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11 J.L. Alperin Local representation theory
12 P. Koosis The logarithmic integral I
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14 S.J. Patterson An introduction to the theory of the
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15 H.J. Baues Algebraic homotopy
16 V.S. Varadarajan Introduction to harmonic analysis on
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17 W. Dicks &l M. Dunwoody Groups acting on graphs
18 L.J. Corwin & F.P. Greenleaf Representations of nilpotent
Lie groups and their applications
19 Ft. Fritsch & Ft. Piccinini Cellular structures in topology
20 H KHngen Introductory lectures on Siegel modular forms
22 M..T. Collins Representations and characters of finite groups
24 ■ H. Kunita Stochastic flows and stochastic differential -equations
25 P. Wojtaszczyk Banach spaces for analysts
26 J.E. Gilbert & M.A.M. Murray Clifford algebras and Dirac
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28 K. Goebel & W.A. Kirk Topics in metric fixed point theory
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33 C. Soule et al Lectures on Arakelov geometry

A Primer of
Nonlinear Analysis
Antonio Ambrosetti
Scuola Normale Superiore, Pisa
Giovanni Prodi
Department of Mathematics, University of Pisa
-■•056529
\ CAMBRIDGE
* UNIVERSITY PRESS

Published by the Press Syndicate of the University of Cambridge
The Pitt Building, Trumpington Street, Cambridge CB2 iRP
40 West 20th Street, New York, NY 10011-4211, USA
10 Stamford Road, Oakleigh, Victoria 3166, Australia
© Cambridge University Press 1993
First published 1993
Printed in Great Britain at the University Press, Cambridge
Library of Congress cataloguing in publication data available
A catalogue record for this book is available from the British Library
ISBN 0 521 37390 5 hardback

Contents
Preface vii
0 Preliminaries and notation 1
1 Differential calculus 9
1 Frechet and Gateaux derivatives * 9
2 Continuity and differentiability of Nemitski operators 15
3 Higher derivatives 23
4 Partial derivatives, Taylor's formula ? 26
2 Local inversion theorems 30
1 The Local Inversion Theorem 30
2 The Implicit Function Theorem 36
3 A stability property of orbits 38
3 Global inversion theorems 45
1 The Global Inversion Theorem 45
2 Global inversion with singularities 54
Appendix 60
4 Semilinear Dirichlet problems 61
1 Problems at resonance 62
2 Problems with asymmetric nonlinearitics 71
5 Bifurcation results 79

Contents
1 Introduction 79
2 Some elementary examples 82
3 The Lyapunov-Schmidt reduction 89
4 Bifurcation from the simple eigenvalue 91
5 A bifurcation theorem from a multiple eigenvalue 101
Appendix 104
Bifurcation problems 107
1 The rotating heavy string 107
2 The Bcnard problem 112
3 Small oscillations for second-order dynamical systems 119
4 Water waves 123
5 Periodic solutions of a semilinear hyperbolic equation 130
Bifurcation of periodic solutions 136
1 The Hopf bifurcation 136
2 Nonlinear oscillations of autonomous systems 139
3 The Lyapunov Centre Theorem 145
4 The restricted three-body problem 153
Problems 160
Bibliography 165
Index 170

Preface
In the last few decades, once linear functional analysis was quite widely
and thoroughly estabilished, the interest of scientists in
Nonlinear-Analysis has been increasing a lot. On the one hand the treatments of various
classical problems have been unified; on the other, theories specifically
nonlinear, of great significance and applicability, have come out.
This book provides an introduction to basic aspects of Nonlinear
Analysis, namely those based on differential calculus in Banach spaces. The
matter is expressed in a geometric style, in the sense that the results
obtained are often a transposition to infinite dimensions of events which
are intuitive in R2 or R3. Indeed, this was a primary characteristic of
the works of Pincherle, Volterra and Frechet.
The topics treated can be divided into two main parts and arc
preceded by a short chapter in which some introductory material is recalled,
and also the main notation fixed.
In the first part, differential calculus in Banach spaces is discussed,
together with local and global inversion theorems.
The second part deals with bifurcation theory which in spite of its
elementary character is, perhaps, one of the most powerful tools used
in Nonlinear Analysis. Our attention is here devoted almost entirely to
the case of simple eigenvalues, but an accurate analysis of hypotheses is
made, in order to include, for example, also the celebrated Hopf theorem.
A specific feature of Nonlinear Analysis is that the theoretical setting
is strictly linked to applications, especially those related to differential
equations, where the power of nonlinear methods is expressed in a more

viii Preface
striking way. Moreover, a relevant fact to be emphasized is that problems
that are often considered of formidable difficulty, once they are framed in
an appropriate functional setting, may be faced and solved quite easily.
It is, indeed, this aspect, peculiar to Nonlinear Analysis, that has
driven us to leave considerable space to applications to differential
equations, including various important classical problems such as Bernard
Problem, the problem of water waves, the restricted three-body
problem and some others. Thus, in addition to more elementary examples
and applications that usefully accomplish theoretical results, in separate
paragraphs and/or chapters, we deal with those problems which require
more care both in formulation and in resolution.
Tools, still of remarkable importance, such as the theory of Leray-
Schauder topological degree, or the critical point theory, which would
require wider theoretical background and more subtle arguments, are
left out in this treatise.
The book in its outlines is self-contained for a reader who, besides
infinitesimal calculus, is acquainted with fundamental results of Linear
Functional Analysis such as the Hahn-Banach Theorem, the "Closed
Graph" Theorem and the Fredholm Alternative Principle. Only some of
the problems dealing with partial differential equations require a certain
knowledge of Sobolev spaces and therefore, in just a few cases, we refer
to results contained in original papers.
This volume is partially based on an earlier booklet, published in
Italian by the Scuola Normale Superiore di Pisa in the series "Quaderni".
The authors wish to thank the Scuola Normale Superiore for the
encouragement.

0
Preliminaries and notation
This chapter contains the notation and some preliminary tools usod
throughout the book. Almost always, the results are not quoted in the
most general form, but in a way appropriate to our purposes;
nevertheless some of them are actually slightly more general than we strictly
need. For more detail we refer to any book of (linear) Functional
Analysis (for example [Y] or [Br] for topics reviewed in sections 0.1-0.4; to
[Br], [KFS], [GT] for 0.5-0.6).
0.1 Some notation and definitions
Rn will denote the n-dimensioiial Euclidian space with scalar product
x ■ y and norm given by |x|2 = x ■ x.
X, K, Z,... denote (real) Bariach .spaces with norm ||.||x. Il-llr, etc.,
respectively (the subscript will be omitted if no possible confusion arises).
B(x*,r) denotes the ball {x € X : \\x — x*|| < r} and B(r) stands for
B(Q,r).
Tf X' is the topological dual of X the symbol (.,.) will indicate the
duality pairing between X and X*.
Let {xn} be a .sequence in X. We say that x„ converges (strongly) to
x € X, written as xn —► x, if ||xn — x|| —► 0 as n —► oo; we say that xn
converges weakly to x, written as xn —' x, if (ip, xn — x) —► 0 as n —► oo
for all ip € X*.
Let X be a Banach space and let V be a closed subspace of X. A
topological complement of V in X is a closed subspace W of X such that
V n W = {0} and X = V © W; V © W is called a splitting of X.

2
0 Preliminaries and notation
Recall also that, associated with such a splitting, there are (continuous)
projections P and Q onto V and W respectively.
0.2 Continuous mappings
We will deal with continuous maps / : U —► Y, where U is an open
subset of X. Continuity means that f(xn) —► f(x) (strongly) for any
sequence xn strongly convergent to x € X. The set of all continuous
f:U^Y will be denoted by C(U, Y).
0.3 Integration
For continuous maps / : [a, 6] —► Y the definition of the Cauchy integral
is given as in the elementary case, as the (strong) limit of the finite sums
£/(&)(*i — U-i) (with obvious meaning).
From
iis</te)(«i - «<-oii < e<ii/(&)(<( - <i-.)ii < Siii/te)ii(«i - <<-■)
there follows immediately the inequality
b b
|y7w<u||</u/(oiid<-
0.4 Linear continuous maps
The space of linear continuous maps A : X —► Y will be denoted by
L(X,Y). The range of A,R(A), is the linear space {A(x) : x € X}.
Sometimes, when Y = X, we will use the notation L(X) instead of
L(X,X). Equipped with the norm
M||=sup{M(x)||:||x||<l},
L(X,Y) is a Banach space. The identity map in L(X) will be denoted
by Ix-
Hereafter, for linear maps, the notation Ax or A[x] may replace A(x).
An eigenvalue of A € L(X) is a n € C such that the equation Ax = fix
has solutions x^O. Any such solution is an eigenvector associated to n
and Ker (fil — A) is the eigenspace associated to /t. We will be mainly
interested in the case when A € L(X) is compact, namely when A is
completely continuous, if this is the case, the following results hold.
Theorem 0.1 (Fredholm Alternative) Let A € L(X) be compact
and n^O. Then
(i) Ker (fil - A) = {0} if and only if R(pl - A) = X,

0 Preliminaries and notation
3
(ii) R(pl -A) = [Ker (pi - A*)}-1 = {u € X : ty,u) = 0 for all if, €
Ker (pi-A*)}.
Moreover one has the following
Theorem -0.2 Let A € L(X) be compact and p ^ 0. Then
(i) Ker (pi — A) is finite-dimensional and R&age(pl — A) is closed,
(ii) the sequence Ker ((pi — A)n) (n € N) is increasing, that is Ker
((pi - A)m) c Ker ((pi - ^)m+1) /or a// m > 1,
(iii) there exists a finite p € N suc/i £/ia£ Ker ((pi — j4)p) =Ker ((pi —
A)q) if and only if q > p.
The (algebraic) multiplicity of p is the dimension of the linear subspace
U„6NKer((^/ - A)n) = Ker((pl - A)p).
It is worth pointing out that the algebraic multiplicity of p is, in general,
different from the geometric multiplicity, defined as the dimension of
Ker (pi — A) (algebraic and geometric multiplicity coincide for self-
adjoint operators on Hilbert spaces). Hereafter, by the multiplicity of
an eigenvalue p ^- 0 of a completely continuous A e L(X) we will always
mean the algebraic multiplicity. An eigenvalue will be said to be simple
if its multiplicity is 1.
0.5 Function spaces
Let £1 be an open subset of R" with boundary dQ and closure Q.
We will use standard notation for spaces of continuous or differentiablc
real-valued functions Ck(Tl) (k > 0), for Lebesgue spaces Lv(Ct) (1 < p <
oo) or l°°(Q). In some cases we will write C(U) instead of C°(U). The
spaces above are Banach spaces under the norms defined, respectively,
by
||u||c=sup{|u(l)| :xeH},
|H|C» = Yl llD"«llc (/3isamultiindex),
O<\0]<k
/m
(the symbol dx will be omitted whenever there is no ambiguity)
||m||l« =ess sup {|w(x)| : i € 0}.
For k > Oand 0 < a < 1, Cfc,a(12) denotes the spaceof Holder functions

4
0 Preliminaries and notation
with exponent a, namely the u € Ck(Q) such that, for all multi-index
0,\0\ = k,
I \z-yr I
For fc = 0 and a := 1, C0,1(£l) is nothing but the space of Lipschitz-
continuous functions on £1.
Equipped with the norm
IMIc*.° = Mlc» +
+ sup U J, ,„ :x,y,en,x?y, \0\ = k\,
Ck,a(£i) is a Banach space.
In some cases, we shall also work with Sobolev spaces HklP(Q) (fc >
l,p € [l,oo)) equipped with the norm
IMI«»..= £ »D"UII"-
The notation Hk will stand for #fc'2 while Hk(Cl) will denote the closure
of Cq°(£)), the space of C°° functions with compact support in £), under
the norm ||u|jHt,a. Among others, let us recall the following result.
Theorem 0.3 (Poincare Inequality) Let Cl be bounded. Then there
exists a constant c = c(Cl) such that
f \u\2 <c f | Vu|2 for all u € flj
(Q).
As a consequence, ||Vu||/,a is a norm in Hq(Q) equivalent to ||u||Hi,2.
In addition to the Poincare Inequality one has that the embedding of
Hq(CI) in L2(Cl) is compact (Rellich's Theorem). Let us recall that X
is embedded hi Y, X •—* Y, if X C Y and the inclusion i : X —► Y is
continuous. If X *—» Y then 3 c > 0 such that
\\u\\y <c||u||x, for all u € X.
If the inclusion i: X —► V is compact we will write -X" ^-*^-* Y.
The following result is a particular case of the "Sobolev Embedding
Theorems".
Theorem 0.4 Suppose that Cl is bounded open set in R", with
boundary dQ of class C0,1, and let k > 1 and 1 < p < oo.
(i) // kp < n, then Hk'P(Ct) ^ Lq(Sl), for all 1 < q < np/(n - kp),
(ii) ffkp = n, then Hk*(SI) ^ L*(SI), for all q € [l,oo).

0 Preliminaries and notation 5
' \
(iii) Ifkp > n, thenHk*(Sl) ^ C°^(Q), wherea = k-n/p ifk-n/p <
1; a € [0,1) is arbitrary if k — n/p = 1 and p > 1- a ■=■ 1 if
k~n/p> 1.
In addition, there result the following.
(i') Ifkp < n, then Hk*(Cl) ^^ L"(Q), for alll<q< np/(n - kp).
(ii') Ifkp=n, thenHk'P(Q) ^^ LQ(0,), for all q G [l,oo).
(Hi') Ifkp > n, then Hk*(Q,) ^^ C(£l).
0.6 Elliptic boundary value problems
Let Cl be a bounded domain (i.e. open connected) in R™ with smooth
boundary dCl (this will always be understood hereafter) and let £ denote
the differential operator
^ ^i-x^i) (oi)
where
atj = ajt € C°°(f7). (0.2)
£ is (uniformly) elliptic if there exists a > 0 such that
53 ayO=)&&-> «|C|2, for all x € £1 and £ € R" (0.3)
l<t,j<7l
Throughout the book, any elliptic operator will be an elliptic operator
with smooth coefficients, namely an £ of the form (0.1) and such that
(0.2)-(0.3) hold.
Consider the Dirichlet Boundary Value Problem (b.v.p. for short)
—Cu = h(x) in £), 1
(0.4)
u = 0 on dil,
where h is given a function on Cl.
Let h € i2(£l); a weak solution of (0.4) is a u € #o(£i) such that
£ /«>£;£;-/^, ***<•*<*&).
If it is a weak solution of (0.4) and u € C2(Cl), then it is a classical
solution.
Theorem 0.5 Suppose C is an elliptic operator. Then the following
results hold.
(i) Let h € Lp(Sl), 2 < p < oo. TTien (0.4) has a unique (weak)
solution it € Hq(Q) O /?2,p(£i) and f/ie following estimate holds:
\M\m„<c\\h\\L,.

0 0 Preliminaries and notation
(ii) If h € L°°(Q) then u € Cl'a(Q) for any 0 < a < 1 and
(iii) // h € C°>a(Q) tfien u € C2'"^) is a classical solution of (0.4) and
Hcv>< c\\h\\co,..
In the above c stands for a positive constant, depending on Cl.
As a consequence of the preceding results, we can define an operator
K : L2(Cl) —► L2(Cl) (the Green operator of —£ with zero Dirichlet
boundary conditions) setting Ku = v if and only if — Cv = u, v € #o(£l).
From the Rellich Theorem it follows immediately that K is compact.
Given a function m € i°°(f)), let us consider the linear eigenvalue
problem
—Cu = Amu in Cl,
it = 0 on dCl.
Ah eigenvalue of (0.5) is a A such that (0.5) has a solution u ^ 0. Any
<j> 7^ 0 satisfying (0.5) is an eigenfunction associated to the eigenvalue A.
If we set n = 1/A and Km(u) = K(mu), problem (0.5) is equivalent to
fiu = Kmu. The eigenvalues Xk of (0.5) correspond, through fik = 1/A*
to the eigenvalues of Km, The"multiplicity of A* is the multiplicity of/i^.
In some cases we will write Ajt(m) or Xk(Cl) to highlight the dependence
of the eigenvalues of (0.5) on m or Cl.
Theorem 0.6 Let m € L°°(Cl), m > 0 and m(x) > 0 in a set of
positive measure.
(i) Problem (0.5) has a sequence
0 < Xi (m) < A2(m) < ... < Xk(m) < ...
of eigenvalues such that Ajt(m) —► +oo as k —► oo. The first
eigenvalue Ai (m) is simple and the corresponding eigenfunctions do not
change sign in Cl. We will let denote 4>\, (sometimes only ¢) the
eigenfunction such that (a) 0 > 0 in Cl and (b) f^tp2 = 1.
We will also let <f>k denote the eigenfunctions corresponding to Xk
normalized by
JM> = si* = {l Ifh^k.
n
When m = 1 we will simply write Xk instead of Xk(l).
(ii) (Comparison property) If m < M in Cl then Xk(m) > Xk(M); if
m < M in a subset of positive measure then Xk(m) > Xk(M). In
particular, ifm< Xk(resp.> Xk) then Ajt(m) > 1 (resp.< 1).
(0.5)

0 Preliminaries and notation
7
(iii) (Variational characterization) There results
Xk(m) = max < / my2 : v € H^(Q), j ^
In n
du du _
3 dxi dxj
fvfa = 0,
for all i = 1,..., Jfc-1 >.
(iv) (Continuity property) Ai(m) depends continuously on m in the
Ln^(Cl) topology.
(v) Let CY be a bounded domain, such that Cl' c Cl. Then Afc(Q') >
Xk(Ct) for all k> 1.
Consider the non-homogenous b.v.p.
—Cu = Xmu+h in Cl, |
n «1 f (0-6)
ix = 0 on aii, J
with, say, h € L2(Cl).
From the Fredholm Alternative Theorem 0.1 we get the following.
Theorem 0.7
(i) If X is not an eigenvalue of (0.5), then (0.6) has a unique solution
forallheL2(Cl);
(ii) if X is an eigenvalue of (0.5), then (0.6) has a solution if and only
if Jn h<f>k = 0 for any k such that X = X^. '
According to Theorem 0.4 (iii) all the preceding discussion can hv. carried
over taking X = C2'Q(H), h € C°>a(Ti) and m smooth.
The arguments above apply to Sturm-Liouville Problems
_A
dx '
a0u(0) + hou'(0) = a\u{it) + b\u'{it) = 0,
a-r-u\+0u = h(x) (0 < x < tt),
where a € C!([0,ttJ), /3 € C((0,tt]), a,0 > 0 on [0,tt], and a0,fao,ai,&i
are such that (og + 6§)(a? + 62) ^ 0.
In fact, it is known [Dl] that for all h € X := C([0,tt]) there exists a
unique u € C2([0,7r]) satisfying (0.6) and hence the map K : h —► K{K)
(is linear and) as an operator from X into itself is compact. It is also
known that such a K has a sequence of positive, simple eigenvalues
^i > f*2 > ■ • ■ > fik • • ■» sucn that fik —► 0 as k —► oo. Correspondingly,

8
0 Preliminaries and notation
the linear Sturm-Liouville eigenvalue problem
~^{a^U)+0U=Xu{x) {°<X <7r)'l
aou(0) + b0u'(0) = oiu(tt) + bxu'{ir) = 0, J
has a sequence of simple eigenvalues A* = 1/^fc —* oo.
Another classical result we will need is the Maximum Principle.
Theorem 0.8 Let Cl C R" be a bounded domain with smooth boundary
and let \<\i. If u € C^fl) U C(U) is such that
—Cu > Au in Cl,
u>0ondfl,
then u > 0 in Ct.

1
Differential calculus
This introductory chapter is mainly devoted to the differential calculus
in Banach spaces. In addition to being a fundamental tool later on, the
treatment of the calculus at this level permits better understanding at
certain aspects, which might otherwise be neglected. ■
We discuss in Section 1 the Frechet and Gateaux derivatives as well
as their elementary properties. The differentiability of the Nemitski
operator is investigated in Section 2 and higher a'nd partial derivatives
are introduced in Sections 3 and 4, respectively.
1 Frechet and Gateaux derivatives
The Frechet-differential is nothing else than the natural extension to
Banach spaces of the usual definition of differential of a map in Euclidean
spaces.
Let U be an open subset of X and consider a map F : U —►Y.
Definition 1.1 Let u € U. We say that F is (Frechet-) differentiable
at u if there exists A € L(X, Y) such that, if we set
R(h) = F(u +h)- F(u) - A{h),
there results
R{h) = 0(||h||), (1.1)
that is
«-0«W-0.

10
1 Differential calculus
Such an A is uniquely determined and will be called the (Fre'chet)
differential of F at u and denoted by
A = dF(u).
If F is differentiable at all u e U we say that F is differentiate in U.
Hereafter, when there is no possible misunderstanding, Frechet
differentiability will be referred to simply as differentiability. A few comments
on the preceding definition are in order.
(i) Let us verify that A is unique. Supposing the contrary, let B €
L(X,Y) satisfy Definition 1.1 and A ^ B. It follows that
1¾^ 0-.1^0. (1.2)
If A ^ B there exists h* e X such that o := \\Ah' - Bh'\\ j£ 0.
Taking h= th*, (eR- {0}, one has
HA(th') - B(th')\\ _ ||Ah'-Bh'|| _ _a_
ll'h'll " IMI "IW
a constant, in contradiction with (1.2).
(ii) If F is differentiate at u then
F(u + h) = F(u) + dF(u)h + o(\\h\\)
and F is continuous at the same point. Conversely if F e C(U, Y)
then it is not necessary to require in Definition 1.1 the continuity
of A. In fact (1.1) yields
A(h) = F(u + h)-F(u)-o(\\h\\)
and the continuity of F implies the continuity of A.
(iii) The definition of differentiability depends not on the norms but on
the topology of X and Y only. That is if, for example, jj.j| and
jjj.|jj are two equivalent norms on X then F is differentiate at u
in (X.jj.jj) if and only if F is in (X,jjj.jj|) and the differential is the
same.
Remark 1.2 The preceding comment (iii) could suggest the idea of
extending the notion of Frechet differentiability to locally convex
topological spaces. The most natural way would be the following: let the
topology of X (respectively Y) be induced by an infinite family of semi-
norms j.jx.i (resp. ||.j|r,j); define the differential of F as the linear
continuous map A with the property that for all j.jy.j there exists a
seminorm \-\x,i such that \F(u + h) — F(u) — Ah\yj = o(\h\x,i)- With
such a definition all the main properties of the differential (below) hold
true. Unfortunately, in dealing with the higher derivatives, there are

1.1 Prachat and Gateaux derivatives
11
strong difficulties and people introduced new classes of spaces, such as
the "pseudotopological spaces", where a differential calculus suitable for
the purposes of analysis can be carried out. These kind of topics,
however, are beyond the purposes of our book.
Examples 1.3
(a) The constant map F(u) ~ c is differentiable at any u and dF(u) =
0 for all u e X.
(b) Let A &L{X,Y). Since A(u + h)-A(u) = A(h), it follows that A
is differentiable in X and dA(u) = A.
(c) Let B : X xY —* Z he & bilinear continuous map. There results
B(u + h,v + k)- B(u,v) = B(h,v) +B(u,k)+B(h,k).
From the continuity at the origin it follows that
||B(M)|| < cllMI ||*||.
Then B is differentiable at any (u,v) € X xY and dB(u, v) is the
map (h, k) -f B(h, v) + B(u, k).
(d) Let H be a Hilbert space witb scalar product (.j.) and consider the
map F : u—> jjujj2 = (u\u). From
\\u + h\f-\\uf = 2(u\h) + \\h\\>
it follows that F is differentiable at any u and dF(u)h = 2(uj/i).
Note that jj.jj is not differentiable at u = 0. For, otherwise, jj/ijj =
Ah + o(\\h\\) for some A e L(tf,R). Replacing h with — h we
would deduce that jj/ijj = -Ah + o(\\h\\) and hence jj/ijj = o(\\h\\),
a contradiction.
(e) If X = R, U = (a, b) and F : U ^ Y is differentiable at t e U,
the differential &F(t) can be identified with dF(t)[l] € Y though
the canonical isomorphism i : L(R,Y) —► Y, i(A) = .-4(1). For
example, if Y = R" and F(t) = (/»(0)i=i.-.,n» dF(') "is" tlie
vector with components dfi/dt.
The main differentiation rules are collected in the following
proposition.
Proposition 1.4
(i) Let F,G : U -> Y. If F and G are differentiable at u e U then
aF + bG is differentiable at u for any a, 6 e R and
d(aF + bG)(u)h = adF(u)h + bdG(u)h.
(ii) (Composite-map formula) Let F : U —* Y and G : V —* Z with

12
1 Differential calculus
V D F(U), U and V open subsets of X and Y, respectively, and
consider the composite map
GoF:U -> Z, Go F(u) := G(F(u)).
If F is differentiable at u e U and G is differentiate at v :=
F(u) e V, then G o F is differentiable at u and
d(GoF)(u)h = dG(v)[dF(u)h].
In other words the differential of G o F at u is the composition of
the linear maps dF(u) and dG(v), with v = F(u).
The proofs of (i) and (ii) do not differ from those of the differentiation
rules in Rn.
Definition 1.5 Let F : U —* Y be a, differentiable in U. The map
F' \U -> L{X, Y), F' : u -> dF(u),
is called the (Frechet) derivative of F.
If F' is continuous as a map from U to L(X, Y) we will say that F is
C1 and write F' €Cl(U,Y).
Let us introduce the concept of variational (or potential) operator.
If Y = R, maps J : U —*■ R arc usually called functionals and J'
turns out to be a map from U to L(X, R) = X* (the dual of X). In
particular, if X = H is a Hilbert space, J'(u) e H* for all u and the Riesz
Representation Theorem allows us to identify J'{u) with an element of
H. To be precise, we give the following definition.
Definition 1.6 Given a differentiable functional J : U —* R the
gradient of J at it, denoted by VJ(u), is the element of H defined by
(VJ(u)jh) = dJ(u)h, for all h€H. (1.3)
A map F :U —* H with the property that there exists a differentiable
functional J : U —*■ R such that F = VJ is called a variational (or
potential) operator.
As for maps in R", we can also define here a directional derivative,
usually called the Gateaux differential (for short, G-differential).
Definition 1.7 Let F : U —* Y be given and let x e U. We say that
F is G-differentiable at u if there exists A e L(X, Y) such that for all
h € X there results
F(u + eh)-F(u) . , ,

1.1 Frechet and Gateaux derivatives
:'13
The map A is uniquely determined, called the G-differential of F at\t
and denoted by dQF(u).
Clearly, if F is Frechet-differentiable at u then F is G-differentiable
there and the two differentials coincide. Conversely, the
G-differentiability does not imply the continuity of F, even: recall the elementary
example F : R2 -*■ R2 defined by
F(s,Q) = Q.
The following result replaces the elementary "Mean-Value Theorem"
and plays a fundamental role in what follows.
For u,v € U let [u,v] denote the segment {tu + (1- t)v : t e [0,1]}.
Theorem 1.8 Let F : U —* Y be G-differentiable at any point of U.
Given u,v e U such that [u,v] C U, there results
\\F(u)-F(v)\\<Sup{\\<iGF(w)\\:w€[u,v}} \\u-v\\.
Proof. Without loss of generality we can assume that F(u) ^ F(v).
By a well-known corollary of the Hahn-Banach Theorem there exists
ij) € Y*, \\ij)\\ = 1, such that
&,F(u)-F(v)) = \\F(u)-F(v)\\. (1.5)
Let j(t) = tu + (1 - t)v, t € [0,1], and consider tlie map h : [0,1] —* R
defined by setting
h{t) = (¢, F[y(t)]) = (¢, F(tu +(1- t)v)).
From 7(( + t) ^, 7(2) + r(u — v) it follows that
h(t + T)-h(t) = / F[7(t+T)]-F[7(0]\
_/tp/h(t)+r(u-v))-F[1(t))\ (lg)
Since F is G-differentiable in U, passing to the limit in (1.6) asr-*0,
we find
h'(t) = (i>,dGF(tu + (1- t)v)(u - v)). (1.7)
Applying the Mean-Value Theorem to h one has
/i(l) - /i(0) = h'(9) for some 9 € (0,1). (1.8)

14
1 Differential calculus
Substituting (1.5) and (1.7) into (1.8) we get
||F(u)-F(i,)||-h(l)-h(0) = h'(9)
= W>, dGF($u +(1- $)v)(u - «))
< IIV-II |jdaF(eu + (1- S)«)j| ||u - »||.
Since jj^jj = 1 and 6u + (1 — 6)v e [it, v\ the theorem follows.
As a consequence we can find a classic criterion of Fr^chet
differentiability.
Theorem 1.9 Suppose F :U —*Y is G-differentiable in U and let
Fa:U^ L(X,Y), F'aiu) = dGF(u),
be continuous at u*. Then F is Frechet-differentiable at u* anddF{u')
= dGF(u*).
Proof. We set
R(h) := F(u* + h) - F(u*) - dcF(u*)h.
Plainly, R is G-differentiable in BE, for e > 0 small enough, and
dGR(h) : k -t dGF(«* + h)k - dGF(u*)/fc. (1.9)
Applying Theorem 1.8 with [u,v] = [0,¾]. wo find (note that R(0) = 0)
||K(h)|| < sup \\d0R(th)\\ \\h\\. (1.10)
From (1.9) with th instead of h, we deduce
\\dGR(th)\\ = ||dGF(«' +th) LdGF(«*)jj.
Substituting into (1.10) we find
\\R(h)\\< sup HdGFK+ihJ-dGFKJIIIIhll.
0<t<l
Since F&is continuous,
sup jjdGF{u* + th) - dGF(u*)jj -> 0 as jj/ijj -> 0
and therefore #(/i) = o(\\h\\).
In view of Theorem 1.8, to find the Frechet differential of F one can
determine dGF and show that Fq is continuous.
Let F € C([a,b},X) and set (see subsection 0.3)
t
*(t) = J F(t)dt
It is immediately verifiable that $ is differentiate and $'(<) = F(t)

1.2 Continuity and differentiability of Nemitski operators 15
(we are using the canonical identification between L(R,X) and X; see
Example 1.3 (e)). $ is called a primitive of F.
From Theorem 1.8 it follows that
II «(0 - ¢(^) II < sup {||F«)|| (t-$):s<(<t}.
Hence, if F(£) = 0 for all £ e [a, 6], one has ¢ =constant. In particular,
$ is, up to a constant, the unique primitive of F.
As a consequence we can obtain the following useful formula. Suppose
that [u,v] € U and let FeC]((7,y). The map Fo j : [0,1] -> V, Fo
7(() = F(*u + (1- 0») '8 C1 and
(FoT)'(*)=.F'(hi + (l-*)») [«-«]•
Integrating from 0 to 1 we get
F(v)-F(u)= IF,(tu + (l-t)v)[u-v}
i\dt
F'(tu + (l-t)v)i
(u-v). (1.11)
Note that in the last integral F' is meant to take values in L(X, Y).
2 Continuity and differentiability of Nemitski operators
In this section we want to study the differentiability of an important
class of operators arising in nonlinear analysis: the so called "Nemitski
operators" we are going to define.
Nemitski operators
Let fi be an open bounded subset of R" and let M(fi) denote the class
of real-valued functions u : Q —* R that are measurable on Cl. Here, and
always hereafter, the measure is the Lebesgue one and will be denoted
by n; all the functions we will deal with in this section are taken in
Let/:£lxR-»Rbe given.
Definrtron 2.1 The Nemitski operator associated to / is the map
defined on M(£l) by setting
u(x) —► f(x,u(x)).
The same symbol / will be used to denote both / and its Nemitski
operator.

