 276
image of a set-valued map 34
indicator of a set 223
inf-compact 281
injective (globally) 209
integrably bounded 326
integral of a set-valued map 327
intermediate cone 126
invariant space 72
inverse image 36
inverse map 34
inverse set-valued map theorem 204
inverse stability theorem 100
inward map 87
Kakutani's fixed point theorem 87
kernel of the derivative 187
Krummungsschwerpunkte 365
Ky Fan's inequality 80
abor value 78
jax's principle 96, 99
jeray-Schauder's theorem" 90
Am sup inf 192
imit of infima 281
imit of set-valued maps 41
imit of sets 17
inear extension of a set-valued map
346
inear process 56
/ipschitz parametrization' 379
jipschitz selection 372
jpschitz set-valued map 41
jcal subdifferential 253
jcal superdifferential 253
>cal uniqueness criterion 103
tcally injective set-valued map 209
>p-sided paratingent derivative 201
/wer Dini derivative 231
wer epicontinuous 285
lower epilimit 276
lower hypolimit 276
lower limit of set-valued maps 41
lower limit of sets 17
lower semicompact 281
lower semicontinuity criterion 49
lower semicontinuous set-valued map
39
lower semicontinuously differentiable
map 188
Lyapunov's Convexity theorem (in
Banach spaces) 340
Lyapunov's Convexity theorem 339
marginal function 48
mathematical striptease 93
maximal monotone map 106
maximum theorem 48, 258
Mazur's theorem 271
measurable (single-valued map) 307
measurable selection 308
measurable space 306
measurable/Lipschitz selection
theorem 375
measurable/Lipschitz map 373
measurable 307
Michael's theorem 355
minimum (bare) 93
minimal map 360
minimal selection theorem 363
minimum (bare) 93
Minkowski's difference of sets 159
Minty's theorem 108
Minty's trick 108
monotone map 104
Moreau's theorem 258
"Moreau-Yosida's approximation 258
Mosco's theorem 290
multifunction 33, 258
multivalued function 33, 270
negative polar cone 63
nonatomic measure 329
nontrivial extended function 222

460
nontrivial set-valued map 34
norm of a process 57
normal cone to a convex subset
139
normal cone to a set 157
open mapping theorem 57
Oresme-Fermat's rule 219
orthogonal 63
outward map 87
paratingent cone to a sequence of
sets 167
paratingent cone 162
paratingent derivative 201
paratingent epiderivative 241
Po'mcare's continuation method 89,
201
point to set map 33
pointwise convergence of set-valued
maps 268
pointwise inverse function theorem
101
pointwise limit of set-valued maps
313
polar cone 25, 63
polar of a viability domain 144
polar set 28, 63
Polish space 306
positive polar cone 63, 146
positively homogeneous function
222
positively homogeneous set-valued
map 56
process 56
product of set-valued maps 37
projection 125
proper map 28
proper set-valued map 43
pseudo-convex function 233
pseudo-convex set-valued map 182
pseudo-convex set 145
pseudo-Lipschitz set-valued map
41
reflexive space 20
residual 44, 163
resolvent of a monotone map 106
restriction of a single-valued map
34, 182
Robinson-Ursescu's open mapping
theorem 58, 194
Rocquefort 321
Sauterne 321
selection of a set-valued map 355
separable space 20
separation theorem 64
sequentially inf-compact 281
sequentially weak upper limit 20
set-valued Graves' theorem 205
set-valued map 33
Shi Shuzhong's theorem 165
simple map 307
simplex 82
sleek set-valued map 182
sleek set 101
smooth Banach space 125
sole of a cone 114
solvability theorem 85, 130
square product of set-valued maps
37
stability assumption 96
stable family of set-valued maps
203
star-shaped set 145
Steiner point 365
strict set-valued map 34
strictly convex space 360
striptease (mathematical) 93
strong convergence of a sequence
in a Lebesgue space 272
strong measurability 307
subdifferential 248
subgradient 248
support function of a set 62
support function of the transpose
75

surjective closed convex process 57
tangent cone to a convex cone 143
tangent cone to a finite
intersection 149
tangent cone to direct image 153
tangent cone to inverse image 148
tangent cone to the interior of a
set 140
tangent cone to a convex set 83,
138
tangent cone to a sleek subset 130
tangentially regular 127
tarte Tatin 321
transpose of a process 67
transpose of a product 68
transpose of the restriction 69
transversality condition 32, 96, 147
uniform boundedness theorem 61,
152
uniformly sleek set-valued map 205
uniformly sleek set 101
uniformly smooth 133
unit ball 17, 158
upper epicontinuous 285
upper epilimit 275
upper hemicontinuous map 74, 175
upper hypo-limit 276
upper limit of a generalized
sequence 19
upper limit of set-valued maps 41
upper limit of sets 17
upper semicontinuous set-valued
map 38
value of a set-valued map 34
variational system 285
viability domain 84
viscosity solution 253
weak contingent cone 252
weak measurability 307
weak-* topology 19
weakened topology 19
Xenophanes of Colophon 119
461
Yosida approximation 111
Zarankiewicz's theorem 23