16
1 Differential calculus
We shall assume that / is.a Caratheodory function. More precisely,
we will say that / satisfies (C) if
(i) s —► f(x, s) is continuous for almost every x € £),
(ii) x —> f(x, s) is measurable for all s € R.
For the purpose of analysis it is particularly interesting when the
Nemitski operators act on Lebesgue spaces IP = IP(Q) (hereafter we
will write IP for IP(Cl)) and we shall discuss this case in some detail.
Let us start by noticing that
f(u) € M(£)) for all u € M(£)). (2.1)
Indeed, if it € M(£)) there is a sequence x„ of simple functions such
that Xn-m a.e. in £). From (C) it follows that
f(Xn) is measurable and f(xn) —► f(u) a.e. in £),
and from this we deduce that /(it) € M(£)).
Continuity of Nemitski operators
Let p, q > 1 and suppose
\f(x,s)\<a + b\s\a, a= ^, (2.2)
for some constants a, b > 0.
Theorem 2.2 Let £) c R" be bounded and suppose f satisfies (C) and
(2.2). Then the Nemitski operator f is a continuous map from Lp to Lq.
For the proof we need the following measure-theoretic result (see, for
example, [Br], Theorem IV.9).
Theorem 2.3 Let /x(£)) < oo and let un —► it in D\ Then there exist
a sub-sequence u„k and h e IP such that
«„fc —► u a.e. in £), (2-3)
\u,lk\< h a.e. in £). (2.4)
Proof of Theorem 2.2 From (2.1)-(2.2) it follows immediately that
/(it) € Lfl whenever it € LP.
To show that / is continuous from IP to Lq, let itn,it € Lp be such that
IN„ - u\\L? —► 0.
Using Theorem 2.3 we can find a sub-sequence {unic} of {un} and h g IP

1.2 Continuity and differentiability of Nemitski operators 17
satisfying (2.3)-(2.4). Since unic converges almost everywhere to it, it
readily follows from (C) that
/KJ-*/(«) a.e.infl. (2.5)
Moreover, from assumption (2.2) and (2.4) we infer
l/KJI < a + b|itnfc|" < a + b\h\a € Lq. (2.6)
As an immediate consequence of the Lebesgue Dominated-Convergence
Theorem, (2.5)-(2.6) yield
jj/Kj - /(it)jjl, = J |/KJ - /(u)|« - o.
n
Since any sequence itn converging to it in LP has a sub-sequence it„t
such that f(unk) —► /(it) in Lq, we can conclude that / is continuous at
it, as a map from LP to Lq.
Remark 2.4 Theorem 2.2 can be proved assuming that / satisfies
(2.2) with a replaced by a(x) € Lq.
Remark 2.5 It is possible to show that, if (C) holds and /(it) € Lq for
all u e V, then / e C{W,Lq). For this and other kinds of arguments
wc refer to [Va], p. 154 and following.
Di'fferentabilrty of Nemitski operators
Our next result deals with the differentiability of Nemitski operators.
First some remarks are in order.
Let p > 2 and suppose / has partial derivative /3* = df/ds satisfying
(C) and such that
|/„(*,S)|<a + 6|Sr2 (2.7)
for some constants a, b > 0. Since /,, satisfies (2.7), Theorem 2.2 applies
and tlic Nemitski operator /« is continuous from IP to V, with r =
p/(p— 2). As a consequence, for the function fg(u)v defined by
/.,(it)y : x —* /s(x,it(x))u(x)
one has that /s(it)y € If for it, v € Lp, where p' = p/(p — 1) is the
conjugate exponent of p.
Theorem 2.6 Let Cl c R" 6e bounded and suppose that p > 2 and f
satisfies (C). Moreover, we suppose that /(x,0) is bounded and that f
has partial derivative /s satisfying (C) and (2.7).
Then f : LP —► V is Prechet-differentiate on LP with differential
d/(it) : v —► f„(u)v.

18
1 Differential calculus
Proof. Integrating (2.7) we find constants c, d > 0 such that
\f(x,8)\<C + d\8\V-\
and another application of Theorem 2.2 yields the continuity of / as a
map from V to V , with j/ = p/(p — 1).
For it, v € IP we evaluate
"(**>*) = 11/0* + «) - /(u) - /.(«MIlf'
/*|/(x,u(x) + y(x)) - /(x,u(x)) - /.(x^xJMx)!"'
n
By the Mean-Value Theorem one has (for almost every x € £1)
j/(x,u(x) +w(x)) -/(x,u(x)) -/,(x,u(x))i;(x)j
i
= K*) /[/.(*,«(*) + <«(*)) - /.(*,«(*) )KI = l«(*) «<(*)l,
0
where
i
*"(*) = fiMxMx) +Cw(x))-/s(x,u(x))]dC.
lg the Holder ine<
/ju(x)IU(x)jp'c
(2.8)
With this notation and using the Holder inequality we get that
r i '/>>'
*j(u, y) ^
' dx
<Hi.HU- r =
Now, the norm jjtu|JLr can be estimated as follows:
l
Mir< fdxf\M*M*)+&(*))-M*M*))\r*C
n o
i
= JdC J\f.(x,u(x) + <«(*)) -/.(x,U(x))|-dx
'/«/.(«+«-/.(«)«;
:¾.
(2.9)
As remarked before, /s is continuous from Lp to Lr. Hence
||/.(u + Co) - /„Mlli- -. 0 as Mi, -. 0, ( € [0,1]. (2.10)
From (2.8),(2.9) and (2.10) it follows that ui(u,v) = o(||«||i,).
For p = 2 the above result does not hold, in general. Indeed, under

1.2 Continuity and differentiability of Nemitski operators 19
the preceding assumptions, the Nemitski operator f is G-differentiable
but, possibly, not Frechet-differentiable. To be precise, let us assume
that (C) holds for /, f„ and
\f*{x,s)\ < const. (2.11)
As before, it follows plainly that f is continuous from L2 to L2 and
the map v —* fs(u)v from L2 to L2 is linear and bounded. In addition
one has the following
Theorem 2.7 Let £1 C R" be bounded and let f and fs satisfy (C) and
(2.11). Then f : L2 -► L2 is G-differentiable and dGf(u)[v] = fs(u)v.
Proof. According to Definition 1.7 we have to show that for all it, y e L2
there results
II/("+«»)-/(")
/,(u)« -tOasi-»0. (2.12)
Ilia
As in the proof of theorem 2.6 one finds
f(u+tv)-f(u)
Letting
}.(u)v = vf[f.(u + C,tv) - /.(u)]d(.
0
1
■- wt(u,v) = /[/> + <*«) - /»M)d<,
\\f(u+ty)-f(u)
■f.(M\
=/.v
: fx,Hx f\l,(u + C,tv) - /,,M|2d<.
When /->0 then ££y —* 0 a.e. in £1 and hence
/,(w + Kv) - f„(u) -► 0 a.e. in £1
Since
\fs(x,u(x) +t£v(x)) - fg(x,u(x))\2 < const,
the Lebesgue Dominated-Convergence Theorem implies
l
f\fs(u + C,tv) - /s(u)|2dC -> 0, as * -> 0, (2.13)
o
and (2.12) follows.

20
1 Differential calculus
The preceding theorem is completed by the following proposition.
Proposition 2.8 Let Q C R" be bounded, suppose f and f„ satisfy (C)
and {2.11) and let f be Ftechet-differentiable at some u* e L2. Then
there exists a(x),b(x) € M(Cl) such that
/{x, it) = a(x)u + b(x).
Proof. Suppose first that it* = 0 and f(x,0) = 0.
Let D(y, 6) denote the ball centred at t/€ 0 with measure 6. Given
x* € Cl and A € R, consider the functions v&(x) e L2(Cl) given by
vs(x) = A, for x e D(x*,6),
vs(x) = 0, for x € fl\Z>(x*, 6).
Obviously v& —*■ 0 in L2 as 6 —► 0.
Recall that if / is Frecliet-differentiable at it* = 0 then /'{0) = dG/(0).
Hence f'(0)v = fs(0)v, and therefore
By a direct calculation one finds
1/2
ll/(w) - /,(Q)«tlU» _ i
MVS
I "'<'■
A)-/.(x,0)A|2
and hence
I
f |/(x,A)-/,(x,0)A|
V
-(0, as 6 -> 0.
(2.14)
V>a(x) =
|/(i,A)-/,(i,0)A|'
(2.14) becomes
7 / V*(x)dx -> 0, as S -> 0.
Since for all ip e L1 and almost every y € £1 there results
- / ^(x)dx —* ip(y), as 6 —* 0,
{2.15)
we deduce from {2.15) that for all A € R there exists a null set N\ such
that for all y g N\ one has tpx(y) = 0.

1.2 Continuity and differentiability of Nemitski operators 21
Taking A in a countable dense subset A of R and letting N = U^eA N\
one infers that for all y g N and all A e A there results ip\(y) = 0 that
is,
/(»,>) = /.(», 0)X. (2.16)
Using the continuity of /(x,.), one deduces that (2.16) holds true for
all A e R and almost every y.
Lastly, let us set
g{x,u) = f(x,u + u*) - f(u*).
One lias that g is Frechet-differentiate at 0 and g(x,0) = 0. Applying
the preceding arguments to g, we find
f(x,u + u*) - f(u*) = f3(x, u*)u
and the proposition follows.
Potential operators
We end this section by dealing with potential operators (Definition 1.6).
The results will not be used in the remainder of this book but are
important in connection with variational problems.
Let Hq = Hq(Q) (wliere Q is a bounded domain of R") denote the
usual Sobolev space (see Subsection 0.5) with scalar product (.).)//',*
and norm jj.jjHi,a. Let n > 2 (if n = 1,2, the arguments we are going to
expand apply as well, with some modifications; see Remark 2.10 below).
Suppose / satisfies (C) and
|/(x, s)\ < a + 6J5J0 with a < ^-±| = T - 1, (2.17)
n — 2
Here 2* = 2n/(n — 2) and, by the Sobolev Embedding Theorem (see
Theorem 0.3 (i)), Hq1 ^ L2' and
HI/,2- < const.jjyjjHi.a
By Theorem 2.2 it follows that
f €C(L2 ,L9) with ¢= — > -^-. (2.18)
In particular one has /(it) € £2"/(™+2) for all it € Hq . As a consequence,
f(u)v € L1 for all u,v € /¾ and the equality
(N(u)\v)Hi-2 - J f(x,u(x))v(x)dx, u,v&Hl, (2.19)
n
defines an operator N : Hq —> Hq. Note that N is continuous. To see
this we evaluate
||JV(U) - 7V(«)||„,., = sup J \J{f(x,u) - f(x,v))wdx\ : \\w\\H>., < 1 I

22
1 Differential calculus
<sup{||/(U) -/(«)IIl»"/(~hi|HIl»* : ||to||«... < 1}
<c||/(«)-/(*)lli»-/<-+.).
If um —* v in H£ one has (by the Sobolev Embedding Theorem) um
in Lr and (by (2.18)) /(».) -. /(«) in L2"/("+2).
Set
x,s)- Jl{x,t)it.
From (2.17) it follows that there exist c, d > 0 such that
|F(x,a)| <c + dja|2". (2.20)
Then F(.,u(.)) € L1 for all u € #<HC £2") and ifc niakes sense to
define a functional $ : Hq —> R by setting
j(u) = / F(x,u(x
))dx.
The functional 0 can be obtained by composition according to the
following diagram t ..
Hi ^ L2* 4 L2"A"+2) 4 L1 -* R
u _► u -> F(.,u(.)) -* F(.,u(.)) -* /F(x,u(x))dx
n
where a and /3 stand for the embedding of Hq into L2*, of £,2nA"+2)
into L1 respectively. Since F satisfies (2.20), Theorem 2.6 applies to F
as a map from L2* to £2"/("+2). From Proposition 1.4(b) (derivative of
the composite map) it follows that <f> is differentiable with
<f>'(u) : v —► f f(x,u(x))v(x)dx.
Then, recalling the definition of "gradient" (Definition 1.6), we have the
following
Theorem 2-9 Let Cl C R" be bounded and suppose f satisfies (C) and
(2.17). Then <f> is aC1 Junctional on Hq with gradient
V0(u) = JV(u),
where N is defined in (2.19).
Remark 2.10 Un =-2 then the same result holds assuming / satisfies
(2.17) with any cr < oo. It suffices to repeat the above arguments using
the stronger form of the Sobolev Embeddings when Cl C R2.

1.3 Higher derivatives
23
3 Higher derivatives
Let F € C(U, Y) be differentiable in the open set U C X and consider
F' :U -* L(X, Y)
Definition 3.1 Let u* € U : F is twice (Fr£chet-) differentiable at it*
if F' is differentiable at u*. The second (Fre'chet) differential of F at u*
is defined as
d2F(u*) =dF>*).
If F is twice differentiable at all points of U we say that F is twice
differentiable in U.
According to the above definition d2F(u*) is a linear continuous map
from X to L(X,Y):
d2F(u*) eL(X,L(X,Y)).
It is convenient to see d2F(u*) as a bilinear map on X. For this, let
L2(X, Y) denote the space of continuous bilinear maps fromlxl —► Y.
To any A € L(X,L(X,Y)) we can associate ¢^ € L2(X,Y) given by
¢,4(1/1,1/2) = [^(ui)](u2). Conversely, given ¢ € L2(X,Y) and /1 e
X, ¢(/1, .):&—► ¢(/1, fc) is a continuous linear map from X to Y\ hence
to any ¢ € L2(X, Y) is associated the linear application X —► L(X, K),
$:/»-> ¢(/1,.) €/,(1,7).
It is easy to see that in this way we define an isomorphism between
L(X,L(X,Y)) and L2(X,Y). Actually, such an isomorphism is an isont-
etry because there results
\\Hl{x,l{x,y)) = sup \\$(h)\\L{X-y)
INI<i
= sup—sup \\$(h,k)\\ = \\$\\Li(X,Y)-
\W\<i ll*ll<i
In the following we will use the same symbol d2F(u*) to denote the
continuous bilinear map obtained by the preceding isomctry. The value
of d2F(u*} at a pair (/1, fc) will be denoted by
d2F(u*)[h,k).
If F is twice differentiable in U, the second (Fre'chet) derivative of F
is the map F" :U -* L2(X,Y),
F" :u-* d2F(u).
If F" is continuous from U to -L2(X,}') we say that F € C2(U,Y).

24
1 Differential calculus
Examples 3.2
(i) If A e L(X,Y) then A e C2(X,Y) and d2A[h,k] = 0 for all
(ii) Let X = C([0,1]) and F : A" -* X, F : u(<) -> u2(*)- ^ €
C2(X, X) and
d2F(u) : (h(0.*(0) -* 2h(t)k(t).
The following proposition can be useful for evaluating d2F(u).
Proposition 3.3 Let F : U —► K be twice differentiable at u* e U.
Then for all fixed h e X the map F/,: X ->y defined by setting
Fh(u) = dF(u)h
is differentiable at u* and dFh(u*)k = F"(u*)[h, k).
Proof. Fh is obtained by composition
U-^L(X,Y)-^Y
u —► dF(u) —► dF(u)h t .
between the derivative u —► dF(u) and the "evaluation map" Sh which
associates to each A € L(X, Y) tlie value A(}{) e Y. Since £h is linear,
the result follows by the composite mapping formula 1.4-(ii).
We have seen that F"(u) can be regarded as a bilinear map. More
precisely one has the following
Theorem 3.4 If F : U —* Y is twice differentiable at u e U, then
F"(u) € L2(X,Y) is symmetric.
Proof. For h,k e X with h, k e B(e) (e small enough), we set
ip(h,k) =F(u + h + k) -F(u + h)-F(u + k) +F(u),
-yh(t)=F(u + h + Q-F(u+t),
and consider, for h fixed, the map (//, : B(e) —► Y,
gh:k^ ij>{h, k) - F"{u)[h, k) = jh(k) - Jh(0) - F"(u)[h, k).
Since F is differentiable in U and F"(u)(h) : k —► F"(u)[h, k] is linear
(as a map from X to L(X, Y)), Theorem 1.8 yields
IWM)-f>)[M||
< sup{\\djh(tk) - F"(u)(h)\\ :0<t< l}|| fc||
= sup{jjdF(u + h+tk)- dF(u + tk)
-F»(h)||:0<*<l}||*j|. (3.1)

1.3 Higher derivatives
25
Since F is twice differentiable at u e U, one has
F"(u + h + tk) = F'(u) + F"(u)(h + tk) + ui(h + tk),
F'(u + tk) = F'(u) + F"(u)(tk) + ui(tk),
with u)(v) = o(jjuli). Hence
F'(u -rh-rtk) - F'(u +tk) =F"(u)(h) + ui(h + tk) - ui{tk). (3.2)
Using (3.1) and (3.2) and taking into account that <jj(v) = o(|ju|j) we
get, that
U(h, k) - F"(u)[h, k}\\ < sup{Hh + tk) - u(tk)\\ :tt<t< l}||fc||
<E(||MI+2||fc||)||fc||, (3.3)
provided jj/ijj and jjfcjj are sufficiently small.
Exchanging the roles of h, fc we get (for jj/ijj, jjfcjj small)
H(k,h)-F"(u)[k,h)\\ <sup{\\uj(k + th)-uj(th)\\:Q<t< 1} jj/tjj
<e(||A:||+2||h||)||h||. (3.4)
Since i})(h,k) = ij>(k,h) we deduce from (3.3) and (3.4)
||F"(«)[fc,fc] -F»[*,h]ll < £(2||fc||2 + 2IIMI2 +2||h|| HMD
<3£(||fc||2 + IIM|2). (3.5)
Inequality (3.5) has been proved for jj/ijj, jjfcjj small enough, but holds
true for all jj/ijj, jjfcjj, because F"(u)[h,k] is homogeneovis of degree 2.
Since e is arbitrary, (3.5) implies that F"(u)[h,k] = F"(u)[k,h] for all
/i,fc.
To define (n + l)-th derivatives (n > 2) we can proceed by induction.
Given F : U —*■ Y, let F be n times differentiable in U- The nth
differential at a point x € U will be identified with a continuous n-liiicar
map from X x X x ... x X (n times) to Y (recall that, as before, there
iH an isometry between L(X,..,, L(X,Y))... and L„(X,Y)).
Let F<"> : U -» L„(X,y) denote the map
F(n) : u -> d"F(u).
The (n + l)-th differential at u* will be defined as the differential of
F^"), namely
d<"+1>F(u*) = dF<n>(u*) € L(X,Ln(X,y)) « Ln+1(X,y).
We will say that F e C"(£/, V) if F is n times (Frechet) differentiable
in U and the nth derivative F^"' is continuous from C to Ln(X, Y). The
value of dnF(u") at (/i],..., /in) will be denoted by
dnF(u*)[/t,,...,h„].
If /i = /ir = ■ - • = hn we will write for short dnF(u*)[h]n.

26
1 Differential calculus
In order to extend Theorem 3.4 to higher derivatives some
preliminaries are in order. Given a map G : U —* Ln(X,Y) and the point
h = (h\,... ,0^) € X x ... x X, we can associate G with the map
G[h] : U -> Y defined by setting
G[h](u) = G(u)[hu ... ,h„).
We can immediately see (see Proposition 3.3) that if G is differentiable
at u then G[h] is differentiable at u and there results
d(G[h])(u):ii->dG(u)[ii,hi,...,h„]. (3.6)
Let F be n times differentiable on C/ and set h = (h2,..., hn).
Applying (3.6) to G = d"~'F, we find that
d(d"-'F[h])(u)[h,] =d"F(u*)[h,,...,h„]. (3.7)
Theorem 3.5 If F : U —> Y is n times differentiable in U, then the
map
(hu .. ■ ,h„) -.d"F(u*)[hi,... ,h„]
is symmetric.
Proof. The result is trvie for n = 2 (Theorem 3.4). By indviction on n,
let the claim hold for n - 1 > 2. Then
d"-lF(u)[h2,..., hi,..., hj,..., h„]
= d"_'F(u)[h2,..., /¾...., hi,..., h„].
Applying (3.7) to h(u) = dn~1F(u)[h2,..., ht,..., hj,..., h,t] we get
that
A"F(u")[h], h2,.. -, ht,..., hj,..., /i„]
= d"F(u')[huh2,...,hj,...,hh..., hn}. (3.8)
Similarly, letting G(u) = d"~2F(u)[hj h„], one hn-s
d2G(u')[h,,h2] = d"F(u")[huh2,h3,... ,h„], —
and from Theorem 3.4 it follows that
dnF(u*)[hi, h2, h3, ■ ■ ■,h„\ = d2G(u')[hu h2]
= d2G(u" )[h2,hi] =d"F(u*)[h2,hi,h3,...,h„]. (3.9)
The symmetry of dnF(u*) is an immediate consequence of (3-8) and
(3.9).
4 Partial derivatives, Taylor's formula
Let us consider two Banach spaces X, Y and let (u*,v*) €XxY. Define

1.4 Partial derivatives, Taylor's formula
27
mappings er„. ;I-»Xx7 and ru- : Y —* X x Y as follows.
ev(u) = (u,v*)i
Tw{y) =(v*,v).
Notice that the derivatives of av. and ru- are respectively, the linear
maps
a : = der„. : h -* (/i,0),
r : =dr11. :fr-> (0,fr)-
Let Q be an open subset of X x K, (u*,u*) E Q and F : Q —* Z.
Definition 4.1 If the map F o ev is differentiable at u* we say that
F is differentiable. with respect to u at («*,u*). The linear map d[F o
<Tu-](w*) E L(X,Z) is called the partial derivative of F at (u*,u*) with
respect to u and denoted by dMF(«*,u*).
Similarly, if F o r„. is differentiable at v* we say that F is
differentiable with respect to i> at (ia*,i;*) and the linear map d[Fo rM-](v*) E
L(Y,Z) is called the v-partial derivative of F at (u*,«*) and denoted by
dvF(u',v').
The preceding definition is equivalent to requiring that there exist a
linear map An E L(X, Z) (rusp. .4,, E L(y, Z)), .snch that
F(n*+h,v')-F(u',v*) = Au(h) + o(\\h\\),
F(v*,v'+k) - F(u*,v') = ,4„(fc) + o(||fc||).
The following result is an immediate consequence of Definition 4.1 and
the differeiitation rule 1.4 (ii).
Proposition 4.2 If F is differentiable at (n*,v*) then F has partial
derivatives with iv.spec.t to u undv at (u~,v*) and we have
d„F(u*,v')(h) = dF(u*.v*)a(h) = dF(w*,w*)(fe,0),
d„F(u',i}')(k) = i\F(u*,t!*)T(k) = dF(u*,«*)(0,fc).
In quite similar way one can define higher partial derivatives. For
example, if F lias u-partial derivative at all (u,v) E Q, we can define
the map Fu : Q -* L(X, Z) by setting
F„(n,L>) = d»F(u,w).
Then the partial derivative dnvF(u*,v") is the u-derivative at (u*,v*)
of Fu, namely
dUlWF(u*,«*) =d„[Fu](u*,t)*).
The map FUtV : Q -* L(Y, L(X, Z)) wil] be defined by setting

28
1 Differential calculus
Moreover, if F is twice differentrable at (u*,v*), then dUjVF(u*,v*) is
the bilinear map from X x Y to Z given by
(h,k) -* F"(u*,v*)[ah,Tk}. (4.1)
The notation d™( vTn_e will be employed to indicate
<C,,„„-«=d£((<cr_'().
The definition of partial derivative given above permits us to obtain
in a rather straightforward way all the classical results of calculus.
For example one can prove the following.
Theorem 4.3 Suppose that
(i) F has U' and v-derivatives in a neighbourhood N of (u*,v*) 6 Q,
(ii) Fu and Fv arc continuous in N.
Then F is differentiable at (u*,v*).
As another example, we can use (4.1) and Theorem 3.4 to show
du,vF(u*,v*)[h,k] = F"(u*,v')[ah,Tk]
'■= F"(u*,v*)[rk,ah] = dWi«F(u*,w*)[fc,/t],
which is nothing else than the classical Schwarz Theorem.
Taylor's formula
Let F 6 Cn{Q,Y) and let u,u + v 6 Q be such that the interval
[it, u + v] C.Q.
Set, 7(() = u + tv, t 6 [0,1] and let <p'. [0,1] -c Y be defined l>y
*(*) = F(7(0)-
Using Proposition 1.4 (ri) and (3.7) it follows readily that the function
(j) is C" ami there result
4>'{t) = dF(u + tv)[v],
<f>"(t) = d2F(u + tv)[v}2,
<j>^(t) = dnF(n + tv)[v]n.
By elementary calculations one has
¢(1) = ¢(0) + ¢-(0) + ^¢-(0) + ■■■ + ^-i—¢(-)(0)
1
+ (^rT5i/<1-'>n",*(n>M*.
0

1.4 Partial derivatives, Taylor's formula
29
and hence
F(u + v)= F(u) + dF(u)[v] + ■■■
i
+ , * , [(l-t)n-lSn)F(u+tv)[v}'ldt.
(n - 1)! J
o
The last integral can be written as
l
. (n-l)!7
= —d,lF(n)[v]n +e(n,v)[v]n, (4.2)
n!
where
i
e(u,v)= 1 [(l-ty^ld^Fiu+^-d^Fiu^dt^Oasv^O.
(n I). J
o
Lastly, let us write explicitly the form of (4.2) when F = F(u,v) is
defined on Q (Z X xY with values in Z and is C", that is, Iras coiitimiovrs
partial derivatives up to order n. We write (it, v) instead of u and set
w = (h, k) = ah + rk. If we rrsc Proposition 4.2 the wjth term in (4.2)
becomes
-i?d(m)F(u,«)Hm = —Srn,iF(u,v)[(7h + Tkyn
■.^d(»>F(u,0)j:(7)H'[rt]M-
!^E(T)d(m>^^)[^]£^]";-
Remark 4.4 (on notation) Hereafter we will often deal with maps
F : R x X —* Y depending on a real parameter A. In such a case
the mixed derivative Fu>^(Xo,u0) is a linear map from R to L(X,Y) :
FUj>i,(\i,u0) E L(R,L(X,Y)). Then, in accordance with what we
remarked in Example 1.3 (o), we can and will identify F11m^(X„,u1,) with
the linear map h —► Fu \(X0, u„)[h, 1].

2
Local inversion theorems
This chapter deals with the local inversion of maps between Banach
spaces. The first section contains a general inversion result,; in the second
maps depending on a parameter are investigated. Air application to the
stability of orbits is given in Section 3.
1 The L^cal Inversion Theorem
For simplicity of notation we will deal with maps F € C{X,Y) where
X, Y are Banach spaces; maps defined on an open subset of X could be
treated with minor chairges only.
Let us start with some preliminaries. Let A €. L(X, Y). A is hivertible
if there exists B £ L(Y, X) such that
AoB-Ty
The map B is obviovisly unique and will be denoted by A-1. We also
set
Inv(X.y) = {A e L(X,Y) : A is invertible }.
Let us recall for future reference that, as a consequence of the "Closed
Graph Theorem", if A e L(X,Y) is infective and R(A) = Y, then
A 6 Inv(X,y).
The following result is also well known (see, for example, [Fi], 3.1).

2.1 The, Local Inversion Theorem
31
Proposition 1.1
(i) lnv(X,Y) is an open subset of L(X,Y). More precisely, if A E
Inv(X, Y) then any T e L(X, Y) such that
is invertible.
(ii) Th,e map J : lnv(X,Y) -> L(Y,X) defined by J (A) = .4"1. w Ck
for allk>l (i.e. C°°j.
It can be remarked that tlie continuity of J' : A —*■ dJ(A) can be
deduced directly by the fact that J is differentiable and dJ(j4)[B] =
-A~loBoA-1.
Let U (resp. V) be an open subset of X (resp. Y). We say that F e
Hom(£/, V) if there exists a map G : V —*■ U, such that
G(F(u)) =u for all «6 U, (1.1)
F(C(i>)) = u for all v 6 V. (1.2)
The map F e C(X,y) is said to be locally inve.rt.iblv. at u" 6 X if
there exist neighbourhoods U of </* and V of »* — F{ii*) e V such thai
F6Hoiii(y,K).
In other words F is locally invurtible at u* if there are neighbourhoods
U and V of ix* and y*, respectively, and a map G : V —*■ U satisfying
(1.1)-(1.2). The map G is called the (local) inverse of F and denoted
byF"1.
The following properties ave direct consequences of the definition.
(a) (Transitivity) If F\ € C(X] ,X2) is locally invertible at a and F2 £
C(X-z, Y) is locally invertible at v = F\(u), then F2 o Fx is locally
hiver!,il)le ni v.
(b) (Stability) If F is locally invertible at u, then it is locally invertible
at any point in a suitable neighbourhood of u.
Let F be locally invertible at u* and suppose F and G = F~l are
differentiate at it* and v*, respectively. Differentiating (1.1) (resp. (1.2))
at u* (rasp, v*) we find
G'(v*)°F'(u*) = Ix,
F'(u*)oG'(v') = IY,
and this means that F'(u*) e lnv(X,Y) with inverse G'(v*) €\nv(Y,X).
The following theorem states that, under suitable assumptions, the
converse is also true.

32 2 Local inversion theorems
Theorem 1.2 (Local Inversion Theorem) Suppose F e Cl(X,Y)
and F'(u*) e \nv(X, Y). Then F is locally invertible at u* with C]
inverse. More precisely, there exist iteiyhbourhoods U of u* and V of
v" = F(u") such that
(i) Fe Kom(U,V),
(ii) F~l 6 Cl(V,X) and for all v € V there results
dF-l(v) = (F'(u))-\n=F-'(v), (1.3)
(iii) ifFe Ck(X,Y),k> 1, thanF-1 eCh(V,X).
Proof, (i) Up to a translation, there is no loss of generality if wc take
u* = 0 and v* = F(0) — 0. Moreover, according to the transitivity
property, it suffices to discuss the local invertibility of A o F, where A is
any linear invertible map. With the choice
A = [F'(O)]-1
we axe led to consider the case in which
F = / + #, where # <=Cl(X,X) and #'(0) =0.
Here I = Ix denotes the identity on X. Let r > 0 be audi that
||*'(p)|| < i, for all ||p|| < r. (1.4)
Using Theorem 1.1.8 we find for all p, q e B(r)
ll*(p) - *(«)ll <sup{||*'(m)|| :w S b.«]}l|P-«ll
<\\\P-Ql (1-5)
Hence # is a contraction and ||*(p)|| < $\\p\\ for all ||pjj < r. For nEl
wc set
$v(u) = u-#(u).
Trivially, $^ is a contraction. Moreover, for all u e B(r) and v e B(r/2)
there results
ll*.(«)ll<ll"ll + ll*(«)ll<r.
Hence, for \\v\\ < r/2, $„ is a contraction, maps B(r) into itself and
therefore has a unique fixed point u S B(r), which satisfies
14= $„(lt) = V- ¢(11),
namely F(u) = v. As a consequence, we can define the inverse F~Y :
B(r/2) —> B(r). To show that F~x is continuous, we let u = F~l(v)
and w = F~x(z), that is,
u + #(u) = v,\
w + ^f(w) = z. J

2.1 The Local Inversion Theorem
33
Using (].5) we get that
11« ~ A\ < h - 4 + ll*(«) - *HII < 11» - 4 + \\\u - w\\
and thus
\\F-\v)-F-\z)\\<2\\v-4-
This proves that F~l is continuous, indeed Lipschitz-continiious with
constant 2. In particular, letting V = B(r/2),U = B(r) f\ F"1 (V), wo
get
F\v e Hom(t/,V).
(ii) Setting u = F-1(u), from u+ &(u) = u, we got
F-1(v)=v-V(F-1(v)).
Since ¢(¾.) = o(\\u\\) and recalling that F-1 is Lipschitz-conthmous, wc
infer that #(F_1(i;)) = o(||u||). Thiti shows that F~l is differentiablc at
v = 0 and
dF_1(0) = I.
Then, if v 6 B(r/1) and u = F~l(v), after a translation that carries v
and u into tlio origins of X and Y, respectively, one can infer that F-1
is differentiablc at v and
dF-l(v) = (F'(u))-\
This proves that (1.3) holds.
Next, we remark that the map (F~1)' : v —* (F'(F~ * («)))"' is
obtained by composition in the following way:
v'-^u = F-l(v)-^F'(u)^L(F'(u))-1 = J[F'(u)}
Since all the maps F~l , F' and .7 arc continuous, it follows thai, F-1 e
C1, proving (ii).
(iii) Lei. F e C*. By induction on k, .suppose thai. F~] e C*'_1.
Repeating the above arguments and taking into account that J 6 CICX)
(Proposition 1.1) we get readily that F-1 e Cfc.
Remark 1.3 The regularity assumption F £ C1 cannot be eliminated,
in general. If X and Y arc finite-dimensional, elementary examples
show that we can drop injectivity. To see tins, let tp : R —>■ R be
non-decreasing and such that
( I 1 1 11
1,a+0(.s2) as «-»0.
Plainly, <p is differentiable at s = 0 (indeed, it can be chosen to be C°°

34 2 Local inversion theorems
rn R — {0}) and there results <fi'(0) = 1, but <fi is not injectrve in any
neighbourhood of s = 0.
In the case of iirfinite-dimensional spaces, the same ip provides a
cormter-example for surjectivity, too.
Let X = Y = C([-l, 1]) and F : X -► Y be given by F(u) =<pou.
Consider the sequence vn € Y,
Vn(t) = -+4
n n2
For siich a sequence there result \\v„\\ —* 0 and vn £ R(F). In foct rf
there exist un £ X such that F(un) = vn one wouM find
»KW) = «.W = ; + S'
Hence there results
¥>(«n(0) > - for (> 0,
<fi(un(t)) < - for t <0.
Then the liionotoriicity of 9? implies
"-,(0 > - +-r^T for (> 0,
«n(0 < 5 for ( < 0,
n 4n*
and Un is not continuous at t = 0, i.e. u,t £ X.
Note that F is differentiable at it = 0 with derivative F'(0) = Ix, but
F$C*(X,Y).
The Local Inversion Theorem is the rigorous justification of the so-
called "procedure by linearization". Roughly, it pcrinitK ns to solve,
locally, a nonlinear problem through the stndy of its linearization.
Sonic easy examples will illustrate the main steps of 1,1 ie procedure.
Further applications are postponed to Section 3 below.
Example 1.4 Let us seek the T-periodic sotntions x = x(t)o(
x" +g(x,x') = zh(t) (1.6)
where g eC'(Rx R, R) and h e (7(R) is T-periodic.
In accordance with (1.6) we set
X = {x 6 C2(R, VL)\x(t + T) = x(t) for all t 6 R},
Y = {h<= C(R, R)|h(* + T) = h(t) for all t 6 R},
and
F(x) = x" + g(x,x').

2.1 The, Local Inversion Theorem,
35
Suppose (/(0, 0) = 0 in such a way that (1.6) has for e = 0 the "trivial"
solution x = 0 and we shall apply Theorem 1.2 with u* = 0
It is immediately verifiable that F E Cl{X, Y) and F'(Q)[w] = w" +
aw' + bw, where a = gx, (0,0) and b = ¢^(0,0).
As an immediate consequence of the Fredholnr Alternative Theorem,
F'(0) is invertible whenever the linear equation w" + aw' + bw = 0,
w E X, has only the trivial solution w = 0. If this is the ease, then there
arc e* aird H > 0 such that for all |e| < e" (1.6) has a unique solution x
with ||.r||.Y < b.
Example 1.5 Let Cl be a bounded domain in R" with smooth
boundary dil. Consider the boundary-value problem
Au + Xu — u;i = h(x), in £);
u = 0, in dCt
where A is the Laplace operator and A E R.
Here we let X = {u E C2-"(H) : u(x) = 0 on dQ}, Y = £7^(57) and
associate to (1.7) the map F : X —>■ Y defined by
F(u) = Au + Au-u3.
Problem (1.7) loads us to seek, for any given h E Y, an6 Ar such
that F(u) = /i. Let us apply Theorem 1.2 with u* = 0.
Plainly / is C°°(X,y) and one has F'(u)[w] = Aw -(- Aw - 3u2w.
Hence for A ^ A^-, the eigenvaiues of the Lapiace operator A on Cl
(with zero boundary conditions), the linear problem
F'(0)H = Aw + \w = 0, w E X,
lias only the solution w = 0. By the Fredholm Altornativc it follows that,
whenever A ^ Xk,F'(0) is oirc-to-one from X onto V. As a consequence
of the Closed Graph Theorem [.F'(O)]-1 (exists and) is conlinuoirH and
wo can conclude that if A -ft A/,, then for till h e Y with noi'in ||/(||v hoiall
enough, (1.7) has a unique solution u — u(h) E X with norm ||ul|V
small. Moreover the correspondence h —> u(h) is C°°.
Example 1.6 Let 0 be a bounded cormected donraiu in R2, with
smooth boundary dCl. and let 7 be a smooth function on dCl. A smooth
function u : Cl —> R is a minimal surface with boundary 7 if
M(U) = AUXX + BUyy — 2UxUyUxy = 0,
where A - (1 + uj) and B = (1 4- a2x).
We shall work in Holder spaces (see Subsection 0.5). Note that, if dCt
(1.7)
(1.8)

36
2 Local inversion theorems
is of class C"'a that rs dCl is locally C"1'"-diffeomorphic to a segment,
we can consider the space Cm,Q(d£l).
Let X = C2-*^), Y = C°-"(n) x ¢^(00) and F : X -> y be given
by
F(lt) = (A4(u),lt|an)-
We are now ready to apply the Local Inversion Theorem at it* = 0.
It in easy to check that F 6 C (X, V) and
.F'(14)[W] = (AWXT + BiJJyy - 2uxuywXy + 2uyuX3!wy
+ 2uxUyyWa- + 2(u.j;Wy + Uy 1DX)UXy , W\d(l)■
For u = 0 it follows that
F'(0)M = (Aw,tu|en).
According to Theorem 0.8, the Dirichlet problem
Aw = h in £), |
w = <£ orr dCl, J
has a unique solution w £ X, provided (h,4>) e K, and uj depends
continuously upon the data h, 4>.
Then Theorem ] .2 yields the existence of neighbourhoods U and V of
0 in X and ^'"(flQ), respectively, such that if 7 6 V then (1.8) has a
unique solution u e £/, and the correspondence 7 —► u is C.
It is worth noticing that (1.8) can have no solutions for some 7 if Cl
is not convex (see [CH, Vol.II, p.167])-
2 The Implicit Function Theorem
Often the introduction of a parameter permits us to extend the range of
applicability of the Local Inversion Theorem.
Lot us consider maps F : A x U —*■ Y, where A and U arc open subsets
of Banach spaces T and -X", respectively, and Y is a Banach space.
We start with the following lemma.
Lemma 2.1 Let (A*,it*) e A x U. Suppose that
(i) F is continuous and F has tlie u-partial derivative in A x U and
Fu : Ax U —* L(X,Y) is continuous.
(ii) Fu(\*,u*) 6 L(X,Y) is invcrtible.
Then the map *:Ax(/-»Txy, given by
*(\,u) = (KF(\,u)), (2.1)
is locally invertible at (X*,u*) with continuous inverse $.

2.2 The Implicit Function Theorem.
37
If, in addition, F 6 C (A x U, Y) then * is C.
Proof. To prove that fl/ is locally invcrtible at (A*,it") it suffices to
repeat, with obvious changes, the proof of Theorem 1.2.
Next, suppose that F e C'(A x U,Y) and let
A = F\(X',u'), B=Fu(\*,u'),
Plainly * e C'(A x U,T x Y) with derivative
*'(V,u*)(t»= (¢,.410 +D[v]).
Tile equation
*'(V,n*)(t» = (7(,1))
yields £ = r/, and
A[ij\ + B[v] = v, (2.2)
Since B is invcrtible (assumption (ii)) (2.2) has a uniqire solution
v = B_1(v - A[i}]). It follows l:hat #'(A*,«') 6 Iiw(7' x/Jx y). An
application of the Local Inversion Theorem to # proves thai (* is locally
invcrtible at (A*,u*) and) the inverse $ is C1.
Remarks 2.2
(i) Under the assumptions of Lemma 2.1 $ has an inverse $ defined
in a neighbourhood © x V of (A*, F(A*,u*)). Owing to the definition
(2.1) of #, tiie first component of $ is the identity. In other words there
results
$(A,u) = (A,y>(A,*)) (2.3)
for some (p:9xl/-tl satisfying
/■'(A, <?(A, (;)) = y for all A e 0. (2.4)
Such a <p is of class Cl and il.s derivatives <p\ and ^,, can be found by
differentiating the identity (2.4):
Fx + Fu o ^A = 0,
Fw o y>„ = /.
It follows that,
<fiX = -[Fu]~1Fx, (2.5)
and
<fi„ = lF„)-\
(ii) The existence of (.tie local inverse <f> of ty cun be proved by taking
F defined on Ax(/,A being an open subset of a topological space T.

38
2 Local inversion theorems
We are now in a position to state the Implicit Function Theorem.
Theorem 2.3 (Implicit Function Theorem) LetF e Ck(hxU,Y),
k > 1, where Y is a Banach space and A (resp. U) is an open subset
of Banach space T (resp. X). Suppose that F(X*,u") = 0 and that
Fu(\*,u') e Inv(X,Y).
Then there exist neighbourhoods 0 of X* in T and U" ofu" in X and
a map g e Ck(Q, X) such that
(i) F(X,g(X)) = 0 for all X e 0,
(ii) F(A, u) = 0, (A, u) e 0 * U", implies a = g(X),
(iii) g'(X) = -[Fu(p)]_1 o Fx(p), where p = (X,g(X)) and X e 0.
Proof. First wc associate to F the map ^ given by (2.1).
According to Lemma 2.1, # is locally invertible at (A*,u*) and #(A*,u*) =
(A',F(A',u')) = (A',0).
Using Remark 2.2 (i), we find that the inverse $ of # satisfies (2.3).
It is also easy to verify that <fi € Ck provided F is Ck.
Setting
g(\)=<fi(\,0) (Ae0),
and using (2.4) we get
F(X,g(X)) = F(A,<?(A, 0))=0, for all A e 0,
which proves (i).
Since $ is one-to-one, (ii) follows, too. Lastly, (iii) is an immediate
consequence of (2.5).
Example 2.4 UT=X = Y='R !,ho above theorem yields a (unique)
C'1 Cartesian curve u = </(A), defined in a neighbourhood of A = A',
sucii that F(X,g(X)) = 0, wliieh is nothing else than the elementary
Implicit Function Theorem.
3 A stability property of orbits
Many perturbation problems in analysis can be handled by the abstract
results of sections 1 and 2.
Here we discuss in detail mi important application to the existence of
periodic solutions of perturbed differential systems.

2.3 A stability property of orbits 30
Non-autonomous systems
Given /eC'(RxRx Rn,R") let us consider the system of ordinary
differential equations
dx
x'=: — =f{s,t,x). (3.1.6)
We suppose that
f(E,f+T,x) = f(E,l,,x) for all(£,/.,3:) e R x R x R". (.1.2)
and
for £ = 0 (3.1.0) has a T — periodic solution y = y{t)- (3.3)
We want to discuss ttie situation if (3-l.e) has, for £ small, aT-periodic
solution ye elose to y.
Let us consider the Cavicliy problem
a' = f(e, i
r)
«(0)=?. ' '
Since / is C, (3.4) has a unique solution a = a(ett,£) defined for je|
small, £ in a neighbourhood of £' = y(0) and t € [0,T]. Moreover, ii.
is well known that a is differential.)]e with respect to the initial value £
mid tlic derivative
da
is the n x n matrix solving the Cixueliy prot)lem
J A
dt,
A(e,0,i) =/= /ru. j
For e = 0 and £ = £* l.lie matrix /1(0,(.,£') will l>o denoted l>y /1,,(1)-
Theorem 3.1 Suppose. (3.2), (:1.11) /loW «w/
"A = 1 is not in the spectrum a o//l,,(7'). {3.5)
Then there exists 6 > 0 and a continuous £(e), \e\ < c, with £(0) = C,
sucli tluit for \e\ < 6 (3.1 .£) jiossesses a mriqru: T-periodic solution y€
with !t(0) = ¢(£).
Proof. It is evident that (3.1-£) Iras a T-pcriodie solution if and only if
there exists £ e RM such that
a(e,r,{) = {.
Heiiec, introducing the map F : R x R" —► R", defined by
F(s,(.) = c(s,T,i)-i,

40
2 Local inversion theorems
we are led to solve the equation F(e, £) = 0.
Notice that F e C*(R x R",Rn). Moreover, since a(Q,t,C) = y(t)
and y is T-periodic, it follows that
F(0, C) = <*(0, T, r) - C = V(T) - C = 0.
In order to apply the Implicit Function Theorem, we evaluate
WO = a*(0,r,O -I=A0(T) - I.
By (3.5) \= 1 & a and therefore .^(0, £*) is mvovtible. Houco for \e\
small F(e,£) = 0 has a viitique sohitiou £ = £(e) near £ = £*, proving
the theorem.
Autonomous systems
The case in which / is independent of time is more delicate and requires
additional study.
In this case system (3.2.e) becomes
x' = f(e,x). (3.G.E)
Let us point out that the period of possible solutions of (3.6.6) is a priori
unknown. This makes the problem more difficult to solve and often the
perturbation results are the only ones that can be achieved in this cane.
We will set /(0, x) = f(x).
Let us assume / e CV(R'\R71) and that
the unperturbed system x' = f(x) has a non — constant
T—periodic solution y = y(t).
Without loss of generality we can suppose thai- y(Q) — (J. Let us
remark explicitly that, since y is not constant, y'(t) ^ 0 for all t.
Remark 3.2 Theorem 3.1 does not apply in the case of autonomous
systems like (3.(S.e). In fiu-t hero .4,, = /1(0. /,,0) sal-isfim
^ = ^(0)^1 (3.8)
-MO) = i, J
and we claim that A — 1 e <J, the spectrum of l,he matrix AQ{T).
To see this, we difFercnl-ial-e
v' = f(y)
and find
y" = f'(y)y'-
Hence, setting v — y' we gel <;(/) =fi 0 for all t mid
v- = f(y)v. (3.9)

2.3 A stability property of orbits
41
Let v* = v(0) and w(t) = Ao(t)v". From (3.8) it follows that
«/ = -^V = f(y)A0(t)v' = f(y)w, 1 (:J
m(0) = v'. J
By the uniqueness of the Canchy problem, (3.9) and (3.10) yield wit) =
v(t). In particular, there results w(T) = tu(0) and hence
A„(T)v" = w(0) = i>*.
Since v" ^ 0. A — 1 is an eigenvalne of ^o(^), proving onv claim.
According to the above remark, v* eKer(j4()(T)—I). We shall suppose
that
A = 1 is a simple eigenvalue of Aa{T) (-^-11)
that is (see §0.4), that, letting M = A„(T) - I and W = R(M). we get
dim(Ker(M)) = codim (W) = 1, (3.11-i)
Ker(M) = Ker(M2). (3.11-ii)
Theorem 3.3 Suppose that (3.7) and (3.11) hold. Trim for \e\ small
there exist continuous majipings h = h(e) and t = t(e) such that
/.(0) = »(0), t(0) = T,
and (3.6.e) has a r(e)-periodic solution y£ satisfying 1/^(0) = h(z).
Remark 3.4 From the geometrical point of view Theorem 3-3 ensures
that the orbit {(/e(£)}teR 's dose to the orhil, {y(£)}teR- Moreover this
solution is the only one having period close to T and orbit close to
{«(')}'«■
Proof of Uieorem 3.3 Without loss of generality we can assume \v* j — ].
Lei, us consider the liyperplane
7{= {he R" : h ■ v* = 0} (^ R"~'y
Roughly, the solution y(t) leaves H at the time £ = 0 from the point,
y(0) = 0 and comes back to the same point (/(T) — 0 at, the time t — T.
For /ieK near 0 the solution a£ = a(e, (, h) of
-dT = /(£'a)'
a(e, 0, h) = h,
will reach H at a certain time r close to T. (See Fig. 2.1)
Our gua] will be to show that there exist I) e K mid reR stici) tlmt
a(e,r, h) = h and this will provide a r-periodic solution of (3.6.e).

2 Local inversion theorems
Figure 2.1
Set u = (/, p) e R x R" = X and let
F:R x X^ X
be the "Poincare map" defined by
F : (£, t, p) -* (j> ■ v*, ot(e, t, p)-p)e X.
If F(e, t, p) = (0,0) for some (e, r, p) e R x X, then
p ■ v" = 0
and
a(e,r, p) = p.
According to the preceding discussion this means that (3.6.e) has a
t-periodic solution y£ and ye(0) — p € K.
We are going to apply the Implicit Function Theorem to F, taking e as
parameter. Tiicrc results F(0,T,0) - (0,^(0,7^0)) = (0,0). Moreover
F eC'(Rx X, X) and the derivative Fu(e, u) e L(X, X) wgiven by
Fu(e,u) : (a, u) —* (v ■ v* ,at(e, t,p)[a\ + a^(e,t,p)[v] — u).
In order to evaluate tlie second component of Fu(Q,T,0) we si.ari, by
recalling that a0(t) = a(0,t,0) satisfies do0/d( = F(a0),a0(Q) = 0.
Thus ao(t) = y(t) and, in particular, one has
at(0,T,0)[o] = ay'(T) = ay'(0) = av'.
As for a^(0,T,0), this is nothing but the matrix j4o(T). In conclusion
there results
Fu(0,T,0)(a,v) = (v-v',av~ + A0(T)[v] - v)
= (v -v*,av* + M(v)).

2.3 A stability property of orbits
43
Next, we need a lemma.
Lemma 3.5 Fw(0,T.O) ehiv(X,X).
Proof. Given (b, z) e X(= R x R"). we have to solve the system
v ■ u* = b, [
av" + M(v) = z. J
By assumption (3.11 (i)) there exists z" e R", |s*| = 1, such that
R" = Rz* © W, with ;• ■ «> = 0 for all w e W. and hence any z e R"
can be written in the form z — ,sz* + w, with s € R, w € W. Moreover
each v € R" can bo written as u = ?V + h, with r € R and H e K.
Substituting into the preceding system and taking into account that
h ■ v* = 0 and M(v*) = 0. we find
r = 6
and
at)* + JW(/() = hz* + w, (3.12)
Taking the projections of (3.12) onto Rz* and 1^, respectively, and
recalling that M(h) e W, we find that (3.12) becomes
u(„'.z*) = .s, (3.13)
«*>* - «(»* ■ z')z* + M{h)-= w. (3.14)
If v" e IV, there exists q e R" (<y ^ D) such that Mq = v" and hence
M2q = Mv" = 0. Then q e Ker(M2), whereas q ¢ Ker(M). This means
that Ker(M) % Ker(Ma), contradicting (3.11(H)). Therefore v" ■ z' ^ 0,
and (3.13) yields
o= ——. (3.15)
»* ■ s*
Substituting into (3.14) one finds
M(1,) = a. —»* + .sz* € W. (3.11))
'(J* ■ z*
Since M is invertible from K to W, (3.16) has a unique solution /t =
<p{$,ut\ with y eontinuous.
Then the solution of equation ^,(0,7^0)^,^] = (6, z) = (b,sz* + w)
is given by a =(3.15) and v = bv* + <p(.s, w) and the claim follows.
Proof of Theorem 3.3 coviplc-hi'.d. Lemma 3.5 enables us to apply the
Implicit Function Theorem to F. Then there exist r(e) and h(e), defined
in a suitable neighbourhood of e — 0, such that F(e,r(e),/i(e)) = (0,0).
In par!,irn];ii\ h{e) e H and
0(£,T(£),h(£)) = A(£).

44 2 Local inversion theorems
Hence y£(t) := a(e,t,h(e)) gives rise to a r(e)-periodic solution of
(3.6.e) such that ye(Q) = h(e).

3
Global inversion theorems
This chapter deals with the extension in the large of the local results
discussed above.
Section 1 contains the Global Inversion Theorem (sometimes called
"Monodromy Theorem"), which goes back !.o Hadainard in the finite-
dimensional case, and to Caceioppoli [Ca] and P. Levy [Le] for general
Banach spaces. Even if such a theorem is a classical result often
understood in the current literature we think useful to have given here an
elementary version, in the frame of Banach spaces.
In Section 2 we deal with mappings F liirtl. possess singularities and
are not global homeomorphisius. Following [AP], we study the ease of
singularities corresponding to a one-dimensional kernel of F'. wliett a
complete, geometric description of the range can be given.
1 The Global Inversion Theorem
In this section we want to investigate conditions under which a map F
is a global hoineomorphism. Since the results we are going to state are
topological in nature, we will consider a map F : M —* N, where M and
N denote metric spaces.
Let F : M —* N and for any subset A of N let
F-^A) = {u e M \F(u) e A}
denote the pre-image of A through F. For brevity, we will write l'~l{u)
for F-1 ({«}).

4G
3 Global inversion theorem.''
We need the following.
Definition 1.1 We say that F is proper if F_1 (A') is compact (in M)
for all compact set K c N.
Let us remark that if F is proper then it maps closed sets into closed
sets.
For all v € N we let [v] denote the cardinal number of the set F~' (v)
Theorem 1.2 Suppose F e C(M,N) is proper and locally invertible
in M. Then [v] is finite for all v € N and locally constant-
Proof. For all v e N, F~l{v) is compact, (since F is proper) and discrete
(since F is locally invertible); hence [v] is finite.
Next, fixing an arbitrary v e N, let
F~'(v) = {ui,U2 Uk-}.
Since F is locally invertible at each it, (i — 1,2,..,, A:) wo can find
neighbourhoods £/j oiu% and V of u such that F\ut € Horn(£/,, V). Notice
that
[?] > A; for all q 6 V, (1.2)
because the equation /^(^) = v 'ias il1' lo^ist A: solutions, one on each £/,.
We claim that there is a neighbourhood If cV of v wueh that
M = fc for all iff e W. (] .3)
If not, there would be a sequence vn —* u, -u,, € V, such that [uH] ^ k.
According to (1.2) we can infer that [v,L\ > k and hence there exist points
Vn ¢ Ul<f<A:£/i, With F(pn) = Vn-
Since F is proper, up to a subsequence, p„ —» p. By continuity F(p) —
u, and therefore p belongs to some £/,, a contradiction because pn £ Uj
and p„ —► p. Thus proves (1.3) and euin]>let(\s the proof of !]ie theorem.
Remark 1.3 If F is not proper, [v] could be infinite and not locally
constant. For example, this is the case if M = N — C and F(z) =exp(z).
Notice that F is locally invertible at each poinl, z e C.
Ak an immediate consequence of Theorem 1.2 one lias the following.
Corollary 1.4 Suppose F e C(M,N) is proper and locally invertible
in M, and let N be connected. Then [v] is constant for all v e N.
In order to improve Theorem 1.2 we give the following definition.
Definition 1.5 A singular point isau ?M where F is not locally

3.1 The Global Inversion Theorem 47
invertible. The set of all singular points of F will be denoted by £. We
will also set
E0 = F-1(F(E)))
M0 = M\£0,
N0=N\F(E).
It is worth noticing that since £ is closed and F is proper F(£) is
closed in N (see the remark after Definition 1.1). Thus So is closed and
Mo and Mj are open.
Theorem 1.6 Let F e C(M, N) be proper. Then [«] is- locally constant
on every connected component of N\F(E).
Proof Let us consider the restriction F" = F\mv
F' : M0 -► No-
Obviously, F", as a map from M(, !,o Mo, is locally invertible at any
u e Mo and is proper. Thcii, applying Corollary 1.4 with M — M0 and
M = Mo, we get the result.
The preceding result can be greatly improved if Mi '^ simply convec.tv,d.
Recall that a topological space T is simply nimieeted if it is aiewise
connected and every closed path a in T is homotopie to a constant, lit
other words, given any
ff€C([0,l],r), witho-(O) = o-(l),
there exist h e C([0,1] x [0,1], T) and tigT such that
h(s,Q) = ct(«) for ajl.s, "j
ft(/*. 1) = « for/ill «, I (1./1)
h(Q,t)= h(l,t) for all (. J
Theorem 1.7 Suppose. F e <?(M, M) is proper and let M() = M\f(£)
be simply connected and M() = M\F~i(F(E)) be arcwise connected.
Then F is a honwomorphism from M(1 onto Mi-
As a corollary, when £ is empty, wo obtain ;i very cliissicil result.
Global Inversion Theorem 1.8 Let F e C(M,N) be proper and
locally invertible on all of M. Suppose that M is arcttti-se connected and
M is simply connected. Then F is a homeomorphism from, M onto M.
The proof of Theorem 1.7 will be carried out through several steps. It

48
3 Global inversion theorems
is convenient to introduce a definition. Let M,N be metric spaces and
let FeC(M,N).
Definition 1.9 Given a path a : [a, b] —► N; we say that the patti
0 : [a, b] —► M inverts F along a if a = F o 6\ namely if ttie diagram
M - >N
commutes. In such a case, we will also say that. F is invertible along a,
with inverse 6.
Remarks 1.10 (i) Let u e M and v e N be such that F(u) = v and
suppose that there are neigtibourhoods U and V of u and v, respectively,
such that F\v e Hom(C, V). Given any path a : \a,b] —► N with
a(a) = v and such that a\a,b] C V, then the relationship F(6(t)) = a{t)
defines a path 0 which inverts F along a, and such that 6(a) = u.
Moreover 6 is the only path with the above properties.
(ii) (Construetron of inverting paths by "pasting") Lot. a : \o,l>] —» -AT be
a path and let c. e (a,b) be given. Suppose that there are two paths 0t
and $2 such that $i : \a,c] —► M inverts F along <t|[,v:j, (¾ : [(:,6] —► A/
inverts F along cr\{c,b}- Moreover let 0\{c) = 0-i(c).
Then, defining 8 : \a,b] —► M by setting 0j[u ,.j = 0, and 9\fc_,,\ = 82, we
have that 6 inverts F along a. Indeed, 6 is well defined and is continuous
at the point c.
Lemma 1.11 Let u" € Ma and v' = F(u') e ^u- Then given any
path cr : [0,1] —► Nq such that <r(0) = v*, there exists a unique path
0 : [0,1] -* JW() that inverts F along a such thatjf(i)) = «'.
Proof. (Uniqueness). Let 0t and #2 he two paths which invert F along
<7 with ¢, (0) = 02 (0) = w* and let
¢ = 8111^6(0,1):^(() = 02(0, ^ [0, .*]}.
According to Re-mark 1,10 (i) £ is well defined and £ > 0 because u* e
Mq. By continuity one has plainly that #i(£) = #2(0- Let us suppose
the contrary that £ < 1 and set
« = »!«) = »2«), t> = F(li).
Since -F is locally invertiblc on Mo, there are neighbourhoods U of u and
V of v such that F\v eHom(£/, V).

3.1 The Global Inversion Theorem
49
Moreover, since 8^ and 82 are continuous, 3 a > 0 such that
0i([€,€ + «])C<y, 62(^ +a]) CU.
On the other side, siuuo F(0j(()) = F(82(t)) = cr(t.), O^t) = 02(t)
for ( e [£,£ + a]. It follows that 8X and 82 coincide on [0,£ + a], in
contradiction with the definition of £. This proves that £ = 1 and the
uniqueness follows.
(Existence.) Let E he the set of all .v <£ [0,1] such that F is invertiblo
along <7| [0,»] with inverse
0< : [0,5] _► M„ such that 8a(Q) - u', F(v') = v' = ct(0).
We are going to show that E is open and closed in [0,1], so that, E
containing at least the point 0, we will have that E = [0, lj. To prove
that E is closed, we set £ = supE, and note that, as before, £ > 0. By
uniqueness, the paths 0S coincide in the intersections of their intervals
of definition; lot 0 denote- the function that they define on (0,£).
Next, let ,s» T £ bo such that a(xn) _► v. Since 8(s„) = F-1(<r(flft))
and F is proper we have that (without relabelling) 8{sn) —► u,, with
F(u) = i'. Let £/ and V bo two neighbourhoods of u and w, respectively,
such that F\,j eHom(£/, V). If m e N is sncli that
¢(.1,,)6^, widff([«,ttl€j)cK
tlum F can be inverted along (t||.Vii^, giving a path 8\ such that 8\ (sm) —
8(&m). According to Remark 1.10 (h), we can say that F is invertible
along <7|[o,£i a11^ this shows that E in closed. The same construction used
above can be employed to prove that E is open. Indeed, if £ < 1, the
path 8\ just introduced could be defined in an interval [sm,£ + rt<], for
some a > 0 sufficiently small, in contradiction with the assumption that
£ = snpH. This completes the proof of Lemma 1.11.
Lot Q = [0, l] x [0,1] and let 8 and a be continuous maps (which wo
will call "2-pathK"):
8:Q -► M, a :Q^N;
as before we say that 8 inverts F along a if
Fo8^a.
The following lomnia is analogous to Lemma 1.11.
Lemma 1.12 Let u* e Mq and v* = F (**>*) € Nq. Then, given any
2-path o- : Q —► N0 such that <r(0,0) = v', there exists a unique 2-path
8 : Q —► Mn that inverts F along a such that 0(0,0) = it*.
Proof (Uniqueness). Let 8\, 82 : Q —► M0 be 2-paths that invert F

50
3 Global invtrsion thcorr.ms
along a with #i(0,0) = #2(0,0) = it*, and let (s,t) be a generic point of
Q. Define <p\2 ■ Q —* Mu and ip : Q —► N0 by setting
^i (A) =^ (As, A(),
02(A) =0a(As, At),
V»(A)=o-(A*,A().
Plainly, <3^i and <p2 are paths which invert F along V; since <zh(0) =
¢2((1) = w* and (/»(0) = f*, Lemma 1.11 implies that <j>\ = <p2- In
particular, sotting A = 1 wc have
el(^t) = o2(s,t).
Let us note that, evidently, the result can be formulated substituting
for Q any rectangle R: if 8\ and 02 arc 2-paths R —► M0 that invert F
along <r|/? and if at some («*,(*) € R one has #i(s*, (*) = 62(8', f), then
¢1 = 82 in /?..
(Existence) Consider a rectangle
RH - [0, s\ x [0,1] C Q,
and let E be the set of alt s e (0,1] such that there exists 99 : Ra —► Mo
that inverts F along <7|fi„ with ^s(0,0) = it*. Clearly 0 € E; indeed, F
is inverUbk* along the ]>ath i, —► (7(0,(), in view of Lemma 1.11. Let £ =
supE; we will show again that £ e E and tliat ¢=1. By uniqueness, all
the 2-paths 0, coincide in the intersection of their domains of definition;
thus we can define a 2-path
0:[O,Ox[O,l]^Mo
Ahich coincides with 0S in Ra. Let us fix any t e [0, l]; since F is
invertibte along the path s —► a{s, t) through a path s —► <p(s), s e [0,1],
such that 0(0) = 0(0, t), by uniqueness we have
4>{z) = 0(z, t) for all 0 < z < £.
If we set </>(£) = (( and <7(£, () = -u, there exist neighbourhoods £/, V of u
and u, respectively, such that F induces a homeomorphism between U
and V. Then we can find a rectangle R (Fig. 3.1) centred on (¢,() and
a 8 : R n Q -» Mo suclr that 5 inverts F along aRnQ with ¢(¢, () = it.
Since 0(2,() = ¢(2:,() for 0 < z < £ (because 6{z,t) = ¢(2), 0(2,() =
(l>(z) for 2 < £) we infer that 6 can be extended to att Rf\Q. In this
way 6 can be extended continuously to all R^ in such a way that the
relationship F 0 0 = a holds true therein. Moreover, one must have
£ = 1; otherwise, if £ < 1, we could cover the segment {(£,():(€ [0, l]}
with a finite family of rectangles tike R, and 0 could be extended to a
rectangle R^+(t with a > 0 smatt enough. This completes the proof of
the lemma.

3-1 The Global Inversion Thvon
1
1
'/////
7///,
////
i
i
i
i
i
a
►
»
Figure 3.1
PtooJ of Theorem 1.7. First from Theorem 1.6 we infer that \v] is
(constant and) > 1 for all <;.€-/V(|. Henee F is onto N(l.
It remains to show that \v] — 1 for all v e No- Arguing by
contradiction, let «(,,», e A/(, and u e JV() he sneh that F(«()) = F(wi) = »
(Fig.3.2). Since Mi, is arcwise connected, we can find a path 6 e
C([0, t],A/o) with 0(0) =«(,,0(1) =u,.
The image of 0 through F,a — Fo8, is a closed curve in the simply
connected space jVfi. Hence there is a continuous hoinotopy h € C(Q,Nq)
satisfying (1.4). Without toss o!" generality, we can assume that h(a, l)
coincide, just with v mid thai,
/i(0,/) = //(1,/.) = " for all t e [0,1]. (1-5)
From Lemma 1.12 we infer their exists a a unique 2-path 0 e C{Q, M„)
that inverts F along />; that is, such thai,
0(0,0) =«o,
F(0(.v, t)) = h(s, t), for all (.s, () e Q.
In particular, from 7^(0(-^,0)) = />,(.v,0) — &{*>)> wc deduce that
e(A-.0)=(T(s), hence
0(1,0) = 0(1)=^. (1.6)
On the other hand, from (1-5) it follows that
F(9(0,0) = /((0, () = ^,
F(e(a,l)) = /t(«,l) = «,
F(e(U)) = A(l,«)=w.

52
3 Global inversion theorems
+ *
Figure 3.2
Therefore, letting
r=({o}x[o,i])u([o,i]x{i})u({i}x[n,i]),
one has 0|p =constant. In particular
0(1,0) = 0(0,0) = 110,
in contradiction with (1.6). This shows that [v] = 1, and completes the
proof of the theorem. ■
An application
Postponing other applications to the next chapter, we wit) discuss briefly
here J)_resu)t concerning a class o!" asymptotically linear Diriohlct
boundary value problems like
-Au(i) = p(«(a)) + h(x), iffi,
u(x) = 0, ie dCt,
where £1 is a bounded domain in R".
We wilt use the notation introduced in subsection 0.6. In particular,
Ai will denote the first eigenvalue of
—Au = Ait in £), it = 0 on dCl,
with associated eigenfunction <j>\, with <f>i(x) > 0 in £1.
Theorem 1.13 Let peC (R) satisfy

3.1 The Global Inversion Theorem
53
(1) p(s) > 0 for alls;
(2) there exist 7 < Ai and b > 0 such that p(s) < 7s + b for all s > 0;
(3) p'(s)<Xl.
Then (1.7) has a unique solution u E C2,a^l) for any h Q C°'a(Cl).
In order to apply the Global Inversion Theorem 1.8 let
X={u€ C2'"<fi) : u(x) = 0 on Oil),
Y = C°>a(Ti),
F:X^Y, F(u) = A(u) + p(u).
Plainly, FeC (X, Y) and, given h e Y, any u e X such that F{u) = /(
is a solution of (1.7). Let us show
Lemma 1.14 F is locally invcriible, on all of X.
Proof. Since F'(u) : v —► Av +p'{u)v, then, according to Theorem 0.7,
F'(u) e \nv(X,Y) whenever the Dirichtet problem
~Av = p'(u)v in £), w = 0 011 0H
has only the trivial solution. This' in actually the case bccnu.se
assumption (3) and the comparison properly of tJie (Ugenvaiues (hoc Tiu'orom
0.6{ii)) imply that A,<p») > 1.
Lemma 1.15 F i.s proper.
Proof. Let hn e Y and un € X lie sequences such that
We claim
(a) there, exists rt > 0 such that u„(:r) > -(■[ for all x £ f).
To see this, let £1* be a bounded domain of R™ such that H C il*. Let
A' denote the first eigenvalue of —A on Cl' with zero Dirichlet boundary
conditions and let <p' be such that <j>'(x) > 0 on Cl* and
-A<£* = A*<£* in Q', <f>* = 0 on dQ'.
Since H C £1* and <t>*{x) > I) on SI* then ft" = »i'mJ!6^<//(j;) > I). Let
us set c" ~ H - (A*^*)—'. Using assumption (a) and the definition of c*
one readily finds
—A{un + c'<p') = p(un) + hn + c'X*4>'
>/i71 + c*A*0" > hn + //>().
Since, in addition, un(x) + c'4>'(x) > 0 for all x e d£l, the Maximum

54
3 Global inversion tlicorcms
Principle 0.8 implies that un > —c*4>' in £), and this proves the claim
(a).
Next we show
(b) there exists c% > 0 such that un(^) < ci for all x e £)-
Indeed, iet K = H + b and let uibea solution of
-~Aw = ^w + K in f), w — 0 on #£i.
Note that such a solution exists because 7 < Ai and w > 0 in £1. by the
Maximum Principle. Setting zn = w — un, one finds
—Azn = 7W + K — p(un) — hn in Cl.
Let £1+ := {x e £i : un(x) > 0}. Using assumption (2) one readily infers
—Azn > ^w + K —yun — b — hn > 72,,. in £1 + .
Moreover, zn(x) = w(:e) > 0 for all x e $£l,t-
Now, Theorem 0.6{v) implies that Ai{£l+) > Ai and therefore, since
7 < Ai, the Maximum Principle 0.8 applies to zn yielding zn(x) > 0 in
£1+. As zn(x) = z(x) — un{x) > 0 in Si — £1+, it follows tliat z„ > 0 in
£1 and this proves (b).
From the preceding steps (a) and (b) it follows that ||u„i|oo < '-:!-
Since un satisfies
—Aun = p(un) -f- hu,
a repeated application of Theorem 0.5 (ii) and (in) implies- ||u„||a' 5 c4-
Then, without retabelhng, un converges in C2 (£1),/)(14,,) + hn converges
in Y and finally, using again equation (1.8), un converges in X. This
completes the proof of the teninia.
Lemmas 1.14 and 1.15 allow us to apply Theorem 1.8 to F and this
suffices to prove Theorem 1.13.
2 Global inversion with singularities
The main purpose of this section is to study the global invertibility of
mappings when the singular set £ is such that Theorem 1.7 does not
apply. Our goat will be a global, rather precise, geometric description of
E, of F{T.) and of the image of F. For this, it witt be convenient to deal
with smooth (C2) maps F : X —► Y where X and Y are Banach spaces,
and substitute for £ the set
£':= {ueX :F'(u)^\nv(X,Y)}.
Clearly £' D E.
Let F e C2(X, Y), u e £' and suppose

3.2 Global inversion with siiigvlaritics 55
(a) Ker{F'{it)) is one-dimensional: let <j> e X — {0} be such that
Ker{F'{u)) = R<£; R(F'(u)) is closed and has codimension one;
(b) There exists 4> € X such that Fa(u)[4>t<ft] £ R(F'(u)).
Recall that a subset M of X is said to be a C1 -manifold of codimension
1 in X if for ail u' e M there exist 6 > 0 and a functional T : Bj,(u*) —►
R of class C1 such that
jW"nBfi(u*) = {ue b6{u') :r» = o}, {2.1)
r'(w')^0. , (2.2)
We anticipate that if M is a cloned, connected, C' -manifold of
codimension 1 in a Banach space X, then X\M has at most two components.
The proof of this fact is postponted to the appendix.
First of all we prove a lemma.
Lemma 2.1 Suppose for all u e £' conditions (a) -(b) hold. Then £
is aCl manifold of codirnension 1 in X.
Proof. Fixing an arbitrary u* e £', we shall describe £' in a
neighbourhood of it*. Let
V = Ker(F'(u*)), a = R(F'(u*)).
From (a) it follows that there exists 4> € X - {0} and V € K* — {0}
(depending on it*) such that V = R<£ and R = Ker(V'), ful(l there exist.
linear subspaces W (resp.2) in X (resp.K) such that
X=V@W, Y = Z®R.
For any u e X there are unique t e R and uj E W such that
7A = (<£ + ?f).
Moreover we let Q and P = I — Q denote the projections onto R and Z,
respectively. Hence, if z e Z is such that, (V1, z) = 1, then P« = (i/j,w)z,
for all i/ e K.
For it near it* we want to see whetlier F'(it) e "lnv{X,Y) or not,. So
we consider the equation Ff(u)(t<f> + w) = v. Applying P and Q we find
PF'(u)(t4 + w) = PvA
QF'(u)(td>+v)) = Qv.j
For it = it* one has F'(u')<j> = 0 and the second of (2.3) becomes
QF'(u')w=Qv.
Now QF'(u') rs invertible, as a linear bounded map from W to R. Since
Inv{W, R) is open (see Proposition 2.1.1), we can find 6 > 0 such that
QF'{u) e lnv(W,R), for all it e Bs(u).

56
3 Global inversion theorems
Setting T = [QF'(u)]_1, one has w = T[Qv- tQF''{u)tj>\ and the first of
(2.3) yields
tPF'(u)<j, + PF'(u)T[Qv - tQF'(u)4,\ = Pv.
Then (2.3) is equivalent to the system
((V, F'(u)<t,) - t(i>, F'(u)TQF'lu)<j,) = (V>, v) - (V>, F'(u)TQv), (2.4)
w = T[Qv-tQF'lu)(j,\. (2.5)
Since (2.4) is uniquely solvable whenever
{AF'{u)4,) - W,F'(u)TQF'(u)<l>) ? 0,
it. follows that u is singular if and only if
(V\-F>M - (i>,F'(u)TQF'(u),j,)=0.
Hence letting
1» = W,F»0 - (i>,F'(u)TQF'{u)<j,),
we get
u<= s'nB6K)or(u)=o, ue BaK)-
Since T is ohivoiisty C1, it remains to show that (2.2) holds. In fact,
with easy calculations one finds
r» = (V,.F>*)K^]). (2.0)
Therefore (b) yields
r'(u-)^ = (v»,F"(u*)[A«^o,
proving the lemma.
To describe F(E') we shall strengthen (b), giving the following
definition.
Definition 2.2 Wo say that u 6 £' is ;ui ordinary singular point if (a)
holds and
(c) F"(u)l<f>,<f>)?R(F'(u)),
where, according to (a), <£ ^ 0 is such that Ker{F'{u)) = R<£.
Lemma 2.3 Let u* be an ordinary singvlar point. Than then: exist
e > 0 and a map # € C'l(B€(um), Y) such that
(i) *'(u*)€ Inv(X,Y),
<ii) #<u) = F(u) for all ueS'n Be(u*).
Proof. We will keep the notation introduced in Lemma 2.1. From that

3.2 Global inversion with singularities 57
lemma it follows that E' n B6(u') = r_1(0). Let # : Bs(u') -» Y be
defined by
V(u)=F(u)+r(u)z.
The map # is C1 and
¢{14) = F<it), for all it 6 E' n B6(u').
Moreover, there results
#'<it*)it = F'(u*)u + r'(u')(u)z.
Setting u = t<j> + w, and using (2.6) we find
#'<it*)it = F'(u')w + tr'(u*)(<j>)z + r'(u*)(w)z
= F'(um)w + t(i>, F"(u*)[4>,4>])z + (i>,F"(um)[w,<f>])z.
It is readily verified that ^'{it*)it = v has a unique solution whenever
(ip,F"(um){<j>,<j>\) ^ 0, Hence, if (c) holds, *'(u*) 6 Inv(X,Y) and (i)
follows.
Corollary 2.4 If every u 6 T! is an ordinary singular point, then
F(E') is a C1 -manifold of codimension 1 in Y.
Proof From the lemma we can find an e > 0 and a neighbourhood N
of F(u) such that # induces a diffeomorphism between Be(u) and N.
Plainly, the functional
- * 7:=ro#-' : N -»R
is Cl and has non-zero derivative. Moreover, Lemma 2.3 (ii) implies
immediately that F(E') n TV =71 (0), proving the corollary.
Assumption (c) allows us to evaluate the "local" number of the
solutions of F(u) — u. More precisely one has the following
Lemma 2.5 Let u* be an ordinary singular point with Ker{F'{u*)) =
R<£, ana\ say,
<V>,F>-)[<M]>>0r
and set v" = F(u"). Then there are e,o~ > 0 such that the equation
F(u) = v" +sz,u g Be(u*), has
two solutions for all 0 < s < a, J
no solutions for all — a < s < 0. J
Proof, (sec Figure 3.3) For simplicity of notation we take it* = 0 and
F{u') = 0 and study the equation F{u) — sz in a. neighbourhood of 0.

5S
3 Global inversion theorems
Figure 3.3
Set A = F'(0) and F(u) = An + lj(u). Substituing u = t<j> + w in
F(u) = sz we find
Aw +u>{t<p + w) = sz.
Applying the projections P and Q we find the equivalent system
Aw + Qu{t<p + w) = 0,
PLj(t<j> +w) = sz.
This procedure (already used, in a linear framework, in Lemma 2.1) is
usually referred as "Liapunov-Schmidt reduction" and wilt be discussed
in greater generality in Section 5.3-
Since u>(0)- = 0, oj'(0) = 0 and A 6 Inv(W,R), we can apply the
Implicit Function Theorem to Aw + Qu>{t<p +w) = 0, yielding solutions
w — w(t), with w of class C2 such that w(0) = 0 and w'(0) = 0.
Inserting in the second of (2.7) we are ted to solve
X(t);= (il>!u(t<f> + w(t))=s.
The map x is C2 (because u> and w are) and there results (we put
ut = ttp + w(t))
X'(t)=ty,u'(ui)[<j> + w'(t)})
X"(t)= (■4>>u"(ut){<t> + w'(t),<t> + w'(t)\) + W>,w'(ui)K (<)])■
Since w(0) = tu'(0) = 0, u>'(0) = 0 and u"(Q) = F"(0), we deduce
X'(0) =0,X"(0) = W,F"(Q)[4,<I,]) > 0,
and the teinma follows.
We are now in position to state the main result of this section.
(2.7)

Appendix
59
Theorem 2.6 Suppose that
(1) F e C2(X, Y) and is proper,
(2) every u 6 S' is an ordinary singular point,
(3) for all v 6 F(E') the equation F{u) = v has a unique solution,
(4) £' is connected.
Then there exist two open connected subsets V0 and Y2 such that
0) y = y0uy2uf(S'),
(ii) the number [v] of solutions of the equation F{u) = v is
(0 ifveYo,
[v] = \l ifv£F(Z'),
L2 ifveY2.
In order to carry out the proof of Theorem 2.6 we need a further lemma.
Lemma 2.7 Let u 6 £'. Then for every neigbourhood U of u there
exists a neighbourhood V of F{u) such that F-1(V) C U.
Proof. Otherwise we could find a neigbourhood U* of u and a sequence
tin ¢ U~ such that F{un) —► F{u). Since F is proper, one has
(without relabelling) that un —► u~ ¢ t/*, with F(it*) = -F(u). This is in
contradiction with assumption (3).
Proof of Theorem 2.6 From Corollary 2.4 F{Tf) is a C'-manifotd of
codmension 1 in Y. According to assumption (4) £' and (hence) F(E')
are connected. The result mentioned before (see appendix), implies that
Y\F(E') consists at most of two connected components. Let u' 6 £'
be fixed and let B£(u*) be the neighbourhood found in Lemma 2.5.
Applying Lemma 2.7 with U = B£{u') we infer that for all v 6 V the
number [v] equals the "local" number of solutions of F{u) = v, with u 6
U, a number that has been evaluated in Lemma 2.5. It follows that [v]
can be either zero or 2. Moreover, [v] is constant on each component of
Y\f(E'), which therefore consists exactly of two connected components
V0 and y2! say, with the properties listed in (ii).
An application of Theorem 2.6 will be discussed in Section 2 of the
next chapter.
Appendix
Here we prove the following.

60
3 Global inversion theorems
Proposition Let M be a dosed connected Cl-manifold of codimen-
sion 1 in the Banach space X. Then X\M has at most two connected
components.
Proof Supposing the contrary, let A\, A2i A3 be nor empty, open,
disjoint subsets of X such that X\M = Ai U A2 U A3. Since X\M is
open, then each Ai is open with respect to X, too. Let r,- denote the
boundary of Ai. For any i = 1,2,3 one has that Tj 7^ 0 (otherwise Ai
would also be closed), is closed and I\ is contained in M. Since M is
a C'-manifold of codimension 1, for any u 6 M there is an £ > 0 such
that B£{u)C\{X\M) consists exactly of two components, say U\ and U2.
As a consequence, only two of the Ai can have non-empty intersection
with Be(u) and thus Be(u) f\ M can be contained in two of the Ti, at
most. Let, for example, U\ be contained in A\ and U2 in A2. It follows
immediately that
r3n[BE(«)nM| = 0. (^1)
Now, let v be any point of 1^. As before, there is a 6 > 0 such that
Bt{v) O [X\Af\ consists of two components. Since v 6 T3, one of those
components has to be contained in A3. Thus all w 6 Bg{v) C\M belong
to T3 and hence T3 is (closed and) open. Since M is connected, M = T$,
in contradiction with (AI).

4
Semilinear Dirichlet problems
In this chapter we wilt apply the global inversion theorems discussed in
the preceding chapter to the study of some classes of semilinear elliptic
Dirichlet boundary-value problems with asymptotically linear nonlin-
earity. In many cases the results concerning the existence of solutions
could be obtained by means of other tools (for example, the topological
degree). In the spirit of what we discussed in Chapter 3 we prefer to
use the inversion theorems. According to their specific nature, these
abstract theorems lead to the finding of precise results under suitable
restrictions. However we shall see that the Global Irfversion Theorem
3.1.8, used in conjunction with the Lyapunov—Schmidt procedure (see
Section 3.2) lets us handle a broad class of "Problems at Rcsonacc"
which are discussed in Section 1. Let us note that the functional setting
is that of the Holder spaces and the results are founds with rather simple
arguments, which are geometrically expressive.
In Section 2 we deal with the so-called "jumping nontinearities" which
are handled by means of Theorem 3.2.6. According to that theorem we
can establish the precise number of solutions for this class of boundary-
value problems. In addition, wc show how the existence results can
be completed by using other tools, such as the method of "sub-" and
"super-solutions".
We are aware that our discussion is far from complete and refer to
Remarks 1.9 and 1.10 as welt as to the cited papers and bibliography
therein for many other results on asymptotically linear Dirichlet
problems.

62
4 Semilinear Dirichlet problems
Notation
Throughout this chapter, Q denotes a bounded domain in R" with
smooth boundary dCt.
1 Problems at resonance
Consider the semilinear Dirichlet problem
Au + p(u) = h(x) in £),
it = 0 on dCt,
where p is asymptotically linear, and h 6 C°,Q{£1).
More precisely, in the present section we consider nontinearities p of the
form
p(s) =as + b(s)
with a 6 R, and b e C{R) bounded. See Remark 1.13 below for a
slightly more general class of problems.
We shall distinguish between the cases a = Xk or a ^ Afc, where Afc
denotes the fc-th eigenvalue of —A with zero Dirichlet boundary conditions
(see Subsection 0.6).
Case a 7^ Afc for all k
Let us by recalling that in the present case (D) is always solvable.
Theorem 1.1 If a ^ Afc for all k, and b is Lipschitz-continuous and
bounded, then for all h € C°'°{£1) (D) has a solution u e C2<a(Q).
For a proof we refer to [Maw]. Let us recall that a main toot is an a
priori estimate tike that stated in Lemma ] .2 below.
._ We want to prove betow an existence and uniqueness result by means
of the Global Inversion Theorem 3.1.8.
We shall work in Holder spaces (simitar arguments could carried over
taking Sobotev spaces). Set
X = {u e C2>a(U) : u = 0 on dQ}, Y = C°^(U),
and consider the map F defined on X by
F(u) =Au + au + b{u).
Plainly, F maps X into Y and is continuous. Let us start with an a
priori bound for the solutions of (D).
Lemma 1.2 Suppose a ^ Afc for all fc, and b is bounded, and let hn 6
(D)

4.1 Problems at resonance 63
Y,un 6 X be such that F(un) = hn. Then \\un\\y = \\un\\co,a is bounded
provided \\hnWy is.
Proof If not, let Zn{x) = un{x)/^un^Y- Dividing the equation F{un) =
hn by iJUnllr we find
b(un(x)) hn ,, ,,
Azn + azn+ \ " =7i—ir. (1.1)
Set Un := -b(un(x))/\\un\\Y+hn/\\un\\Y; since Un is bounded in £),
Theorem 0.5 (ii) implies that jj^jjci.o <const. and, up to a subsequence,
zn —►z' in C*(Cl). Since |J2njjy = 1, then \\z'\\y = 1; in particular z' is
not identically zero.
Multiplying (1.1) byw€ Co°(£l) and integrating we find
- / VwVzn + a wzn = / wUn- (1-2)
In the right-hand side one has that Un —► 0 (uniformly) because jj/injjy <
ci, b is bounded and \\un ||y —► oo. Then, passing to the limit as n —► oo
in (1.2) we get
- f VwVz' + a I wz' = 0, for alt w 6 C%°(Cl). ~
This means that weakly, and by Theorem 0.5 (iii), strongly, Az* +az* =
0. This is a contradiction, because a ^- Xk.
From Lemma 1.2 we deduce the following.
Corollary 1.3 The map F : X —>Y is proper.
Proof. Let h^ —► h in Y and un € X be such that F(un) = hn. From
the preceding lemma one has that ||itn||y < c\. Setting 6n = —aun —
b{un) + h„, we find that |j0n||y < c2. Since -Au„ = 0n, Theorem 0.5 (Hi)
yields ||un||x ^ C3. Therefore un —►it* in C^Q) (up to a sub-sequence).
As a consequence, 6n converges in C0,a and un —» W in X, proving the
claim.
We are now in position to state the following result.
Theorem 1.4 Suppose that
(bl) b e Cl(R) and there exists M > 0 such that \b(s)\ < M for all
seR,
(b2) either a + b'(s) < Xi for all s 6 R, or Xk < a + b'(s) < Xk+i for
alls<=R.

64
4 Semilinear Dirichtet problems
Then for all h e C°'a(£l) problem (D) has a unique, classical, solution
Proof. Note that now F is of class Cl. Moreover one has
F'{u) : v —► An + av + b'{u)v.
In order to apply the Global Inversion Theorem 3.1.8 to F, it remains
to prove the following lemma which is simitar to Lemma 1.14 of Chapter
3.
Lemma 1.5 F is locally invertible on X.
Proof. Let u 6 X and set m{x) = au(x) + i/(u(x)). From (b2) it follows
that
Xk < m < Xk+i (or m < Ai)
and the comparison property of eigenvalues (Theorem 0.6 (ii)) yields
Afc{m) < 1 < Xk+i(m) (resp. A,<m) > 1)
and this means that the linear b.v.p.
—Av — mv in Cl,
v = 0 on dCl,
has the trivial solution v = 0 only. According to Theorem 0.7 this
suffices to show that F'{u) is invertible.
Proof of Theorem 1.4 completed . Lemma 1.5 and Corollary 1.3 allow
us to apply the Global Inversion Theorem 3.1.8.
Case a = Afe (Problems at resonance)
If a = Afe then problem (D) becomes
Au + Afeit + b(u) = h in Cl,
it = 0 on dCl,
which is usually called problem at resonance. It is clear that, in contrast
with the results stated in Theorems 1.1-1.3, (PR) may now have no
solution at alt: this is indeed the case if b = 0 and J"n fufik i1 0. It is
easy to extend this non-existence result to the semilinear case. Let Afc
be simple and (for a fixed fa) set
f)+ = {x 6 Cl : fa(x) > 0}, Cl- = {x € Cl : fa(x) < 0}, (1.3)
m~ = inf {b(s) : s 6 R}, m+ = sup{6{s) : s 6 R},
(PR)

4.1 Problems at resonance
65
and
A = m~ <pk+m+ 4>k, B = m+ I <f>k + m~ I <f>k-
Proposition 1,6 Let Xk be simple and let b{s) be bounded. Then a
necessary condition for (PR) to have a solution is that
t>k < B. (1.4)
A<Jl
Pi-oof. Let it be a solution of (PR). Multiplying by 4>k and integrating
one finds
fuAfa + J (Xku + b(u))<pk = J
hd>k-
Since A<£fc + X^ipk = 0, we deduce
j b(u)<pk = fupk-
From this, the result follows immediately.
Let us remark explicitly that (1.4) remains the same if <pk is replaced
by -<£fe-
The question whether condition (1.4) is alsosuffieient for the existence
of solutions of (PR) was first investigated by Landesman & Lazer [LL],
In the rest of this section we wilt prove some results concerning this
problem, under .some restrictive conditions, such as the simplicity of A^
and (1>3) below. These assumptions allow us to use rather elementary
arguments and to point out some interesting phenomena. The discussion
follows [AMI].
Suppose that A^ is simple, and let W denote the L2-orthogonal
complement of R<£fc, namely
W=z{weX:(w\4>k)=Q}.
Here and always below we use the notation (u\v) = j^ uv. Every u 6 X
wilt be written in a unique way in the form u = tfa +w, with w 6 W
and t = (u|<£fc). Substituting into (PR) we find readily
Aw + Atm +b{t<pk +w) =h. (1.5)
We wilt employ the "Lyapunov-Schmidt reduction", as in Lemma 3,2,5,
Letting P denote the projection onto W, namely
Pu = u - (u\4>k)4>k(= w),

66 4 Svmihnear DiTichlet problems
(1.5) turns out to be equivalent to the system
A™ + Xkw + Pb{t<pk +w)= Ph, )
(b(t<pk+w)\<pk) = (h\<pk). J
The first equation in (1.6) can be solved as in the non-resonant case.
To be precise, iet$:RxW-»ynW denote the map defined by
¢{(, w) = Aw + Xkw + Pb(t4>k + in).
Lemma 1.7 Suppose b satisfies (bl) and
<b3) Afc_, < Xk+b'(s) <Xk+1(k > I) for all sell (or\i+V(a) < X2).
Then for all h e V there is a unique w{t, Ph) (for brevity we will omit
hereafter the dependence on Ph) such that
(I) $(t,w(t)) = Ph,
(ii) t ->w(t) isC\
(iii) ]]«;(*) ||c»,<* <const.
Proof. We want to show that the Global Inversion Theorem applies to
¢{(,.). The properness of w —► ¢{(,^) follows in the same way as for
F in Theorem 1.5. In order to show that, $w(t,v>) is invertible on W it
suffices, according to Theorem 0,7, to prove that the linear homogeneous
problem
Az + Xkz + Pb'{t<i>k +w)z = 0, ze W, (1.7)
has the trivial solution 2 = 0 only. Setting m{x) = b'{t<pk{x) + w{x)),
one has that Pb'{t<pk + w) = mz — {mz\<fik)<fik and (1.7) becomes
Az + (Xk + m)z - (mz]4>k)4>k = 0, z 6 W. (1.8)
Let us consider the case k > 1 (the case k = 1 requires some changes
in the notation only) and denote by Wi (resp. W2) the space spanned
by {¢1,...,^-1} (resp, by {^+,,^+2, ...}), in such away that W =
W,®W2, and z — zt +z2, withzi 6 W*. Multiplying (1.8) by Zi{i = 1,2)
and integrating we find
(zi\Az) +Xk(zi\z) +(zi\mz) =0, i = 1,2.
Setting z = z\ +22 since {z\\z2) = {VZ1IV22) = 0, we get
- /|V«il2 + f(Xk + m)z2 + fmz1Z2=0 (i = 1,2)
and hence
- /|VZ)|2+ f(Xk+m)z* = - f\Vz2\2+ f(\k + m)4, (1.9)

4.1 Problems at. resonance 67
If z is not identically zero, (1,9) and (b3) imply
J\Vz2\2-J\Vzi\2=J(\k + m)(zl-z>)
n n n
<\k+ljzl-Xk-x fz*. (1.10)
Note that (-1)|0,-) = 0 for alt i = 1,2,..., k. Hence from tho variational
characterization of Afc+i (see Theorem 0,6 (iii)) it follows that
J\Vz2\2>Xk+1Jzl (1.11)
On the other hand, from Z\ = £i<i<fc-iCi<fo one has readily
f\Vzi\2<K-xJ4. (1-12)
Subtracting (1,12) from (1.11) we find a contradiction with respect to
(1.10). This proves that 2=0. Therefore $„, is invertible and (i-ii)
follow from Theorem 3.1.8. As for (iii) we can argue as follows. Let us
write wt for w(t). FronvAuJt + XkWt = Ph— Pb{t<pk +u>t)> and since b
is bounded it follows" (see Theorem 0.5 (ii)) that ||uj(|jc>.» <const. Then
3a > 0 such that \\Ph- Pb{t4>k + wt)\\c^ < «■ Using Theorem 0.5 (iii)
we get the result
Set T(t) = jnb{t<pk +w{t))<pk- According to the previous lemma T ;
R —► R is continuous. Moreover, the preceding discussion shows that to
find a solution of (PR) it suffices to solve the one-dimensional equation
T(t)=Jh4>k. (1.13)
n
Roughly, we will «ee that the behaviour of T as \t\ —► oo is closely related
to that of b(s) as \$\ —► co.
We suppose that
(b4) b(s) —►b+(b~) eR as s —► +oo (-co, respectively)
and set
A'=b~ ! 4>k+b+ ! 4>k> B'=b+ f 4>k+b~ J fa,
n+ n- n+ n-
where f)+ and Cl~ have been defined in (1,3). Without toss of generality,
we can take A' < B',
Theorem 1.8 Suppose A* is simple and that b satisfies (bl), (b3) and

68 4 Senulinear Dirichiet problems
(b4). Then (PR) has a solution provided
A' < (h\d>k) < B'
Proof. Let tn —► +00 and set wn = w(tn). Using Lemma 1,7 (iii) we
infer that wn converges uniformly to some w' (up to a sub-sequence).
Then
tn<j>k(x) +wn(x) —► +co{—00) for alt x 6 £)+{resp. Ct~).
As b is bounded, an application of the Lcbesgiie Dominated-Coiivergence
Theorem yields
T(tn)^b+j 4>k+b- j4>k = Bf (tn_n-oo).
n+ n-
Simitarty, one has
T{tn)-+b- J fa+b+ J <j>k = A' (tn->-oo).
n+ n-
Since T is continuous, it follows that (1.13) has a solution t* provided
(1.4) holds. The corresponding u' = t'<pk +w(t') gives rise to a solution
of (PR).
Remarks 1.9
(i) Assumption (b3) and the simplicity of Xk can be eliminated.
Following [AM2] the argument is, roughly, as follows. Let i,j 6 N be such
that Xi < a < Afc + tf{u) < c2 < Aj for alt u 6 R. Set
V = span {<fc+i» ■ ■ ■ i^j-i}
and let W denote the L2-orthogonal complement of V with projections
Q and P respectively. As before, any u 6 X can be written in the
(unique) form u = v + w, with v = Qu 6 V and w = Pu 6 W and (PR)
cati-be again replaced by the equivalent system
PF(v+w) =Ph,
QF(v+w)=Qh.
The first equation (on W) can be uniquely solved finding w = w(v), and
we are ted to the finite-dimensional equation P{F(v +w{v)) = Ph. This
tatter can be studied by means of the Brouwer topological degree.
(ii) For other results on problems at resonance, see for example the
book by Fucik [Fu] which contains an extensive bibliography. Problems
at resonance where the linear part (together with the boundary
conditions) gives rise to Fredhotm operators with positive index have been
investigated in fSche] and [AAM].
0

4.1 Problems at resonance
69
In the rest of this section we wilt discuss some of the possible
phenomena arising in the study of (PR). The following result deals with a
case in which (1.4) is not satisfied.
Theorem 1.10 Suppose that Xk is simple and (bl) and (b3) hold.
Moreover let us assume that
(b5) sb(s) —► a > 0 as |s| —► oo.
Then (PR) has a solution provided (h\<j>k) = 0.
Proof, We keep the same notation as before. In particular, Lemma 1.7
holds true, and taking h such that (h\<f>k) = 0 we are led to solve the
equation T{t) = 0, where T is defined in (1.13).
To use (b5) it is convenient to write tT(t) in the following form;
tT(t) = /b(td>k +w(t))(tfa+w(t)) - fb(td>k+w(t))w(t).
Let tn be any sequence such that jt„j —> co and set wn = w(tn) and un =
tn4>k +u>n- Note that, as before, wn —► w' uniformly in Cl, Moreover,
setting
n';« £1+U£T,
one has un{x) = wn(x) for all x 6 Cl', and hence
tnT(tn)= fb(un)un- fb(un)wn.
Furthermore, there results
|un(x)| = \tn<j>k(x) + wn(x)\ —► oofor \tn\ —► coaiid x ¢ iY.
Using this and (b5) wc deduce
f b(un)un -> a\Cl'\. (1.14)
n-
Moreover, since b(s) —► 0 as |s| —► oo we also find
J b(un)wn -»0. (1.15)
ii'
Lastly (1.14)-(1.15) yield
tnr(tn) -»(r|f)'|.
Since, plainly, \Q{\ > 0, the preceding limit is positive and the equation
T{t) = 0 has a solution.
Remark 1.11 The same arguments show that there exists e > 0
(depending on Ph) such that (PR) has at least two solutions provided

70
4 SemiUnear Dirichlet problems
0 < |{/i|<£fc)| < e. In addition, in a way simitar to that sketched in
Remark 1.9 (i), the same existence and multiplicity result can be proved
relaxing (b3) and the assumption that Afc is simple; see [AM2|.
We end this section with a uniqueness result.
Theorem 1,12 Consider (PR) with Afc = Xx and suppose b satisfies
(bl), (b3) with k = 1, (b4) and is such that tf(s) ^ 0 for all s (for
example, let us take b'{s) > 0).
Then (PR) has a unique solution if and only if
b~ I' fa < I' htfn <b+ f 4>s. (1.16)
Proof Note that (1.16) is nothing but (1,4): in fact now Q = f)+ and
Q~ = 0. We wilt prove the theorem showing that T is strictly increasing.
For this, let us recall that w is differentiabte with respect to t (Lemma
1.7) with derivative denoted by w'. From
Aiv + Aim +Pb(t<j>! +w) =Ph ,
it follows that w' satisfies
Aw' + Aiu/ + Pbf(t<f>i + w)(4n + wf) = 0. (1.17)
As for T, one has that T is differentiable with derivative
r'(t) = Jb'itfa +w(t))(<f>y +w'(t))4>1.
n
Then (1.17) becomes
Aw' + Xtw' + b'itfa +w)(4n +w')-Tf(t)<p1 =0. (1.18)
Suppose the contrary, that 3t" such that r'(2*) = 0. Then, setting
u' = t'tpi +w(t') and z~ = <j>\ +w{(t'), from (1-18) we infer
Az' + Xxz' + f(u')z' = 0.
This means that z" is a solution of the linear b.v.p. Az' + m'z' = 0,
where m* = Ai + b{(u')- Notice that z' is not identically zero (in fact
(z'\<j>\) = (<£i|<£i) = 1) and therefore Afc(m*) = 1 for some integer
k > 1. By assumption m* < A2 and hence the comparison property of
eigenvalues (see Theorem 0.6 (ii)) yields A2(m*) > 1. Thus one must
have Ai(m") = 1 and z" is either > 0 or < 0 in Cl. Since b' > 0
and 4>i > 0, it follows that T'{t') = jilb'{u')z'4>i is > 0 (resp. < 0)
according to the sign of z'. This is a contradiction, proving the theorem.

4.2 Problems with asymmetric nonlinearitics 71
Remark 1,13 Alt the preceding results could be easily extended to
elliptic problems like
Cu + au + b{u) = h,
u\aa = 0,
where £ is an elliptic operator with smooth coefficients (see subsection
0.6) and b such that b(s)/s —► 0 as |s| —► oo. Moreover, h could be taken
in £2{£)): in such a case one should work in Sobolev spaces instead of
Holder spaces and one would find weak solutions.
2 Problems with asymmetric nonlinearities
In this section we consider problem (D) in the case when p(s) has two
different asymptotes as s —► ±oo. More precisely we start by assuming
that p satisfies
(pi) p € C2<R),p<0) = 0 and p"(s) > 0 for all u 6 R,
(p2) p'(s) —► Y (resp. 7") as s —► -co (resp. +00), and there results
0 < 7' < Ai < 7" < A2.
We keep the same notation as in the preceding section, letting X =
{ueC2'a(n) :u = Q on dil},Y = C°•<*(&) md F(u) = Au+p(u). Note
that (pi) implies that F 6 C2(X,Y). The main abstract tool will be
Theorem 3.2.6.
We start by proving the following.
Lemma 2.1 F is proper.
Proof. Let un € X be such that F(un) = hn is bounded in Y. We claim
that ||u,i||v is bounded. Since the arguments are very close to those of
Lemma 1.1, we wilt indicate the main changes onty."t*t supposing the
contrary, |ju„||y —► 00; then letting zn = un/\\un\\y one has
^+^¾ w
where
VW \p'(0) ifu = 0.
Since tp is bounded, (2.1) and Theorem 0.5 (ii) imply that
IWIc'-" < const,
and, up to a sub-sequence, zn —* z* in C1 (fi) with ||2* \\y = 1. Multi-

72
4 SemiUnear Dirichlet problems
plying (2.1) by w e Co°(£l) and integrating we find
-jvwVZn + jw<(,(un)zn = J-^l-. (2.2)
n n n
Note that if z*(x) < 0 (resp. > 0) then un(x) —> -co (resp. +oo).
Thus, letting
{T' if z*(x) < 0,
y if 2*(x) > o,
p'(0) if 2*(x) = 0,
one has that <p(un(x))zn(x) —► m(x)z"(x) pointwise in Cl. By the
Lebesgue Dominated Convergence Theorem, we infer from (2.2)
- / VwVz* + / mwz* = 0, for all w e C£°(£l).
Therefore A2* + mz* = 0 and Afc(m) = 1 for some integer k > 1.
Since m < j" < A2 it follows from Theorem 0.6 (ii) that A2(m) > 1
and hence one has \\(m) = 1. As a consequence z* does not change
sign in Cl. Then m equals either 7' or -y" and in both cases we reach
a contradiction. This proves the claim. This a priori estimate implies
readily that F is proper as shown in Corollary 1.3.
Next we want to study the singular set
X' ={ueX : F'(u) £ Inv(X, Y)}
. Notice that here F'(u) is nothing but the map v —> Av + p'(u)v and
hence u e E' whenever the linear b.v.p.
Av +p'(u)v = 0 in £),
(2.3)
= 0 on dCl, '
has solutions other than the trivial one, or in other words, whenever
A*.(p'(m)) = 1 for some integer k > 1. Since, by assumption, p'(s) <
l" < A2, the comparison property (Theorem 0.6 (ii)) implies that 1 is ,
if an eigenvalue, the first eigenvalue of (2.3).
Lemma 2.2
(i) E' is not empty, closed and connected.
(ii) Every u e £' is an ordinary singular point.
Proof. To prove (i) we will show that E' has a Cartesian representation
on a linear subspace W of X of codimensiou 1. Fixing z e X7z(x) > 0
for all x e £), let W be any linear subspace of X such that z £ W. Every

4.2 Problems with asymmetric nonlinearities 73
u e X can be written in the form u = oz + w, with a e R and w e W-
Let m,,. = p'{o~z + w) and consider the first eigenvalue A1(m(T) of
Av + Xmav = 0 in £),
it = 0 on d£).
According to the preceding remarks, the point az + w belongs to the
singular set E' if and only if Ai(mCT) = 1. Since z(x) > 0, ma > m^
whenever a > p and hence \i(ma) is a decreasing function of a (see
Theorem 0.6 (ii)). Moreover, from z(x) > 0 in £), we also infer that
ma —* Yi"/") as a —> -co (resp. +co), pointwise and in Lq (for all q)
because p' is bounded. Using the continuity property of the eigenvalues
(Theorem 0.6 (iv)) wc deduce that
Ai(mCT) —* -^1(7') = —- > 1 as a —* -co,
7
Ai(m„) —► Xi(j") = — < 1 as a —*■ +00.
Since A^m,,) is decreasing, it follows from (2.4) that there is a unique
a" such that Ai(mCT-) = 1. Since the corresponding it* = a*z+w belongs
to 57,-(0 follows.
(ii) Let u e E'. Then \\(p'(u)) = 1. Since the first eigenvalue is simple,
Condition (a) of Section 3.2 holds. In particular, Ker(F'(it)) is spanned
by a non-zero function 0, which does not change sign on £); moreover,
h e R(F'(u)) if and only if /n 4>h = 0 (see Theorem 0.7 (ii)) so that
R(F'(u)) = Ker(^), where
1}) : h —* I (ph.
a
To prove that Condition (b) of Section 3.2 holds, we notice that
F"(it)[u,n/| = p"(u)vw. Then we have
(^,F"(it)[0,0])=yP»03,
n
which does not vanish because p"(s) > 0 (Hypothesis (pi)) and <j> is
either positive or negative on £). This proves that (ip,F"{u)[$,0]) ^ 0
and hence that v is ail ordinary singular point.
Lemma 2.3 For all h e F(E'), the equation F(u) = h has a unique
solution.
Proof. Let 2 e E' be such that h = F(z). Suppose the contrary that
(2.4)

74
4 Semilinear Dirichlet problems
there is w ^ z with F(w) = h. Set
u(x) = / (p(w(a:)) -p(2(x)))/(tu(x) - z(x)) for all x : w{x) ^ z(x),
\ p'(z(x)) for all x : w(x) = z(x).
From F(2) = F(w) (that is , A2 +p(z) = Aw + p(w)) it follows
immediately that v := w ~ z is a (non-trivial) solution of
An +uv =0 in £), )
ti = 0on dCl. J
In other words, there results Afe(u>) = 1, for some integer k > 1. Since
7' < u> < 7" < A2, then we must have Ai(u>) = 1 and v(= w~~z) does not
change sign on Cl. Suppose, for example, that v > 0, that is, w > z (if
v < 0 the argument is similar). Since p" > 0, a>(x) < p'(.z(x)) on £) and
hence (1 =)A,(a>) > Ai(p'(z)). But zeE' implies that A,(p'(z)) = 1, a
contradiction.
Lemmas 2.1, 2.2 and 2.3 allow us to apply Theorem 3.2.6. yielding a
precise description concerning the solutions of (D).
Theorem2.4 Let p satisfy (pl-2). ThenY = C°'a(U) = Y0UY1UY2
with the following properties:
(1) Y\ is a C1 -manifold of codimension 1 in Y and (D) has a unique
solution for all /1 € Vi ;
(2) V0 and ^2 ore disjoint open subsets of Y such that for all h e V2
(D) has exactly two solutions, while for all h e Yq problem (D) has
no solution.
Theorem 2.4 is taken from [Ap]. A different proof has been given by
Berger & Podolak [BP] by using arguments more on the line of those
discussed in the preceding section for problems at resonance. Roughly,
a Lyapniiov-Schmidt procedure yields the reduction of problem (D) to
the study of a one-dimensional equation T(t) = (/1,0i); moreover the
properties of p allow one to show that T(t) —► +00 as \t\ —>■ +00 and
that r"(*) > 0 for all t.
Problems like the preceding have been called problems with "jumping
nonlinearities" to point out the fact that the nonlinearity p has two
different asymptotes at +00 and at -co.
Also in the case of problems with "jumping nonlinearities" the
existence (and multiplicity) results could be obtaind by different methods
and could be improved. Below we will expound a rather general result
due to Kazdau h Warner [KW]. First some preliminaries are in order.
To highlight the different behaviour of p at ±00, we will take p of the

4.2 Problems with asymmetric nonlincarities 75
form
p(s) = 0s* ~ as~ + b(s),
where a, 0 eR,/?^ a (we will take below a < 0),s+ — max(s,0), s~ =
s+ ~ s, and b is bounded. Note that the nonlinearity p in Theorem 2.4
is of this form with /3 = j" and a = j'.
It is also convenient to set
~h(x) = t<t>(x) + n(x), with (n\<f>) = 0,
where 4> = 4>\ > 0, b(x7 s) — b(s)+rj(x) and p(x,s) = 0s+—as~+b(x,s).
With this notation (D) becomes
~Au = p(x, u) + 20 in £), u = 0 on #£2. (Pt)
The index ( is used to highlight that the results will depend on t.
First of all, lot us note that if the interval [a, 0\ does not contain any
eigenvalue Afc, then (Pt) has solutions for all (6R. This could be proved
by topological degree arguments, but will not be discussed here.
The situation is different when some Afc e (e*, /3), a case in which
nonexistence or multiplicity results can arise. More precisely, we prove the
following.
Theorem 2.5 Suppose that
(p3) b is Holder-continuous and \b(x, s)\ < M, for all (x, s)6(lxR,
(p4) -oo < a < Xi < P < +oo.
Then there exists T* e R such that
(i) for all t <T* problem (Pt) has a solution,
(ii) for all t >T* problem,(Pt) has no solution.
The proof of Theorem 2.5 will be carried out using sub- and super-
solutions. Recall that x G C2(Cl) is a sub-solution of the problem
—Ait = q(x,u) in £1, I
„ „ ' (2-5)
a — U on d£2, J
if there results
—Ay < g(x, y) in £1, I
X < 0 on dQ. J
A super-solution ij> is defined similarly, reversing the inequalities in (2.6).
It is well known (see for example [Ama]) that the following result
holds.
Proposition 2.6 Let p be continuous in Cl x R and suppose that (2.5)
has a sub-solution x and a. super-solution ip such tliat x < ip on Cl. Then
(2.5) has a solution u with x < u < iff.

76
4 Semilinear Dirichlet problems
Let us start by noticing that from a < 0 and \b\ < M it follows that
p(x, it) > au+ — au~ — M ~ ctii— M. (2.7)
For any (eR, consider the boundary-value problem
-Au = au - M + t ¢, u\du = 0. (2.8)
Since a < Ai, (2.8) has a unique solution xt- Using (2.7) it follows
immediately that
-Axt =axt-M + t$ <p(x,xt)+t$-
Therefore, we can say that the following lemma holds
Lemma 2.7 For all t eR, Xt is a sub-solution for (Pt).
Next we prove another lemma.
Lemma 2.8 If ip is any super-solution of (Pt) then ip > Xt-
Proof Since ip is a super-solution of (Pt) and xi solves (2.8) one infers
-Aty - xt) > p(x, ip) + t<f> + AXt
= p(x, ¢)-+ t<f> - (aXt ~ M + t<f>)
= p{x,ip) + M -axt-
Using (2.7) we deduce that
-A(V> - Xt) > <*(ip - xt) in ^.
Since a < Xt and ip—xt > 0 on dCl the Maximum Principle 0.8 applies
to w = ip — Xt yielding ip — Xt > 0 in Cl, as required.
The next step is to prove a third lemma.
Lemma 2.9 There exists t" e R such that for all t < t*, (Pt) has a
super-solution ipt.
Proof. Let us fix a > 0 and take m > 0 satisfying
m > max{p(x, s):i€fi, 0 < s < a}.
Moreover, let Cl' CC £1 (i.e. Cl' has closure contained in Q) be such that
|fi'| < e, where e > 0 is a number to be determined later on, and set
Q* = Cl\Cl'. Fixing Cl" CC £)', consider a function h e C°°(Cl), such
that (i) 0 < h < m in £), (ii) /i(x) = 0 in £)", (iii) h(x) = m in Q* and let
us denote by ^ the solution of Aip = /i, ^|©n = 0. From the Maximum
Principle (Theorem 0.8) ip > 0 in Q. Moreover from Theorem 0.5(i) one
has
IMU»- < CiIIMIlp < cie,/p for all p > 1.

4.2 Problems with asymmetric rwnlinearities 77
Taking p > n/2 (where n is the dimension of £)), we deduce, using
Theorem 0.4(iii),
Halloo <C2eI/p.
Therefore, if e is taken in such a way that C2e1/,p < &, then one has
0 < ^(x) < cr and hence
p(x,#r)) < m. (2.9)
Let 0o = min{0(x) : x e ^'}(> 0) and set t" = —mffa. We claim that
t*<p + m </iinf) (2.10)
Indeed, if x € Q' then
2*0 + m = ——0 + m < 0 < h,
while if x e Cl* one has 2*0 + m < m = h, and (2.10) follows.
Using this and (2.9) and (2.10) we find
-Aip = h>t'$ + m> p(x, ip) + t*<p.
Lastly, for all t < t* one has
-Aip > p(x, ip) + t*<p > p(x, ^) + (0,
proving the lemma.
Proof of Theorem 2.5 Define T* by setting
J" = sup{2 e K. : (Pt) has a super-solution}.
From Lemma 2.9 it follows that T* is well defined. Moreover it is easy
to see that T* < +oo. In fact, if (Pt) has a super-solution w, that is
—Aw > p(x, w) + t<f>, then multiplying by <p and integrating on 1¾ we
find
<pp(x,w) + t
= / <p[(iw+ — ocw~] + / 06(x, w) + (.
Hence
( < Ai <pw — / 0[/3w+ -qu)"] + c,
n n
where c = M J"n 0. Since Aiw +<ra" — /3w+ — Ai(uj+ — w~) + ctw~ —
0w+ < 0 because a < Ai < /3. Then it follows that ( < c and J" < +oo.
For all t < T* (Pt) has a super-solution ipt and a sub-solution xt
(Lemma 2.7); moreover from Lemma 2.8 it follows that Xt < ipt- Using
Proposition 2.6 we infer that for all ( < T* (Pt) has a solution u with

78 4 Semilinear Dirichlct problems
Xt < u < iftt- It remains to show that (P() has a solution for t = T".
Take a sequence tk —*■ T",tk < T*. Problems (P*) corresponding to
values t = tk have solutions u*. From the preceding construction it is
easy to check that the Uk converge to some it* which solves (Py-).
This completes the proof of Theorem 2.5.
Remarks 2.10
(i) Improving Theorem 2.5, it has been shown [AmaH] that for all
t < T"(Pt) has at least two distinct solutions (the second one is found
by degree-theoretic arguments). See Problems (1) and (2) below for
another multiplicity result in this direction. For an extensive discussion
of elliptic equations with jumping nonhnearities we refer to [Fu].
(ii) A geometric description of the range of a differential operator more
in the spirit of Theorem 2.4 can be found in [McS].

5
Bifurcation results
The structure of the solution set of a nonlinear functional equation can
be very complicated and often it could be convenient to assume a
"genetic" point of view, seeking for when new solutions are generated, near
a given one, after a small perturbation. A convenient device consists
in finding (or introducing) a parameter A, and studying an equation
F(X,u) = 0 which possesses a fixed solution for all values of the
parameter. An interesting phenomenon is when there is a "branching" of
new solutions of F(X,u) = 0 in correspondence with some value of the
parameter. This is the object of the "Bifurcation theory" we will discuss
in this chapter in its more elementary aspects.
1 Introduction
Let X,Y be two Banacli spaces. We are interested in studying equations
of the type ..._.,.._
/f(X,u) = Q [) (1.1)
where
F-.-Rx X-*Y_
is a map depending on a real parameter A.
As we will see in the next chapter, equations like (1-1) model a broad
class of problems arising in applications, where the parameter A often
has a physical interpretation: it can be the intensity of the loading in
some elasticity problems, the Rayleigh number in hydrodynamics, and
so on.

80
5 Bifurcation results
-Vy\C ^? »
Figure 5.1 This and the following bifurcation diagrams
are to be interpreted as suggestions only.
In this chapter we will always assume that F e C2(R x X,Y) and
that "~
J^VQj= 0 for all A g R.
If this is true, then (1.1) has for all A the solution u — 0, which will
be referred to as the tjivial sdluliofT?)
S = ((A,u) e R x X : u ^ 0,F(X,u) = 0} will denote the set of
non-trivial solutions of (l.l). "" ~ ' ' '
It can happen that for some values of the parameter there are one
or more solutions of (1.1) that branch off from the trivial one. These
values of A are called the bifurcation points of (1.1) (Figure 5.1). More
precisely, we give the following definition:
Definition 1.1 We say that, A* is a bifurcation point for F (from the
trivial solution) if there is a sequence (An, un) eRx! with un ^ 0 and
F(K,un) = 0such that
(A„,«7l)->(A*,0).
Another, equivalent, way to define a bifurcation point, is to require
that (A*,0) belong to the closure (in R x X) of 5, that is, that in any
neighbourhood of (A*,0) there is a point (A,it) e $■
Let us begin our discussion by stating a result that follows immediately
from the Implicit Function Theorem.
Proposition 1.2 A necessary condition for A* to be a bifurcation point
for F is that the partial derivative Fw(A*,0) is not invertible.
Proof. If Fu(A*,0) e Inv(X,Y) then Theorem 2.2.3 applies and there

5.1 Introduction
81
exists a neighbourhood 0 x V of (A*,0) such that
Therefore A* is not a bifurcation point for F.
An interesting case is when X = Y and
F(X7u) = Xu~G(u). (1.2)
In such a case F„(A*,0) ~ A*/— G'(0) and Proposition 1.2 becomes
the following.
Proposition 1.3 If X is a bifurcation point for F of the form (1.2)
then X belongs to the spectrum a[G'(0)] of G'(0).
It is quite natural to ask whether or not Proposition 1.3 can be
inverted:
i/Ag <t[G'(0)], is A a bifurcation point for F ?
We anticipate that, in this generality, the answer to the preceding
question is negative.
The particular case when F lias the form (1.2) with G = A e L(X) is
particularly enlightening. Note that, if G is linear, then G'(0) — A and
the relationships between the bifurcation points for
F=XI~A (1.3)
and the spectrum <r(A) of A can be established in a precise fashion.
First of all, it is clear that the eigenvalues of A are bifurcation points
for F= (1.3).
Moreover, the following result can readily be proved.
Proposition 1.4 Let A e L(X) and F(X,u) — Xu~ A(u). Then A* is
a bifurcation point for F if and only if A* belongs to the closure of the
eigenvalues of A.
Remarks 1.5
(i) As a consequence of Proposition 1.4, wc deduce that in general,
there might be points X belonging to the spectrum of A that are not
bifurcation points for F of the form (1.3).
(ii) From Proposition 1.4 it also follows that A* can be a bifurcation
point for XI ~ A without being an eigenvalue of A.
We have seen that when F has the form (1.3) all the eigenvalues of A
are bifurcation points for F. The following example shows that in the
nonlinear case a value A* can be an eigenvalue of G'(0) without being a
bifurcation for F(\,u) = Xu~ G(u).

82
5 Bifurcation results
Example 1.6 Let X = Y = R2, and consider the application G :
X —>■ X defined by
G(x,y) = (x + y3,y~x3).
The value A* = 1 is an eigenvalue of g'(Q) = I, but it is not a
bifurcation for
F : (x,y)-> X(x,y) - G(x,y).
For let (x, y) be a solution of F = 0. From
Ax = x + y3, "t
Ay = y - x3, J
it follows that
x* + yA = 0,
and hence (x, y) = (0,0). Therefore F = 0 has only the trivial solution
and there are no bifurcation points for F.
2 Some elementary examples
In this section we will discuss a couple of simple examples, where the
existence of bifurcation points can be proved in a rather elementary way.
As a first example, let us consider tire boundary-value problem
d^+A(it-U3)=0, *€[0,tt], (2.1)
, u(0) =u(tt) =0. (2.1')
For all values of the parameter AeR (2.1)-(2.1') has the trivial solution
u(t) =0. To find other possible solutions we can work in the phase plane
and argue as follows. Multiplying (2.1) by
Au
p=di
one finds immediately that any solution of (2.1) satisfies the energy
relationship
\p2 + A(y - y) = c (= constant). (2.2)
The integral curves in the phase plane (u,p) are represented in Figure
5.2.
We can distinguish between two families of integrals, which are
separated by curves that pass through the singular points (1,0) and (—1,0).
In particular, the closed curves correspond to periodic solutions of (2.1)
and we are interested in the arcs of those closed integrals that start from

5.2 Some elementary examples 83
Figure 5.2
points (0, p) and again reach the p-axis after a time equal to ir. From
the symmetry, such a time is an integer multiple of the semiperiod T.
Let r^ be an integral curve crossing tire it-axis at it = £, withO < £ < 1
(see Figure 5.3).
Putting p = 0 and it = £ in (2.2) we can find the value of c
corresponding to P^:
c=A(k~^4)- (2-3)
Let T"(A,£) be the semiperiod of P^. From the symmetry, T"(A,£) is
given by
7
where 7 denotes the arc of T^ with p > 0.

84
5 Bifurcation results
Figure 5.3
Taking into account (2.2) and (2.3) one finds with easy calculations
dx
r(A'?)-2/vT2c-^2-^)]
J x/[2c - \((V - ^V)]
= 2 /- - ^
(2.4)
VPme-^-HiV-ii'y'
Va/
d!/
x/[l - i?2 - !/2(l " |?2!/2)] '
Let us note explicitly that (2.4) albws us to extend T{A,.) at £ = 0
by setting
^0) = 7a/^
dy
(2.5)
-!/2) xA
u
As anticipated before, 1^ gives rise to a solution of the boundary-value

5.2 Some elementary examples
85
6 &
Figure 5.4
>0,
problem {2.1)-(2.1') whenever £ is such that
mT(A,0 = T {2.6)
for some m e N.
For example, if £ is such that T(A, £) = 7r, then 1^ corresponds to
a positive solution of {2.1) with u(0) = 0 and p(0) = u'(0) = p^ {see
Fig.5.3), u(n) = 0 and p(n) = u'(n) = -p$.
To discuss equation {2.6) we first deduce from {2.4) that,-for any fixed
A, there result
dT(\(j) .
di
T(A,0-»+ooi»£Tl.
Taking {2.5) also into account, we get that
(a) if A < 1 then T(A,£) > tt for all £, and (2.1)-(2.1') has only the
trivial solution u = 0,
(b) if A = 1 then T(A,0) = tt and £ = 0 is the only solution of {2.6),
and hence {2.1)-(2.1') has again the trivial solution only,
(c) if 1 < A < 4, then (2.6) (with m = 1) has a unique solution £ ^ 0
which corresponds to a positive solution u^ of (2.1)-(2.1'). (More
precisely, as A | 1 the solution £ = £(A) as well as p$ tends to 0;
correspondingly (2.1)-(2.1') has a family uA = U£{X) °f (positive)
solutions, depending continuously on A, such that ||ua((c ~* 0
as A | 1, and we can say that the (positive) solutions of (2.1)-
(2.1') bifurcate from the trivial solution at the value A = 1 of the
parameter).
The discussion can be carried over showing that (see Figure 5.4)
(d) if k2 < A < (Jfe + 1)2 then (2.6) has k solutions £i,...,£fc ^ 0,

86
5 Bifurcation results
Figure 5.5
satisfying
hT(\£h) = ir (h =1,...,¾).
Each £h corresponds to a solution of (2.1)-(2.l') with precisely h— 1
nodes in (0, it). In particular we can say that there is a continuous
family of solutions u\ of (2.1)-(2.1'), with k — 1 nodes in (0,7r), such
that |[ua|[ci -»■ 0 as A i k2.
It is worth noting that, from the abstract point of view, the boundary-
value problem (2.1)-(2.1') gives rise to a functional equation of the type
(1.1). Here X is the Banach space of C2(0,7r) functions vanishing at
t = 0 and t ~ 7T, Y = C(0, tt), and F : R x X -» Y is given by (see also
Example 2.1.5)
F(X,U):
(1¾ w
u3).
There results
d2u
Fu(X,0) : u — -^- + Au,
and the values A^ = fc2 are precisely the eigenvalues of the linear problem
d2u
d<2
+ Au = 0, u(0) = u(n) sr 0.
As a second example, we consider the buckling problem for an elastic
beam of length L.
We suppose that one edge of the beam is hinged, while the other one
is variable on the x-axis. The beam is compressed at the free edge by
a force of intensity K > 0. Denote by (x(s),j/(s)) the coordinates of a
point A on the beam, as a function of the length s of the arc 0^4, and let
0(5) be the angle between the tangent to the beam at A and the x-axis.
See Figure 5.5.

5.2 Some elementary examples
Figure 5.
In accordance with the Euter-Bernoulli theory, the curvature of the
beamvat any point is proportional to the momentum of the applied force.
Then there result
fCy =—k— (k = constant), -^- = sine
ds as
together with the boundary conditions
y(0) = y(L)=0.
From (2.7)-(2.7') we deduce
d2^
ds2
+ Asin^ = 0, s e (0,L],
(2.7)
(2.7')
(2.8)
*'(0) = 4(L) = 0,
where A = K/k > 0.
To study (2.8) we can proceed as before. We refer to Figure 5.6.
If we let p = d^/ds, conservation of energy yields
-p2 — A cos 4> = c = —A cos ^o,
and the semiperiod of the (closed) curve passing through (<£o> 0) is given
by
r(A,<M=2|^
ff/2
2 r ie
\A J \/(l-W2sm20)'

88 5 Bifurcation results
An arc of a closed curve joining two points of the z-axis corresponds
to a solution of (2.8) whenever
hT(X, <pQ) = L,
for some /ieN.
Since T(A,.) is stilt strictly increasing and T(A, <fo) —► +oo as <fo | it,
we deduce:
(a) For 0 < A < tt2/L2 (2.8) has only the trivial solution ¢ = 0.
(b) For fcV/L2 < A < (k + 1)2tt2/L2 (2.8) has k nontrivial solutions
4>i, ■ ■ ■, <Pk- In addition, for fixed k = 1,2,..., there exists a
continuous family <px of nontrivial solutions of (2.8) whose Cl -norm
tends to zero as A | A& = k2ir2/L2.
The bifurcation diagram is drawn in Figure 5.7.
As for the preceding case, we can see that the functional equation
corresponding to (2.8) is
F(^4>)-= ^4+Asin^ = 0
as2
whith <f> e X, the Banach space of functions of C2(0, L) such that <£'(0) =
#(L) = 0.
The linearized equation i*0(A, 0)^/; = 0 becomes
ds2
+ \4> = 0,rp'(tt) = rp\L) =0,
(2.9)
whose positive eigenvalues are just

5.3 The Lyapunov-Schmidt reduction 89
Note that A = 0 would also be a bifurcation point of F(X,<fi) = 0, F
given by (2.9). Indeed, for A = 0, the equation F(Q, <fi) = 0 has the family
of nontrivial solutions <j> =const. However, these solutions correspond to
the trivial solution j/sOof the physical problem (2.7).
3 The Lyapunov-Schmidt reduction
In this section wc dicuss a general procedure, introduced by Lyapunov
[Ly 1-2] and Schmidt [Schm], which will be a basic toot hereafter, but
can also be useful in several other situations. It is a method we have
already used in Sections 3.2 and 4.1, for the specific problem studied
there.
Let F e C2(R x X,Y) be such that
F(A,0) =0.
According to Proposition 1.2 the possible bifurcation points for F arc
the values A* such that Fti(X* ,0) is not invertibte. We sot
L = Fu(A*,0),
V = Ker(L),
R = R(L),
and suppose that
(a) V has a topological complement W in X.
This means that there exists a closed subspace W of X such that
X= V®W (3.1)
and any u e X can be written in the form
u = v + w, v e V, w € W. (3.2)
On the range R of L wo Jissumc
(b) R is closed and has a topological complement Z in Y.
This means that Y = Z ® R, with Z closed and such that Z C\R= {0}.
For example, (a) and (b) hold true when V is finite-dimensional and R
has finite codimension, that is, when L is a Fredfiolm operator.
Next, let P and Q denote the conjugate projections onto Z and /?,
respectively.
Using (3.2) and applying P and Q one finds that F(X,u) = 0 is
equivalent to the system
PF(X,v + w) = 0, (3.3')
QF(X,v+ w) = 0. (3.3")

90
5 Bifurcation results
For later use, it is also convenient to set
F{\, u) = Lu + ip(\, u).
Using (3.2) and recalling that Lv = 0, one has
F(A, u) — Lw + <fi(X, v + w).
Recalling that Lw € R, we get QLw = Lw. Then (3.3") becomes
Lw+Qip(Xtv + w) =0. (3.4)
We set
$(A, v, w) = Lw + Q<p(X, v + w)
and note that $ € C2(R x V x W, R). Moreover
$™(A*,0,0) : w -> Lw+Q<pu(\*,Q)w.
Since, by definition, <p(\, u) = F(A, it) — Lu, there results
^U(A*,0) = FM(A',0) - L = 0. (3.5)
In other words, <^U(A*,0) is the zero mapping in L(X,Y) and therefore
it follows that $U,(A*,0,0) = L\w.
We remark that the restriction L\w of L to W, as a map from W to
R, is injectivc and surjective, Since R is closed, (L|w)~' is continuous
from R to W, namely
L\w e Iso(W,fl). (3.6).
Hence $„,(A*,0,0) e Iso (W, ft), the Implicit Function Theorem applies
to $ and (3.4) can be uniquely solved, locally, with respect to w. To be
precise, there exist
(i) a neighbourhood A of A*,
(ii) a neighbourhood V of v = 0 in V,
(iii) a neighbourhood W of w = 0 in W, and
(iv) a function j € C2(A x V, W), ,
such that the unique solutions of (3.#") in A x V x W are given by
(A,«,T(A,«)).
In particular, for future reference, we remark that results
7(A,0) =0 for all A 6 A. (3.7)
Moreover one has
7V(A',0) = 0. (3.8)
To see this, we can use the Implicit Function Theorem or else we can
take into account that
Ly(\,v) +Q<p(\,v +y(\,v)) = 0 for all (A, v) 6 A x V.
Differentiating with respect to v at (A*,0), and letting T = yv(\*,Q), we

5.4 Bifurcation from the simple eigenvalue 91
find
Lrx+Q<pu(\*n(\*,Q))\x + rx] =Ofor all z € V.
Since 7(A",0) = 0 and using (3.5) we get that LTx — 0 for all x e V,
and hence Tx eVHW. Thus Tx = 0 for all x e V.
After these preliminaries, we can substitute
w = j(X,v) (3.9)
in (3.3') getting
P(F(A,v + 7(A,v))) = 0. (3.1())
The equation (3.10) in the unknowns (\,v) e A x V is called f/te
bifurcation equation and, together with (3.9), is equivalent (in A x V x W)
to the initial equation F(X,u) =0.
Obviously, the preceding reduction is useful if the bifurcation equation
is simpler than F = 0. This is the case when L is a Fredhohn operator:
if dim(V) =p and codim(ft) = dim(Z) = g, then (3.10) is a system of q
equations in the unknowns (A, v) e R x Rp.
4 Bifurcation from the simple eigenvalue
In Section 1 we saw that the possible bifurcation points of F(X,u) ~ 0
are those A* such that F„(A*,0) is not invertibte. To find sufficient
conditions for A* to be a bifurcation point, some restrictions arc in order. In
this section we wilt study the case in which L = F„(A',0) is a Fredholm
map with index zero and with one-dimensionai kernel. In the case of
equations tike "" """""
G(w) = Au,
with G e CX{X, X), C(0) = 0 and G'(0) compact, this corresponds to
the case when A* is a simple eigenvalue of G'(0).
Let us take a map F e C2(R x X,Y) satisfying F(A,0) = 0 for all
A. We note cxpticitty that the condition F € C2 could be weakened
assuming that FeC'fRxI, Y) and has mixed partial derivative FUi\,
see Remark 4.3 (i).
The hypothesis that L = .Fu(A*,0) satisfies assumptions (a) and (b)
of Section 3 needs to be specified here.
Keeping the notation of the preceding section, we set V =Ker(L), R =
R(L) and let W and Z denote complementary subspaces of V in X and
R in Y, respectively.
We will say that L (or F) satisfies Assumption (I) if

92
5 Bifurcation results
(I-i) V is one-dimensional: Bit* € X, u* ^- 0 such that V = {tu* : t e
R},
(I-ii) ft is closed and codim (ft) = 1.
According to (I-ii) Z is one-dimensional and there exists a linear
functional ip e Y*, iff ^0, such that
^{1,^:(^)=0}.
Wc also use symbols P and Q to denote the projections onto Z and R,
respectively.
With this notation, the bifurcation equation (sec (3.10)) becomes
W,F(\tu' +7(A, («*))) =0. (4.1)
It is convenient to set A = A* + /i, and
/3(/i,r.) = W'.-FO^* + /M«* + 7(** + /!<'"*)))■
Note that /3 is a real-valued function defined in a neighbourhood i/ of
(0,0) e 1R x R and is of class C2 there, because F and 7 are C2.
The following properties of /3 will be used later (subscripts denote
partial derivatives):
(/31) /3(/1,0) = 0 for all /1; in particular,
(/32) /3„ (0,0) =/3,,,,,(0,0) =0;
(/33) A (0,0)=0.
To prove (/31) we note that
/3(/i,0) = (V,F(A',T(A*+/,,0))).
Since T(A,0) s 0 (see (3.7)) and F(A*,0) = 0, (/31) and (/32) follow.
Next, to prove (/33) we differentiate /3 with respect, to t yielding
A(m> 0 = (¢, F»(>>" + m, (u* + t(a* + /», <«*))("* + 7» (>* + /*> («*)«*)).
Letting ( = 0 and taking into account that 7(A* + /1,0) = 0, we find
/3,(/1,0) = i>,F„(A* +//., T(A* +/!,0))K +7„(A* +//,())«*])
= (V,F„(A*+/1,0)K+T„(A* + /»,0)U*)). (4.2)
Since 7„(A*,0) = 0 (see (3.8)), we infer
/3,(0,0) = (V>,F„(A*,0)u*) = (ip,Lu') =0,
proving (/33).
Furthermore, from (4.2) it follows that
&A»,0) = (V.-Fu,a(A* +/i,0)(u* +7v(A* +/i,0)u*])
+«., F„(A* + m,0)T„,a(A* + M,0)(u*J).
Here and always hereafter, we identify, according to Remark 1.4.4, the
mixed derivative such as FUi\ or jVi\ with linear maps.

5.4 Bifurcation from the simple eigenvalue 93
Letting /, = 0 one finds
/3,,,,(0,0) = »,F»,1(A*,0)|u-+T,(A*,0Kl)
+ (V,Fu(A-,0)T„,A(A',0)K])
= (fP,FUiX(X',0)\u'}) + «>,.Fu(A-,0)T„|A(A*,0)K]).
Finally, since i>\R = 0 and F„(A*,0)7„iA(A*,0)(irj 6 R,
(V,F„(A-,0)T„,A(A,,0)(u-))=0,
and we infer
(/34) /3,,,.(0,0)= G0,F„,,(V,O)(u'l).
Furthermore, let us remark for future reference that, with direct
calculations, one finds
(/35) /3,.,(0,0) = (V,F„iU(V,0)(u*,u')>.
We are now in position to state the main result of this section.
Theorem 4.1 Suppose F e C2(R x X,Y) be such that F(X,0) = 0 for
all A € R. Let X" be such that L = FU(X*,Q) satisfies assumption (I).
Moreover, letting M denote the linear map Fu A(A*,0), we assume that
Mu* ¢ a. (4.3)
Then X* is a bifurcation point for Ft In addition the set of non-trivial
solutions of F = 0 is, near (A*,0), a unique Cl cartesian curve with
parajnetric representation on V.
Proof. According to the preceding discussion wc have to solve the
equation
P(H, t) = 0,
where fi is C2. In order to use the elementary Implicit Function
Theorem, we need to "desingularize" /3. For this, let us introduce the function-
lM' ^1/3,(/1,0) for ( = 0.
Using properties (/31-4) it is easy to see that h is C1,/,(0,0) = 0 and
that
M°>°)=/3,,„(0,0),
M0.0) = ^/3,,,(0,0).
Sotting
o := fi„(0,0) and b := (,,(0,0)

94
5 Bifurcation results
and using (/34) and (/35) we find
a = (tp,Mu'),
b=±W,FUtU(\*,0)\u*,u*}).
In particular, from Assumption (4.3) one deduces
a = (ip,Mu*) ^0. (4.4)
Therefore the Implicit Function Theorem applies to h = 0 yielding a
neighbourhood (—£,£) of t = 0 and a unique function ft £ C '(—£,£)
such that ^(0) = 0 and h(n(t),t) ~ 0 for alt t e (-£,£). Since the
equation /t(^, () = 0 is equivalent for t ^ 0 to /3(^, () = 0, it follows that
the bifurcation equation (4.1) has been solved uniquely by ft ■= fi(t).
Then, according to the results of Section 5.3, one finds that
F(X" +ft(t),tu* + j(X* +n(t),tu*)) = 0 for all t e (-e,e).
Note that tu* + t(A* + n(t),tu*) ^ 0 provided t ^ 0. Therefore the
set S of nontrivial solutions of F(X, u) — 0 is given, in a neighbourhood
of (A*,0), by the (unique) cartesian curve
A = \* +fi(t), 1
u- tu* +t(A* +n(t), tit*), J
where t € (—£,£),£ ^0. This completes the proof of the theorem.
Theorem 4.1 becomes particularly expressive when Y = X and
F(X,y) = Au-G(u),
where G e C2(X,X) is such that G(0) = 0. As already seen in Section 1,
the possible bifurcation points of f are points of the spectrum of G'(0).
Here we will show that, when G'(0) is compact, any simple eigenvalue
A 7^ 0 of G'(0) is in fact a bifurcation point. A statement of this sort
will provide a first answer to the question posed in Section 1, after
Proposition 1.3.
Theorem 4.2 Let G e C2(X, X) be such that G(0) = 0 and such that
G'(0) is compact. Suppose tiiat A* ^- 0 is a simple eigenvalue o/G'(0),
in the sense that
dim(Ker(A*/-G'(0))) = 1, (4.5)
Ker(A'/ - G'(0)) n R{X*I~ G'(0)) = {0}. (4.6)
Then X" is a bifurcation point for F(X, u) = Xu — G(u).
Proof HereL = Fu(A*,0) = A*/-G'(0), V =Ker(A*/-G'(0)) = {tu* :

5.4 Bifurcation from thr simple eigenvalue 95
t € R} and R = R(X*I— G" (0))- Since G'(0) is compact, assumption
(I) follows immediately from (4.5).
Moreover, owing to the specific form of F, one has M = FUj\(\*tQ) is
the identity map, and therefore a = (i/j,u*) where, as before, it* denotes
a vector spanning V. According to (4.6), u* ¢ R and thus a / 0, proving
(4.3). Then Theorem 4.1 applies and the result follows.
Remarks 4.3
(i) As anticipated, F can be assumed of class C1, with continuous
mixed partial derivative FUi\. For a proof, which requires some
technicality, sec [PA]. Similarly, one can show that Theorem 4.2 holds provided
GeCl(X,X).
(ii) The bifurcation results stated in Theorems 4.1 (or 4.2) do not
hold, in general, if wc only assume that L satisfies (I), namely that
the dimension of Ker(L) and the codimension of R(L) are 1 without
assuming (4.3) or (4.6). To sec this, wc can slightly modify Example
1.6. To be precise, let X = Y = R2 and
The same calculations performed in Example 1.6 show that, if
F(\,x,y)=Q,
then there results
y2+yA+x*=Q.
Henqe F(\,T,y) = 0 has the trivial solution, only, and there are no
bifurcation points.
Here, the derivative of F with respect to v = (x) evaluated at (A, 0,0)
is the map
/Xx - y\
\yj V xv
Then, for X* = 0, we have that L can be identified with the matrix
'-[: -.']■
Tims V - R =span{</*}, with u* = (J), and (I) holds. On the other
hand, M is the identity and hence Mu* —u* € R. Note that Theorem
4.2 does not apply either. Indeed, the algebraic multiplicity of A = 0 is
2, not 1 (in other words, (4.6) is not satisfied).
It is worth noticing that Theorem 4.2 applies only to maps of the
specific form F(\, u) = Xu — G(u). We mean that when X ~Y but F
is not in the form XI — G, one has to use Theorem 4.1. In such a case,

96
5 Bifurcation results
the condition V f\ R = {0} does not piay any role. To explain this, let
us consider the map F:RxR2-t R2,
As before, for A = 0, L = [° ~g] and V = R =spanQ, but now the
mixed derivative Fu ^(0,0,0) is the matrix M = [n] and M : (0) —►
(,) ¢ R, so that (4.3) holds true and X" — 0 is a bifurcation point for
F. With a rather elementary calculation one can solve the system
Xx-y-y^Q, 1
X(x + y) + x3 = 0, J
showing that the bifurcating branch has equation
y = -z3 + ..., 1
(iii) When a = (ip, FUi\(\*,Q)[u*]) = 0 several different situations can
occur and a more careful analysis is required. In the analytic case, it
can be useful to employ the Newton polygon method to solve the
bifurcation equation. For mote details on this matter we refer to [VT]. The
Newton polygon method has been extended to differentiable functions
by Dieudonne [D2].
(iv) Assuming F more regular (say C°°, for simplicity) one can
complete Theorems 4.1 and 4.2 by some calculations which will allow us to
specify the behaviour of the bifurcating branch near (A*,0) (see Figure
5.8).
Since n(t) solves h(n, t) = 0, there results (see the proof of Theorem
4.1)
'J(U) Mo,o) «'
where (see earlier)
o=W,F„,i(A*,0)(u*]),
» = iw,F„,„(A',0)(u-,u']).
Therefore if b ^ 0 we have
X = A* - - t + o(t)
and the bifurcating branch (A,it) € S can be parametrized (for \X — X*\
small) in the form
u~ —-(A — X")u" + o(A — A").

5.4 Bifurcation from the simple eigenvalue 97
(a) case 6^0 (b) case 6 = 0, c > 0 (c) case 6 = 0, c < 0
(transcritical) (supercritical) (subcritical)
Figure 5,8
We note that when 6^0 the equation F — Q has nontrivial solutions
both for X > A* and for X < X* (transcritical bifurcation).
When 6 = 0 one finds
2c := /«"(0) = ~(Vi, Ft(Ut,(V,0)K]3). (4.7)
If 6 = 0 and c ^ 0 the bifurcating branch has the form
/\-\*\l/2
u = ±l j .u* +0(\-\').
Note that the preceding formula shows that if c > 0 (respectively, r < 0)
then the bifurcating branch emanates on the right (respectively, left) ol
X* (supercritical, rospeclively subcritical, bifurcation).
It is worth remarking that, when F(X, u) — Xu~G(v) and G is smooth,
the values of b and c are given by the formulas
» = -^».G"(0)K,«*])
and
c = ^«.,G"'(0)(«*]3).
Remark 4.4 When F(X,u) = Xu — G(?i), with G compact that is,
G(u,t) is relatively compact in X for any bounded sequence {u„}. it is
possible to use the~Ceray-Sehauder topological degree and Theorem 4.2
can be greatly improved. Results of this sort arc outside the .scope of this
book and cannot be discussed here. However, owing to their relevance,
we shall give a review of the most important ones in a short appendix
at, the end of this chapter.
Postponing further examples to the next chapter we discuss here some
problems related to those studied in Section 2.
Example 4.5 (Sturm—LiouviUe problems) Let J = [0,7r],o- €
Cl(J),PGC(J),a,P >0on J, PeC2(JxRxR) and let ao,6o,a,,6,
be such that (a£ + b%) (af + tf) ± 0.

5 Bifwcation results
Consider the Sturm-Liouville b.v.p.
d / d \ , / dii
-£U:=-E^-Uj+/3U = Au+P^,U,£J,iej, (4.8)
o0u(0) + 60"' (0) = 0,11(71-) + 6iti'(7r) = 0, (4.8')
where A is a real parameter.
Setting X = {« 6 C2(J) : 11 satisfies (4.8')}, Y = C(J), define F ■
R x X -> Y by
F(A, u) = £u 4- Am + p(«) (4.9)
(as usual we are using the same symbol p to indicate the Nemitski
operator associated with the real-valued fvmction p) so that the solutions
of (4.8)-(4.8') are the pairs (A, u) 6 R x X such that F(A, u) = 0.
Suppose that p = p(x, s, £) satisfies
p(x, 0,0) s 0, p„ (1,0,0) = 0 and K (1,0,0) = 0.
As a consequence, one lias
.F(A,0) =0 for all A,
F„(A, 0) : u -> £11 + Am.
Recall (see subsection 0.6) that the linear problem
- Cu(x) =Au(i) (ieJ|,
..]
(4U1)
«0^(0)+6oit'(0) =ai?i(7r)+btw (tt)-ft '
has a sequence Afc of positive, simple eigenvalues, such that A*. —* oo
as fc —* co. Let <^fc be an eigenfunction of (4.10) corresponding to A*,
normalized by '■
IT
y^.dx=i.
0
Let, us apply Theorem 4.1 with A* = A*, tint I u* = ft.-- According to
subsection 0.4, one nas
IT
V=KerfF„(Afc,0)]=R^fc, R= R\Fu(Xk,0)] = {u eY : l uyk&x = Q).
fl
Therefore (I) holds true. Furthermore wc can define i}) by (ij),u) =
£ utpk<\x.
Since FVtx(\k,0) : v -► v, a = (ip,<pk) = Jo <fitdx = 1, proving (4.3),
In conclusion, applying Theorem 4.1 wc infer that each Afc is a bifurcation
point for F = (4.9). Hence
For each k = 1,2,..., there is a continuous family u\ of noTitrivial
solutions of (4.8)-(4.8') such that ||u\||ca —* 0 as A —* \k.

5.4 Bifurcation, from the simple eigenvalue 9Q
Example 4.6 (Dirichlet Problems) Let Q be an open bounded
domain in R" and consider the boundary-value problem
- A?x = Xu + p(x, u, V?x) in £),
u = 0 on dCl,
where p 6 C2(R x R x R") satisfies p(x,0,0) = 0, pB(x,Q,Q) = 0 and
Pz(x,Q,Q) = 0.
Since the discussion does not differ from that of the Sturm-Liouville
problem, we will be sketchy.
Let X = {u 6 C2'"^) : ix = 0 on dQ), Y = ¢7^(57) and F(A,m) =
Aw + Au + p(u); one has that F(A,0) = 0 for alt X and Fu(A,0) is the
map v —* Av + Xv. Hence Fu(A,0) has a nontrivial kernel provided X is
an eigenvalue of
-An = Xv in Q,
u = 0 on d£l-
If Afc is any simple eigenvalue of (4.11) with corresponding ci gen function
ifk, normalized by ffttptdx = 1, then (I) holds true. As before, one
has R = R(Fu(Xk,0)) = {u 6 Y : (^,u) = fnu<pkdx = ()}; since
FUlx(Xk,0) : v —> v,a = {ip,<fik) = 1 and (4.3) tiolds, too, Therefore,
frorn Theorem 4.1 it follows that
any simple eigenvalue of (4.11) is a bifurcation point for
F(X, u) ~ Au + Xu + p{u).
We note that, in particular, this result applies when we take the first
eigenvalue of (4.11).
To know the behaviour of the bifurcating branch we refer to Remark
4.5. For simplicity, let ns take a (smooth) Honlinearity p depending on
u only.
Since here F1K„(Xk,0) : (v,w) —*p"(0)vw, (4.4) becomes
n
If p"(0) = 0, one uses (4.7) yielding
c=-^p"'(0)yVfc<is-
n
For example, if p(ix) = -ix3, then 6 = 0 and c > 0; hence the bifurcation
is supercritical that is, occurs for X > Afc, while, if p(ix) = ix3, then c < 0
and the bifurcation is subcritical (see Figure 5.9).
We end this section with some further remarks on the geometric
character of Theorems 4.1-4.2.
(4.11)

100
5 Bifurcation results
=e
9
(a) Bifurcation portrait
for — An = Xu— u3
(b) Bifurcation portrait
for - Au = Xu + ix3
Figure 5.9
After the Lyapunov-Schmidt reduction, the problem of finding the
bifurcation points of F{A, u) = 0 is reduced to the search for the zeros
of a real-valued C2 function /3 = /3(^, 2), with the properties that
0(^0) =0forallM,
/3((0,0) = 0,
&,t<0,0)^0.
The proof we have carried out led us to find two branches of solutions:
that of the trivial zeros, and that of the nontrivial solutions, giving rise
to the bifurcation branch. Suppose now that F is perturbed through F,
with \\F - F\\ci < e, with e small. Perturbing F through F will affect
the bifurcation equation in the sense that /3 = 0 will be replaced by a
perturbed bifurcation equation /3 = 0, with ||/3 - /3l!c2 small. In general
it is possible to prove that the zeros of /3 become two branches that
do not cross themselves, in general, but are merely "close" (see Figure
5.10).
This kind of perturbation phenomenon arises, for example, wiTen one
deals with bifurcation problems from the point of view of numerical
analysis: approximation or truncation procedures can be viewed as
perturbation.
For a discussion of this kind of problems, we refer to the paper by
Golubttsky and Schaeffer [GS].
5 A bifurcation theorem from a multiple eigenvalue
In this section we will discuss a result dealing with a case in which
Ker(L) is, possibly, not one-dimensional.

5.5 A bifurcation theorem
101
(«) Solutions of 0 = 0 (b) Solutions of 0 - 0
Figure 5.10
For simplicity, we consider an F 6 C°°{R x X,Y) and assume that
(a) find (b) of Section 3 hold true. Keeping the notation of §3, we set
L = F„(V,0) and write X = V®W,Y = Z® R, with V = Ker(L) and
R = R(L).
Let M denote the linear map FUi^{A*,0) (recall Remark 1.4.4) and B
the bilinear map F„iM(A*,0); then, setting X — X" + /z we find that the
equation F — 0 becomes
Ln + ftMn+ -B(u,n) + tf(\* +/*, u) = 0, (5.1)
where ip is smooth and such that
ip(\,0) = 0, 0„{A*,O) =0, ^Ui„<r,0) =0, ^aiU{A*,0) =0. (5.2)
We seek solutions of the form u = fi{v + w), with v 6 V and w 6 W.
Substituting into (5,1) we find
fiLw + fi2M{v + w) + -fi2B\v + w, v + w] + ^(X* + ^, n(v + tu)) = 0.
Then, according to the Lya])uuov-Schmidt reduction, the equation /*' —
0 is equivalent to the system
H2PM(v + w) + -ti2PB\v + w,v + w] + Pij}(\* + n,n(v + w)) = 0, (5.3')
liLw + li2QM(v + v)) + -ii2QB{v + iu,v+(o] + QiJ,(y+fi,n(v+w)) = 01
(5.3")
where P and Q indicate, as usual, the projections onto Z and R.
According to (5.2) we can write
l}>(\* + {!.. ft(v + V>)) = 1^1/)((1-, v, w)
where if> is smooth.

102 5 Bifurcation tcsilILh
Hence (5.3')-{5.3") are equivalent for /i ^ 0 to
PM(v + to) + -PB\v + to, v + to] + liPip(\' + n, /i(v + to)) = 0, (5.4')
Lra + /iQM(v + to) + -mOB(» + to, « + to] + i?Q^>(>.' + n, /i(» + to)) = 0.
(5.4")
With $ = $(/*, v,w) denoting the left-hand side of (5.4") there results
¢{0,0,0) = 0 for all v 6 V as well as ¢,,,(0,1),0) = L\w\ hence, for
any fixed v' 6 V, we can solve (5-4") uniquely will] respect to w in a
neighbourhood of /*, = 0, v = v*, from (5.4") it follows readily that
w = /17(/1, v)
with 7 smooth. Substituting into (5.4') we find the bifurcation equation
N(fi1v) := PM{v + /17(/1, v))+ ^PB{v + /17(/1, v),v + /17(/1, v)}
+/iJV(V + /1, n(v + /17(/1, i)))) = 0. (5.5)
Note that N is smooth. Moreover, let us point out that 7 depends on
v* 6 V. We wiLl show that if v* can be chosen in a suitable way then
(5.5) can be solved, giving rise to a bifurcating branch for F = 0. More
precisely one has the following.
Theorem 5.1 Suppose that V = Ker(L) has a topological complement
in X and R — R(L) is closed and has a topological complement in Y.
Moreover, letting M = FUt\{\*-,0) and B = Fuu(\*,0), suppose there
exists v* 6 V, v' ^- 0, such that
(a) PMv* + ±PB(v*, v\) = 0,
(b) the linear map S : V —♦ V, Sv = PMv + PB(v*tv) is invertible.
Then there is a brunch of nontrivial solutions of F = 0 bifurcating from
(V,0) with equations
A - A* , . , .
(5.6)
:<m)» J
u;/ifire ^(0) = 0 and x'(0) = V.
Proof. From (a) it follows that N(Q,v*) = 0; moreover Nv{Q,v') = S,
which is invertible by (b). Then the Implicit Function Theorem applies
to N(fi,v) = 0. To be precise, there exists v = v(/i), defined for |/i|
small, such that v(0) = 0 and
JV(M,»(M))=0.
Hence we find a bifurcation branch of the form
u(/i) = n(v(fj.) + M7<M, «(/*)))■

5.5 A bifurcation theorem 10!1
Setting x{&) '■= m(u(m) + M7(Miu(m))), we get x'{0) = v'. As a
consequence, u(n) ^ 0, |^| small and > 0, and u = x(p) gives rise to a branch
of nontrivial solutions of F = 0. This proves that X' is a bifurcation
point for F = 0 and completes the proof of the theorem.
Remarks 5.2
(i) The branch found in the preceding theorem might not be unique:
cither because v* might be ttot uniquely determined, or because there
are other nontrivial solutions of F = 0, not in the form /i(v + w).
(ii) The equation of the bifurcating branch is parametrized with
respect to ii and thus indicates that Theorem 5.1 gives rise to a transcril-
ical bifurcation.
Theorem 5.1 can be used to find a sufficient condition for the existence
of a bifurcation when dim(V) =dim(2) = 1, but Theorem 4.1 does not
apply.
Let V = Ru", and suppose that PMu' = 0. Then conditions (a) and
(b) of Theorem 5.1 become
(a') PB\u*,ur}=0,
(b') the linear map v —* PB\v*,v] from V to Z is* invcrtiblc.
If {a')-(b') hold true then an application of Theorem 5.1 yields the
following.
Theorem 5.3 Suppose. F e C2(R x X,Y) is such that F(X, 0) = 0 for
all A g R. Let X' be such that L = FU(X*,0) satisfies Assumption (1)
and let V = Ru*. Moreover, we set M = FUix(\,Q), B = FUiU(V,0)
and we assume that Mu* 6 R and that, {a')-(b') hold true. Then X* is
a bifurcation point for F.
An application
Let us apply Theorem 5.1 to the following problem: given a continuous
27r-periodic function h, to find 27r-periodic solutions of
u" + Xu+ hu* = 0, (5.7)
We set X = C%„, Y = C27r, where C^ (resp. C2sr) denotes the
space of 27r-periodic Ck functions (resp. continuous functions), and let
F : R X X — y,
F(X, u) = u" + Xu + hu2. (5.8)
Here Lv = v" + Au, Mv = v and B[u, v] = 2huv. For A = A* =

104
5 Bifurcation results
A;2, V = Ker(L) is two-dimensional and spanned by {cosH,sin kt).
Moreover, Z = span {cos kt, sin kt}, too, and R = R(L) is L2-orthogonat
to Z. As for the corresponding projection P '. Y —* Z, one has that
Ph = {a,k cos ktf bk sin kt) where
1 f
ah — ~ I h(t) cosktdt
n J
a
2ir
bk = ~ / h(t) sin ktdt.
o
If we write v = A cos kt+B sin kt condition (a) leads us to find nontrivial
solutions of the system
2ir
A + - (Acoskt + B sin kt)2h{t) cos kt = 0,
o
It:
B+ - (Acoskt+ Bsin kt)2h(t)sin kt = 0.
o
Tins system is of the form
)=0,/
(5.0)
A + V(A,B) =
B+Q(A,B)--
whcre V, Q are homogcneoiis polynomials of degree 2, whose coefficients
depend on h. From the geometrical point of view, the solutions can be
thought of as the intersections of two conies crossing through the origin
transversatty to each other. So they intersect in another point in the
projective plane. This intersection is not on the "line al infinity', (liul
is, A, B 6 R, provided the system
V(A,B)--
Q(A,B)--
has the trivial solution A = B = 0 only. It is easy to see that tins is
the case for att h 6 Y\Y0, for some thin set Yq (in the sense of Baire).
Then, for a "generic" h (5.9) has a nontrivial solution (A*,B") 6 R2;
as for condition (b), it also holds for alt h up to a thin set. Then we
can conclude that, for att h 6 Y, up to a set of first category in Y, each
\ = k'2, k ~ 1,2,..., is a bifurcation for F given by (5.6); each bifurcating
branch gives rise to a family of 27r-periodic solutions of (5.5).
::}

Appendix
105
Appendix
In this short appendix we want to review some very important
bifurcation results, which require tools other than the Local Inversion Theorem.
We will deal with equations of the type
F(X,u)=^Xu-G(u) = 0, (Al)
where G satisfies
(Gl) G e C(Xt X) and is differentiate at u = 0, with (compact)
derivative A = C"(0),
(G2) G is compact.
It is always understood that G{0) = 0.
Theorem Al (Krasnoselskii, [Krl]) Suppose that (Gl-2) hold and
let A* be an eigenvalue of A with odd (algebraic) multiplicity. Then A*
is a bifitrcut.iou point for F.
Roughly, (lie proof relies on the following arguments. If, supposing the
contrary, A* is not a bifurcation point then there exist a ball D around
u = 0 and s > 0 such that
F(A, u) £ 0 for all X 6 \X - e, X + 5], for all u 6 dD. (A2)
In view of (A2), it makes sense to consider the Lcray-Schandcr
topological degree, d(Fx, D, 0), of Fx := F(A,.), with respect to D and w = 0,
and, by the houtotopy invariancc of the degree, one lias
d(FA._e,Z>,0)=d(Fv+e,Z>,0). (A3)
On the otltor hand, if necessary taking D smaller, the degree of Fx can
he evaluated by linearization: more precisely, if X is not an eigenvalue
of A — C'(0), then one has
i\(Fx,D,Q) = d(\I-A,D,Q) = (-l)k, (A4)
where k denotes tJic sum of the (algebraic) multiplicities (see subsection
0.4) of the eigenvalues fi of A, with p > A.
Let VI, denote the sum of the (algebraic) multiplicities of the
eigenvalues fi of A, with n > A*, and m* that of A*. If necessary taking e
smaller, we can assume that A* is the only eigenvalue of A in the interval
[A — e, A + e}. Then from (A4) we infer
d(PV-K,£,0) = (-l)"\
d(FA._e,Z>,0) = (-l)m+m\
Since nC is odd, it follows that d(Fv_e,Z>,0) / d(Fv+e,Z>,0), in
contradiction with (A3). This proves that A* is a bifurcation point.

5 Bifurcation results
X
(a) Possible bifurcation
diagrams in case (i)
(b) A possible bifurcation
diagram in case (it)
Figure 5.11
Actually, the global nature of the topological degree can be used to
improve Theorem Al as follows.
Theorem A2 (Rabinowitz, [R2]) Suppose that (Gl-2) hold and let
A* be an eigenvalue of A viith odd (algebraic) multiplicity. Then from
X* there branches off a continuum (namely a closed connected set) £ of
nontrivial solutions of F = 0 such that either
(i) £ is unbounded, or
(ii) £ meets another eigenvalue fi^ X* of A.
Theorem A2 applies to a large variety of problems. Among others, wo
mention Sturm-Lionville problems [CrR], existence of positive solutions
of nonlinear eigenvalue problems [AH], existence of vortex rings in mi
ideal fluid [AmiT].
A last result which is worth recalling deals with the case in which G is
a variational operator. To be precise, let us assume that X is a Hilbert
space and that there exists g : X —* R, such that G = V<7- Note that
in such a case (Al) becomes Vg(u) = Au, whose solutions can be found
as critical points of g on the Htlbcrt sphere \\u\\ — pf the parameter X
playing the role of the Lagrange multiplier.
Theorem A3 (Krasnoselski, [Krl]) Suppose G £ Cl(X,X) is a
variational operator and satisfies (G2). Then any eigenvalue of A =
G'(Q) is a bifurcation point for F = 0.
For a proof using Morse Theory, sec [MP]. Improvements can be found
in [Bo] and [Mar].

6
Bifurcation problems
There is a broad variety of problems arising in applications that can
be handled by the Bifurcation Theorem stated hi Section 5.4. In the
present chapter wo will discuss some of thent. We have* tried to choose
problems that are relevant from the physical point of view but that, do
not need too much technicality. Only one of them, the Benard Probloin
discussed in Section 2, is not seJf-contained. Indeed, the analysis of
linearized equations requires some delicate tools that would need much
more space. Ncvcrthless, the relevanee of the problem has driven us to
include it in this chapter, even if we had to be sketchy in several points.
1 The rotating heavy string
Following the formulation of Kolodncr [Ko], we consider a string with
uniform density p and length = 1, hung at the origin of the coordinates
(the 2-axis will he considered to be pointing downwords) in R3. The
points on the string will be parametrized through the arclength s € [0,1]
and denoted by x(,s,t) = (x(s,t),y(8,t),z{s,t)). It is convenient to take
s in such a way that x(l,2) = (0,0,0) is the fixed endpoint of the string.
The equations of the motion are
pxtt^pg+(Txa)a (1.1)
together with
W2 = 1, (1-2)

108
6 Bifurcation problems
where g = (0,0,,9) **> the acceleration of gravity, T denotes the tension
and subscripts denote partial derivatives.
We look for solutions corresponding to a string moving with constant
angular velocity uj > 0. More precisely, we will seek for x.y, z and T of
the form
x = x{s.,t) = r{s) cosujt,
V = y(M) =r{s) sinurt,
z = z(s),
T = T(s)t
In addition we will also require that z'(s) < 0.
There result
xtt = —no2 cosbjt, ytl = —ruj2sinujt,
x, = r cosut, yK = r' sinut
where i denotes d/ds. Substituting in (1.1)-(1.2) one finds easily
(7Y')' + /xj2r = 0,1
(7V)' + A9 = 0, \ (1.3)
<r')2 + (*')2=l. J
The* boundary conditions to be added to (1.3) arc
1-(1)=0,1
3(1)=0, \ (1.4)
T(Q) = 0. J
It is convenient to modify (1.3) to obtain a single equation. First of
all, the second of (1.3) is independent of r, and can be integrated; taking
into account that T(0) = 0, one litis
T(s)z'(s) = -pgs. (1.5)
In particular, .since z'(s) < 0, one has thai, T(s) > 0 for all ,s > 0.
From (1.5) and the last of (1.3) we infer
^ = ^(1-^) = (^.
Setting
.(.,) _2£m
P9
one finds
T2-(pgu)2 = (pgs)2, (1.6)
That is, T2 = p2g2{u2 + s2). Recalling that T > 0 for all s > 0, we find
that (1.6) yields
T = pg%/(u2 + s2).

6.1 The rotating heavy string
109
On the other hand, from the first of (1.3) and letting A = uj2/g > 0 we
deduce
u' + Ar = 0. (1.7)
Differentiating, we find
,, u
0~u + Ar = u + Xpg—f
and finally
y/( U2 + 3*)
As for the boundary conditions, one has u(0) = T(())v'(0)/pg = 0, while
from (1.7) and r(l) = 0 we infer u'(l) = 0.
In conclusion, the problem given in eqns (1.1), (1.2), (1.4), can be
written in the form
u" + X n l\ i\ = 0 for 0 < s < I, I
VV + *2) \ (1.8)
u(0) = 1/(1)=0. J
If (1.8) has a non-trivial solution u(s), corresponding to a certfiin value
A > 0, then we can go back to the solutions of (1.1), (1.2), (1.4) by
-M--u'<s>
1
We remark that from the preceding formulas one finds
z'(i») = -,s'(u2(.s) + s2)~l/2 < 0 for all s < 0,
and hence the solution is consistent with our setting.
In order to employ the bifurcation results of Section 5.4, wc fix the
functional setting, taking X = [a e C2([0,1]) : u(0) = 'u'(l) = 0} and
Y = C([0,1]). It is convenient to introduce some further notation. For
u € X we denote by u(s) the function defined by
u(s)/s for 0 < s < 1,
/(0) for * = 0,
in such a way that we can write
u{s) u{s)
y/\u(S)2 + S2\ = vlH(s)2+l]'
We also let $(u) denote the function
g(»)
*<•>-{#
VWs)2 + l]'
(1.9)

110
(i Bifurcation problems
and define F : R+ x X -» Y by setting
F(A,u) = u" +A$<u). (*)
Plainly, if u € -X", A > 0 and F(X,u) = 0 then u is a solution of (1.8).
Lemma 1.1 $ e C°°(X, Y) and &(u)v is the function
V U2V
S^ (U2+1)V2 " (u2+ 1)3/2-
/n particular $'{0)u is 2/ie function s —► u{s) = v(s)/s.
Proof. It suffices to note that, $ is obtained by composition between the
linear continuous map u —► it (from X to Y) and the C°° map
w-nu/(w2+ 1)1/2
(from y to itsolf).
From Leiiuna IJ we infer that the linearized problem v"—A$'{0)t; = 0
is given by
» i (1.10)
,,(0) =//(1)= O.J
Tt ia well known that the equation v" + Xv/s = 0 is related with the
BcskcI equations.
In fact, setting
t = 2y/Xs, v(s) = tw(t), (1.11)
one readily has
„_ 4A2 /d2w Idtu _ w\
v ~~r\~de + tdt t*j-
Then v" + Av/s = 0 hecomcK
^+«£ + («._1)to_0, (1,2)
which is the /tth Bessel equation t2z" + tz' + (t2 — k2)z = 0, with k = 1.
In the following JK denotes the reth Bessel Function of the first kind (see
[Sa] Cap.III, §6). For example, J\ is defined by
Recall that there results J'K(t) = JK-i(t) - (K/t)JK(t),n > 1 (see [Sa],
p.175). In particular, one has
J[{t) = Mt) - jJi{t). (1.13)

6.1 The rotating heavy string
111
Lemma 1,2 Tfie eigenvalues of (1.10) are given by
K = K/2)2,
where on denotes the, nth zero of the Bessel Function Jq. If A = A„
then the corresponding eigenfunctions of (1-10) are of the form u(.s) =
cipn(s), ceR, where
<fin(s) = 2-v/A^s ■ Ji(2^/Xns).
Proof. Through tlie change of variable (1.11) the equation i/' + Xv/s = 0
translates into the Bessel equation (1.12). Prom the theory of the Bessel
equations [Sa] it follows that the general solution vi(t) of (1.12) sucli
that tw(t) —► 0 as t —► 0 is given by
w(t) = cJi(t), c€ R.
Hence the general solution of
1,(0) = 0, J
is given by
,,(,^) = 2c^/Xs-Ji (2^/Xs).
From
v'(s) = Cy/- . J}(2s/Xs) + 2r.XJ'i(2s/Xs)
and using (1-13) we get that
v'{s) = 2c\Ja(2-y\s).
Then the boundary condiUon v'{~\) = 0 gives risie to
2cXJ0(2x/X) = 0.
If 2^/A ^ an then a = 0 and ■;; = 0. Hence (1.10) has non-trivial solutions
if and"only if A„ = (crn/2)2 and the lemma follows.
We are now in position to state the following.
Theorem 1,3 For n = 1,2,..., let an denote the -n-th zero of the
Bessel function Jo and let A,, = ((7,,/2)2. Than any X„ (.s o bifurcation
point for F = 0, where F is given by (*).
Proof We are going to apply Theorem 5.4.1 (jointly with Remark
5.4.3(i)) to F(X,u) = u" -r A$(u) (where $(u) is given by (1.9)) with
A* = A„. According to Lemma 1.1, F 6 C'(R x X,Y). Keeping the
notation of the preceding chapter, we set L = Fu(Xn70)7 V = Ker(£)

112
6 Bifurcation problems
and R = R(L). By Lemma 1.2 it follows that V is one-dimensional and
spanned by tpn.
As for R, we can argue as follows. The equation v" + Xv = h (!i 6
Y) can be written in the form v = XG(v) — G(h)7 where G denotes
the Green operator of —d2/ds2 with the boundary conditions v(0) =
v'(l) = 0. Then the Fredholm Alternative Theorem 0.1 applies and R
has codimension 1 in Y.
The preceding arguments show that Assumption (I) of Theorem 5.4.1
holds true.
Lastly, a = {^^^{An.O)^]) = Q tp\ > 0 and Theorem 5.4.1
applies yielding a branch of nontrivial solutions of (1.8) bifurcating from
(A„,0).
Remarks 1,4
(i) As remarked before, nontrivial solutions of F = 0 correspond to
nontrivial solutions of (1.8). Moreover, since F(\,u) = F{X,—u),
solutions arise in paira (ut—u). One could also readily show that the
bifurcation is supercritical. Then, recalling also the discussion iu Remark
5.4.3 (iv), we can say that for X > A„, A near A,,,' there exists a
family (w„(A), — un(X)) of pairs of nontrivial .solutions o/(1.8), continuously
depending on A, such that ||u„{A)||^^ —► 0 as X I Xn.
(ii) The preceding result can be improved: indeed, uriing shooting
methods, it has been shown in [Ko] that for any Xn < A < A„+i the
problem (1.8) has exactly n pairs of nontrimal solutions ±u\ ± 112,. ■ ■ ,±u„
such that uj has j isolated zeros in [0, l].
2 The Bernard problem
Fluid dynamics is a typical area, of application for bifurcation theory. In
this section we will discuss a very classical question: the Benard Problem
dealing with convective motions in a heated fluid. We shall restrict our
treatment to the case of planar stationary flows; such a problem, even
though much simpler than the general one, nevertheless is still physically
interesting and quite nontrivial from the mathematical point, of view.
For more complete results on the Benard Problem, see, for example,
(R3](Ve|.
Consider a viscous fluid conducting heat, situated between two
horizontal walls, kept at a constant temperature, and suppose the lower
wall is warmer than the higher wall. If the difference in temperature in
small, the fluid remains at rest and the transmission of heat occurs only

6.2 The Benard problem 113
by conduction; but if the temperature gradient passes a certain critical
threshold, convective motions are triggered. This is indeed what was
experimentally observed by Benard around 1900.
The mathematical model we are going to consider (due to Boussinesq)
is based on the following physical assumptions:
(a) the fluid has constant density, constant, heat conduction and
satisfies the Navier-Stokes equations;
(b) the only forces acting on the fluid are those of buoyancy (vertically
directed and depending linearly on temperature).
Let us remark that the variations of density due to temperature are
important for the generated buoyancy forces (condition (b)), but they
are neglected in the dynamics of the fluid (condition (a)).
With u denoting the velocity of the fluid, p the pressure, 8 the
difference between the temperature and the linear function interpolating the
values on the bounding walls, one finds the following system:
u( + (u ■ V)u + Vp - Au - ROe = 0, "]
V-u^0, \ (2,1)
6t + P(u ■ V)0 - A0 - e ■ u = 0. J
Here
- e is a unit vector vertically directed,
- R (Rayleigh number) is a positive constant, and is proportional
to the gradient of the temperature,
- P (Prandtl number) is another positive constant.
The form of system (2.1), containing two constants only, has been
obtained by choosing suitable units. We can also suppose that the
bounding walls are represented by planes at level —1 mid +1, respectively. On
these planes both the velocity field u and 0 vanish.
For the sake of simplicity, we will be interested in planar stationary
solntions only. Let us introduce cartesian coordinates ('■'.,y) and set
u = (v,w). Since u is a planar, solenoidal field, V ■ u = 0, we can
introduce a stream potential tp by setting
_ dip _ _<fy
dy' dx
Substituting in (2.1) one readily finds the system
-A0+^ + PB(^, 0)=0, J
where A is a quadratic function in the derivatives of tp up to the third
order, and B is bilinear in the first derivatives of ip and 0. The region

114
6 Bifurcation problems
where the system (2.2) is considered is the strip
S= {(x,y) eR2 : -co < x < oo,-l < y < 1}.
The boundary conditions become
ip(x, 1) = $(x,-l) = 0, "I
^(x,l) = i>v(x, -1)=0, 1 (2.3)
0(x,l) = 0<:e,-1) = 0. J
We remark that the physical conditions of the problem demand that t/'
be constant on dS, but we further require that the flow through aiiy
vertical section be zero: this implies that tp has the same constant value
on dS (obviously, such a value can be taken to be 0).
We look for nontrivial solutions of (2.2)-(2.3), which are (2ir/a)-
periodic in the x variable, where a is a positive, fixed value, The search
for periodic solutions is suggested by the fact that, experimentally, eon-
vective motions have - in many cases - a roll pattern, repeated
periodically.
Let us introduce a further restriction to the class of admissible
functions; more precisely, let us assume that
ip is an even function of x and 8 is an odd fimictlon of x.
It is convenient to introduce the following function spaces:
H* is the space of functions ip defined in S, (27r/a)-periodic in x, even
in xt having fourth-order derivatives in L2(S) and such that
1P(X,±1) =1py(X,±l) = 0,
H2 is the space of functions 0 defined in S, (2ir/a)-periodic in x,
odd in x, having second-order derivatives in L2{S) and such that,
9(x,±l) = 0.
L'a the space of functions defined in S, (2tv/(i)-periodic in x, even in
x, and squaxe-integrable,
L'a the space of functions defined in S, (27r/a)-periodic in xt odd in x,
and squaxe-integrable.
Let X = Hi x H% and Y = L'a x L"a. An analysis of the terms
contained in A(ip) andB(^,0) shows that if (¢,0) eH^xHl then A( V')
and B(tp,8) belong to H^. If we take into account that the operator
is elliptic, it follows by standard regularity theory that the solutions
(¢, 8) e X of (2.2) are in fact smooth.

6.2 The Binard problem
115
In addition, the same analysis shows that the map
GM)-(.4(0), B(iM))
is of class C°° as a map from X to Y. Then the left-hand side of (2.2)
defines, for all R, a map Fr : H* x H% —► L'a x L"a which is of class C°°.
Taking the Rayleigh number R as bifurcation parameter, we are led to
the equation F(R,ip,0) := FR(ip,0) = 0. Plainly F(#,0,0) = 0 for all
In order to apply the bifurcation results of Section 5.4 we have to
study the system obtained by linearization from (2.2)-(2.3), namely
- A0 +
*T
dx
dj>_ _
dx
= 0,
(2.4)
¢(1, ±1) = V„(:r, ±1) <=8{x,±l\) =0.
We want U> show that (2.4) has a simple eigenvalue R such that
Theorem 5.4.1 applies to F.
After the substitution ■</> = y/R ■ ¢, (2.4) becomes
■^■fc-
0,
■ AS + yfl ■ ^ = 0,
(2.5)
V>(a:,±l) =0y(x,±l) = <?(:£, ±11) =0.
The left-hand side of (2.5) contains an operator La which can be written
in matricial form as follows:
r a2 -VR-i
+ y/R-
0
o
-A+y/R-B
The operator A, as an operator on the Hilbert space L'a x L" with
domain of definition H* x H%, is self-adjoint and invertible. It is easy to
verify that the operator B is symmetric; since, for real A big enough, the
operator Lr + A/ (still with domain H* x H^) is invertible, we deduce
that Ln is self adjoint *. Therefore, Range(L) is the subspace orthogonal
to Kor(L).
' Here we use the following abstract result. Let H be a Hilbert space and
let A, B two linear maps defined on a dense domain T> C //. Suppose that
(i) A is a self-adjoint; (ii) B is continuous and symmetric, in the sense that
{Bu\v) = {u\Bv) for all u, v E T>, and (iii) T= XI+ A + B is invertible from
1? to // for some real A. Then A + B is self-adjoint.

116
6 Bifurcation problems
In order to verify the assumptions of Theorems 5.4.1 it remains to
prove that, for suitable values of R, K&t{Lr) is one-dimensional.
In view of the properties of the spaces H*, H%, we can set
<j>(x, y) = 4>0(y) + Efc>i cos{fca:c) <j>k{v),
8(x,y) = Efc>i s\n(kax)8k(y).
Substituting into (2.5), one readily obtains the following system of
infinitely many equations:
Mla4>k -y/RkaOk=Q, -J
A/fco0fc - ■*/# - fca & = 0, > (2.6)fc
<j>k(±l)=4>k(±l) = 9k(±l)=0,\
where Mka - -d2/dy2 + k2a2, and k = 1,2,....
Let us consider, in general, the system
Mk2a<j>-\0 = 0, "J
Mka9-\4>=0, \ <2.7)*
4(±l)=t'(±l) = 8(±l)=0,\
for k = 1,2,... . We shall show that (2.7)¾ has a smallest, simple,
eigenvalue. The proof of thin fact requires some technicality and we will
limit ourselves to giving the outline of the arguments, only. First one
proves a lemma.
Lemma 2,1 The operator M^a with the boundary conditions
<t>(±l)=<t>'(±l)=Q
is invertible with compact positive- inverse (that is maps positive
functions into positive ones).
A proof of Lemma 2.1 can be found (in a more general form) in [Ve].
In addition, an elementary argument yields the following.
Lemma 2,2 The operator Mka with the boundary conditions
0(±l) =0
is invertible with compaet positive inverse.
Let G ~ Gka be the inverse of M^a (with the boundary condition
8{±\) = 0) and let H = Hka be the inverse of Af£a (with the boundary
conditions <p{±l) = <£'{±1) = 0). Then (2.7)fc is equivalent to
6 = \2GH(8).
Plainly, by regularity, we can work in C{\— 1, l]). Since Gand H are
compact and positive, an important result by Krasnoselski {[Kr2]), Chap.2,

6.2 The Benard problem
117
Theorems 2.5 and 2.10) yields the existence of a unique eigenvalue with
positive eigenfunction 6 > 0, and this eigenvalue is simple. The only
assumption that remains to be verified is that GH is uo-p°sitive, that
is, that there exist uo > 0, «o ^ 0, ^1^ a constant a > 0, such that
GHuq > ana.
For this, it suffices to set u0(y) = (y + l)2(y — 1)2 (or else any function
positive in (0,1) with zero derivative at the endpoints): indeed, one shown
readily that for all continuous nonnegative ?;, v ^ 0, the function Gv has
positive (respectively, negative) first derivative at the point —1 (rcsp. 1).
In conclusion we can state the following result.
Lemma 2.3 For each k = 1,2,..., pTvblem (2.7)a.- has a smallest
positive eigenvalue X(ka), which is simple.
We remark that, if (2.7) has an eigenvalue A with eigenfunction {¢, 0),
then it lias also an eigenvalue —A, with eigenfunction {<$,—$).
In addition, we need the following estimate.
Lemma 2.4 There exists a constant C > 0 such that \(ku) > Ck2n?.
Proof. Let ns act
1
-I
and '
1
^(A ») = \ /½. + 2*V# + *W + b\ + k?aW)<iy.
-1
System (2.7)/,- can be .seen as the Iiiilw Lagrange equation
grad.F(&0) = A grades, 0)
where A plays the role of the Lagrange multiplier. An a consequence,
one can readily see that the value fi(ka) = 1/A(fca) has the variational
characterization fi(ka) = msx{Q{4>, 6) : ^(¢, 8) = ]}. Hence one has
,6(<M)
fi(ka) = sup
^(¢,9)1
Since, plainly,
ew,0)<^yV2 + 02)d!,

118
G Bifurcation, problems
and
l
-1
then one has
cum . 5/<*2 + 92>*
Q{v,») I -i
FI&B) ~ k'2a2 J
From the Poincare inequality we deduce that
-r < constant,,
/(2^ + 92)^
-1
and Urns /i(ka) < C/k2a2t proving tlie lemma.
We arc now in position to go back to problem (2.5). Let Ra be snch
that
JR.-**.
a
Let {4>{,Q*) denote an eigenfunction of (2.7)i corresponding to A{a).
From Lemma 2.3 it follows that \/RZ is ;iw eigenvalue of (2.5) with
eigenfunction
<P*[x*v) = ^i (y) cos ax, 1
^k?/) = ^{y)sin«x. J
Suppose that
A(a) A(fca) , . , „ ,„ v
--^ < --^-, for all integers k > 2. (2.8)
a fca
Since X{ka)/ka is the smallest positive eigenvalue of (2.6)^, it is clear
that X(a)/a is different from any eigenvalue of (2.6')fc corresponding to
an integer k > 2. If {¢, 0) e Ker(Lfi) then the Fourier components <pk,Qk
of <£, 0 satisfy
MkaOk-\(a)k<j>k = 0. ]
But the preceding arguments show that \{a)k is not an eigenvalue of
(2.7)t for k > 2. Hence ¢1 = q^J, «1 = 9J and <j>k = 6k = 0 for all
fc > 2. This proves that Ker(Lft) is one-dimensional and spanned by
(01, #1). In this case Theorem 5.4.1 yields the existence of a branch of
solutions of (2.2-3) bifurcating from R= Ra, ¢ = 0, 6 = 0.

G.3 Small oscillations for 2nd-order dynamical systems 119
If (2.8) does not hold then from Lemma 2.4, there exists an integer
k* such that
X(k*a) X(ja)
~ '- < -4-1 for all j > k'. (2.9
k*a ja ■ '
Setting a* = fc*a, from (2.9) we get that
_i__i < _i L for tali k > 2
(7* A'fl*
wliich is the same as (2.8) with a replaced by a". It follows that there
exists a bifurcation of solutions, with period It;/a"', from R", \JR* =
\{a")/a". Since Inja" is a sub-multiple of 2iv/at these solutions are
(27T/a)-periodic, too. In conclusion, we can state tlic following.
Theorem 2.5 For any fixed r > 0, there exists R^- > 0 suck that the
system (2.2)-(2.3) possesses a family of solutions (^n,0n), r-periodic
with respect to the. x variable, depending continuously on the Raylcigh
number R for R in a neighbourhood of RT, Moreover, as R —* Rr,
(</>«,*«) ^<0>0) inH\ xHl
3 Small oscillations for second-order dynamical systems
In some cases the problem to be studied inherits a specific symmetry
and the eigenvalues of tlie linearized equation are not simple. However,
it can still be possible to apply the bifurcation theorems of Section 5.4,
working in suitable invariant subspaces. In this and in the next section
we will discuss two such problems.
Here we will discuss a result due to Hopf [Ho] dealing with .second-
order autonomous systems of the type
^ = /(-), (S,
where u e R" and / : R" -► R".
If /(0) = 0, (S) has the trivial solution u = 0 and we look for periodic
solutions of (S) "near" u = 0.
To state in a more precise way the problem we want to address, it is
convenient to put in evidence in (S) the dependence on the period. For
this, let us perform a change of scale of time letting s = u>£. Then (S)
becomes
^ = /w. (¾)
If, for some u> > 0, u = u{s) is a 27r-periodic solution of (Sw), then
u(t) = u(ut)

120
6 Bifurcation problems
is a T-periodic solution of (S), with T = 27r/u>.
The problem of the existence of small oscillations near u = 0 can now
be made more precise.
Let u>* > 0 be a value with the property that there exist
(a) a sequence ujn —► a/,
(b) a sequence un of 27r-periodic solutions of (SWn) such that ||w„ ||oo 1
0.
Defining un(t) = un(wnt), we find that u„ is a sequence of {27r/t±;„)-
periodic solutions of (S), witlTjjSnUoo | 0.
In conclusion, we can give the following definition.
Definition 3.1 If oj* > 0 is such that (a) and (b) hold true, then we
will say that (S) possesses small oscillations in correspondence with the
frequency u>* .
Problem (Sw) fits into the framework of bifurcation theory, the
frequency u> playing the role of the bifurcation parameter. Roughly, we can
explain this claim by noticing that the solutions (oj, u) of
F(u,u):=u^-J(u)=a, (3.1)
(u> > 0, u in a suitable space of smooth 27r-pcriodic solutions'; sec later
on) correspond to solutions of (Sw). There results F{u>,0) = 0 and,
moreover, if u>* is a bifurcation point of F = 0, then, hy definition, there
exists a sequence (LJn,un) —* (oj*,0) such that u„ ^ Oand F(ojntun) = 0.
This means that (tjjntun) satisfy (a)-(b). Since tlie converse is also true,
we can conclude that (S) possesses small oscillations in correspondence
with the frequency u>* if and only if u>* is a bifurcation point of (3-1).
We will suppose
<f0) /eC2{R",R"), /(0) = 0,
(fl) A := /'(0) is non-singular"and has r,r > 1, negative real
eigenvalues
—U>?,-U>2,. . .,— b%,
with, say, 0 < u>i < u^ < ... < u>r.
Theorem 3.2 Suppose f satisfies (f0-l). Let Wj be such that
(f2) — a)? is simple (i.e. has algebraic multiplicity = 1),
(f3) uj8/ujj is not an integer, for all s ^ j.
Then (S) possesses small oscillations in correspondence with the
frequency Uj.

6.3 Small oscillations for 2nd-order dynamical systems 121
Proof. Let C^ denote the space of 27r-periodic functions of class
Cfc<R, R"), C27r = C$K, and let F : R x C^ -+ C2jr be defined by
F(W,u) = «ajg?-/(«). (3.2)
According to the preceding arguments, we want to show that u>j is a
bifurcation for F. This will be done using Theorem 5.4.1 with A* = ljj.
For this purpose, we have to study the derivative L = Fn{ujjt 0),
i d2fi
L : u —* u>,- —— — Au.
3 ds2
It is easy to see that L has a two-dimensional kernel. For example, if
n = 1 and A = —u/j then Ker(L) is spanned by eos£ and sin2. To
overcome this difficulty, it is convenient to restrict F to the subspaces
X = {x 6 C|, : u(-t) = u(t)},
Y={veC2l,: v(-t) = v(t)}.
The restriction F\x will still be denoted by F. Let us note that F{ujt.)
maps X into Y because (S) is an autonomous system and does not
contain u'. Formally the Fourier expansion of any u € Xt v e Y is
U = lio + U\ COS 2 + U2COs2£ + ... = YjlifcCOS kt,
k.>(l
V = u0 + u, COS t + V2 cos 2t+ ... = 2_j Vk cos ^'» '
k>0
with ufc,?)fc e R"-
For fee N wesetXfc = -k2Lj]l-A and M =Mi = -u$I-A. By
assumption Ker{.M) is one-dimensional; we let £ e R" denote a vector
spanning Ker{.M) and II the corresponding spectral projector, namely
the projector of R",
x —► £{77, x) = IIx,
where r/ is the functional that is zero on R{M) and such that (7/,£) = I.
We need the following
Lemma 3.3
(i) Ker(L) = span{£eos£},
(ii) Range(L) = {v e Y : Uvi = 0} where vi = £ j v(t)costdt.
Proof. Consider the equation Lu = u, with u e. X and v G.Y. If Uk,vk
denote the Fourier coefficients of it, v, than Lu = v becomes
-k2u]uk - Auk = Mk{uk) = vk, (k e N). (3.3)fc

122 6 Bifurcation problems
If Lu = v then Uk,Vk satisfy (3.3)^; conversely, if u e X and v e Y and
Uk,Vk satisfy (3.3)¾ then Lu = v.
We claim that (3.3) has a unique solution for all k ^ 1. In fact, if
fc = 0 then Mo is nothing but A, which is invertible; if k ^ 1 and k ^ 0,
then from (f2) it follows that —k2LJ2 is not an eigenvalue of A and hence
Mk{k 7^ 0,1) is stiU invertible. In conclusion (3.3)^/1 has a unique
solution given by
uk=Ml\vk) (fc^l).
For k = 1 (3-3) becomes
JjUi — Aui = Mui = v\,
which is solvable whenever Ilui = 0.
After these preliminares we can now prove (i) and (ii).
First, if v = 0 then the solutions of (3.3) are given by
uk = 0 if k ^ 1, 1
ui = a£ (a e R). j
Hence one has Ker(L) = span{£ cost}. This proves (i).
To prove (n), we first let v = Lu, for some u € X. The preceding
arguments imply that Ilui = 0 and hence {v e Y : Uvi = 0} 3 R =
Range(L). Conversely, let v e Y be such that Ilui = 0. Then, formally,
Lu = v has a solution of the form
u{t) = A~lvq + ui cos t + v(t) (3.4)
where M{u\) = Hi and
v(t) = 7]A4fc x(vk)coskt.
k>i
Wc claim that v is of class C2. To sec this, we first notice that
where
Aft(x) = 0(^) for all x<=R" (3.5)
Then it follows that
v(t) = w(t) + z(t)
with

6.4 Water waves
123
and
z(t) = y^A/fc(ufc)cosfcf.
k>1
Since w(t) is nothing but the iterated integral of the continuous function
-% /,Vk cos ^)
then w € C2.
As for 2{(), we remark that, v being continuous, (3.5) yields
N-k(vk) = o(±).
Hence z is also of class C2. In conclusion, u given by (3.4) is C2 and is
a solution of Lu = v. This completes the proof of (ii).
Proof of Theorem 3.2 completed. From Lemma 3.3 it follows that
assumption (I) of Theorem 5.4.1. holds true. As for hypothesis (4.3), it
suffices to notice that here
and hence*{1.2) is satisfied too. Then Theorem 5.4.1 applies and the
result follows.
4 Water waves
In this section we will discuss a bifurcation result dealing with the
classical problem of the existence of periodic waves of a heavy, 'niviscid
fluid with infinite depth. One of the first results on this matter is due to
Levi-Civita [LC], and Krasovski (sec [Be] for an outline of the Krasovski
result. Periodic water waves with infinite depth have been studied by
Nekrasov [Ne]). For other problems concerning waves in fluid dynamics,
see, for example [T].
Let us consider a two-dimensional fluid with uniform density p = 1,
say and infinite depth, having a free bounded surface C, We will take the
x-axis in the horizontal direction and the y-axis upward directed. We
will look for waves moving with constant velocity c> 0. It is convenient
to take a reference frame which is in translatlonal motion with velocity
c. Then the profile of the wave becomef? independent of time and the
velocity field q = (u,v) will be a function of x and y only. Moreover,
since the fluid is assumed to be at rest at great depth, it will move with
velocity —c with respect to the new reference frame, and this leads to
q -f (-c,0) as y -»■ -oo, uniformly with respect to x. (4.1)

124
6 Bifurcation problems
Let us also assume that C has equation y = 0{x).
The fluid is assumed to be irrotational and incompressible so that
there exist two functions <p(x, y) (velocity potential) and ip{x, y) (stream
function) such that
\7<p = (u,v), Vi/) = (-v,u).
Such <p and ip are conjugate harmonic functions.
Let ns introduce the following notation:
z = x + iy,
w = u — if,
/ = <p+iip,
where i is the imaginary unit (i2 = —1). There result
™*=u+iu and \w2\ = u2 +v2 = \q\'2.
Moreover, the profile C of the wave is a streamline ip =const. and we
will assume that
i/>(x,y) = Oforall(x,y)e C.
The Bernoulli law yields (recall that p = 1)
-\w\2+gy + p = mast, (4.2)
where g is the gravitational constant and p is the pressure. Since p
is also constant on the upper surface C (actually p is the atmospheric
pressure), (4.2) implies
-\w\2 + gy = const, on C. (4.3)
We will look for periodic waves, that is for velocity fields that arc
periodic with roypect to x with period h € R; in terms of w this means
wc are looking for w satisfying
w(z + h) = w(z) for all z e € (4.4)
for some (unknown) real h,
As a consequence the holomorphic function / has derivative d//dz =
w which is /i-periodic. Hence from (4.4) it follows that
f(z + h)- f(z) = const. (4.5)
To find the value of this constant, it suffices to put z = iy and let
y —► —oo in (4.5), using the integral representation formula of <p + iip.
So there results
l(z + h)-l(z) = ch. (4.6)
The above problem is a free boundary one, in the sense that the
region filled by the fluid A = {y < 0{x)} is unknown. To overcome this

6.4 Water waves
125
difficulty one looks for a (holomorphic) transformation of coordinates
so that such a region becomes a fixed domain. To this purpose, it is
convenient to use the same f{z). This choice is suitable for a change of
coordinates provided |/'{z)| = \w\ > const. > 0 for all z t A. If this is
the case, the map z —► / is globally invertible with inverse denoted by
¢. The region A is transformed into the region {/ = <p + lip : ip < 0}.
(See Figure 6.1.)
The function w is holomorphic with respect to z. and therefore with
respect to /. As a function of /, w is periodic wlfli period ch. To see
this fact, we start from (4.6), that Is f(z + h) = f(z) + ch, and apply $
to find
$</ + c/t) = z+ h = ¢{/) + h.
This implies $'{/ + ch) = $'{/). Since $' = dz/d/ = 1/w, we conclude
that
w(f + ch) = w(f)
as claimed.
It remains to translate the equation (4.3). To do that, it suffices to
note that one has
-\w\2 + gy = const, on ip = 0.
Differentiating with reypect to <p we find
1
21
Note that dy/d<p is the imaginary part of dz/d(p. Since there results
dz _ dz _ 1 _ 1 u + in
dip d/ d/ /dz it — in u2 + v2'
it follows that
>i2+»s
= 0 on i)> = 0.
dtp \w\2

12G
6 Bifurcation problems
and (4.3) becomes
H3"5-H +9v = 0on 1/) = 0. (4.7)
A last change of variables will allow us to work with functions with fixed
period. If we set
w becomes 27r-periodic in £ and (4.7) transforms into
M3^M + ?u = 0 on t/ = 0. (4.8)
In conclusion we can formulate our problem in the following way:
Find a junction w = u — in of the variable £ + \T}, such that
(i) w is holomorphic in the half-plane £) — {£ + iT;: 7} < 0},
(ii) w is 2iv-periodic with respect to £,
(iii) w satisfies (4.8) on 7} = 0, and
(iv) w —> c aST}—> — oo, uniformly in £.
In other words we will look for functions it, v defined in £), such that
(i') u and v are harmonic conjugate functions,
(ii') u and v are 2iv-periodic with respect to £,
(iii') u andv satisfy (4.8) on t] = 0,
(iv') ost/-> — oo, u-tc and u-*0.
To translate the preceding conditions into a functional equation, some
preliminaries are in order.
Let E denote the class of real-valued functions u defined on the half-
plane £) such that (i) u is harmonic in £), (ii) u is 27r-periodic with
respect, to £, (iii) the function £ —* u{£,t/) is bounded in L2(Q,2iv) for
rj e (-00,0).
Lemma 4.1 Any u e E has the form
u(A, v) = t + 5Z(afc ^ fc£ + bk sin k0ekr>,
k>\
where 7 e R and YL(ak + ^k) < °°-
Proof For fixed 77 < 0, the function u{.,t}) has Fourier expansion
u(£, »7) = *(*7) + S<<**<T7) ™s fc£ + /3fc<7/) sin fc£).

6.4 Water waves
127
Since u is harmonic,
e"(i) = o, I
0kM = #0k(v), J
whence
e(r)) = mr) + e0, i
"0k(v)=bkekT, + bke-k^ J
Since 14(.,77) *s bounded in L2(Q,2iv) as?j-> —00, one has
m = 0, afc = 0, 6fc = 0,
as well as £q = 7 and
£(a£+^)efc*<L.
From this last inequality it follows readily that £(aj[ + b2.) < L, as
claimed.
Remark 4.2 From Lemma 4.1 we infer that any it e E has, in addition
to (5)-(55), (iii), the following properties:
(iv) as rj —* -00, u(£,tj) —* 7, uniformly in £,
(v) u has a trace on 7} = 0, denoted by u# = u#(£), which is 2/ifi
L2-limit ofu(,,r}) as rj —* 0.
It is worth noticing that 7 is nothing but the mean value of u#. In the
sequel the mean value of a function z e L2(Q,2ir) will be denoted by [z\.
Let us introduce the following function space:
H — {uq € L2(R) : uq is 27r-pciiodic and [uq] ~ 0}.
With this notation, for any u e E one has that u# = 7 + uo, with
uo e # ■
The solution it we arTi" seeking is therefore a function in E of the form
«(£, V) = c + 53K cos fc£ + bk sin fc^e*" (4-9)
fc>i
and its trace on 77 = 0 is given by it# = c + uo, with uq e H.
The other unknown v is such that —v is the harmonic conjugate of it,
and v —* 0 as rj —* -co, uniformly on £. Hence, if it is given by (4.9)
then v will have the form
"(£, V) = ^2(~bk cos fc£ + ak sin fcOefc*.
The trace of such a v has mean value equal to zero and therefore belongs
to H. It is convenient to introduce a map K : H —► H in the following

128
6 Bifurcation problems
way: if z € H is of the form £fc>](pfcCosfc£ + (jfcsinfc£) then Kz is given
by'
Kz = ^(-qk cos fc£ + pfc sin k£). (4.10)
fc>i
Plainly, K is a linear continuous map from H into itself (indeed, an
isometry).
According to the preceding remarks, the solutions of our problem will
be u and v given by
u{i,v) = e+ y^(afcCQsfcg + bksmk£)ekJ),
k>\
v(£,tj) = V^(-&fccosfc£ + afcsinfc^)efcTi.
fc>i
The trace of it on t; = 0 will be of the form e + uq, with uo € H, while
that of v will be vq = K{uq). With this notation there results
\w\2 = (e + uo)2 + v2 = (c + u0)2 + (Ku0)2 on rj = 0. (4.11)
To write equation (4.8) in terms of uq and 1¾ we shall introduce some
more notation: for z e L2, we indicate with Hz the primitive of z such
that [Tlz] = 0. '
In other words, if
z(0 = 2(Pfcc"osfc£ + <?fcsinfc£)
fc>i
and, by (4.10),
UKz(£) = ^(-?± cos k£-^shik A . (4.12)
fc>i
After these preliminaries, let us integrate (4.8) with respect to £; one
finds
tIH4 + Tr-n^" = const, on 7j = 0. (4.13)
4 27T
Taking into account (4.11), and dividing by c3, we find that (4.13)
becomes
4^ ((e + u0)2 + (Kuq)2)2 + XHKu0 = const., A = ^.
Taking the mean value of both sides we find that the constant In the
right-hand side is nothing but (l/4c3)[((c + uq)2 + (Kuq)7)12]. In
conclusion, letting F ; H —> Ht
F(X,uo) = ^((c+uofHKuoff-^lttc+uof+iKuofW+XTlKuQ,

6.4 Water waves
129
we shall solve F(X,uq) = 0.
Plainly, F is smooth <C°°) and F<A,0) = 0. The derivative FUo(X,0)
is the map
z -» z + XUKz.
Taking into account (4.12) we get that
2=£fc>i (pfccosfc^ + gfcsinfc^) e Ker{FUo{A,0))
whenever
* \ (*= 1,2,...).
As a consequence, Ker{FUo{A,0)) ^ {0} if and only if A = 1,2, —
Moreover, for A = k e N, such a kernel is two-dimensional and spanned
by (cos k£, sin k£). In view of this fact, we can proceed as in §3 and
consider the subspace X of H consisting of those z that are even. The
restriction F* = F\x maps X into itself, because UKz is an even
function provided z is. Let LK = F* (k, 0). According to the preceding
discussion it is easy to check that, for each n = 1,2,..., Ker(L^) is one-
dimensional and K(LK) has codimension 1. Moreover the second nrixed
partial derivative F*u^x(k, 0) is the map z —* UKz and hence assumption
(4.3) of Theorem 5.4.1 holds true as well.
In conclusion, we can state the following.
Theorem 4.3 For all n = 1,2,.,. there is a branch of nontrivial
solutions of F(X,Uq) = 0 bifurcating from (n, 0). Moreover, each nontrivial
solution (A, ito) gives rise to a solution w = u — in of the problem (4.8').
Theorem 4.3 can be completed by some remarks. We shall be sketchy
here, leaving the details to the reader.
According to Remark 5.4.3(iv) one can verify that the branch
bifurcating from (k, 0), parametrized with respect to ^, has equation"
uotK(n,0 = hcosk£ + uk(h,£),
XK(H) = n + yK(n),
where wK(p,.) is even, wK = 0{fj.2) as fj. —* 0, uniformly with respect to
£, and 7* is even. Of course, the corresponding VQiK has the form
where Xk{^, ■) is odd and \k = 0{p2) as ^ —► 0.
It can be also checked, that for n = 1, there results
7i(M) = -^+0(M4)-

130
6 Bifurcation problems
Moreover, let us point out that the solutions on the branch TK
emanating from (k, 0) correspond to the same wave profile as that of the
solutions on the first branch IV To make this claim more precise,
consider the solutions bifurcating from (1,0),
"o.i(/*»0 = /* cos £ + wi (/*,£)»
Ai(/*)=l + 7i(/i),
that satisfy
F(Ai(/i),u0,i) = 0.
Letting
«o,*(/*,0 = «o.i(/*»*0»
K(n) = K + K7l{^),
one can immediately verify that there results
Since TK is the unique branch of nontrivial solutions bifurcating from
(k, 0), it follows that
untK — uo,« = voti(p,K£),
XK(H) =XK(fj) = n+ivyi(fi).
Hence the solutions on TK are given by
"0^(^.0= /*CO8*€ + Wl(/i,K0,
K(n) = «+ «7i(/0»
and therefore they give rise to tlie same wave profile as that
corresponding to the solutions on IV
5 Periodic solutions of a semilinear hyperbolic equation
This section deals with the existence of periodic solutions of a class of
semilinear wave equations. We anticipate that the problem we are going
to discuss cannot be handled directly by the abstract results of Section
5.4; however, it fits into that general framework and will be solved by
means of a suitable Lyapunov-Schmidt reduction.
Let us consider the strip 5 = {{x, t) e R2 : x e [0,w],—00 < t < +00}
and a function /:SxR-»R, such that
/<x,* + 27r,u) = f(x,t,u) for all (x,t,u) eSxR. (5.1)
We will look for solutions u = u(x,t), (x,t) € S, of the following

6.5 Periodic solutions of a semitinear hyperbolic equation 131
semilinear hyperbolic problem
Au + e f(x,t,u) = 0 for all (x,t) e S, "]
u(Q, t) = u(tv, t) = Q for all * g R I (5.2)
u(x, t + 2iv) = u(x, t) for all (x, t) e S, J
where e is a real parameter and A denotes the wave operator:
>_&___&_
d\? dx2 ■
For e = 0 (5.2) has the trivial solution u = 0 and our goal will be to
show that, under suitable assumptions on the nonlinear term /, (5.2)
has a unique nontrivial solution for all e ^ 0 sufficiently small.
A result of this sort was first proved by P. H. Rabinowitz [Rl].
Subsequently, L. De Simon h G. Torelli [DST] gave a new proof of this
existence result. The arguments of [DST] not only arc very simple, but
also point out clearly the bifurcation framework that is beyond the
problem. We shall expound only some of the material contained in [DST],
referring to that paper for more details as well as for additional results.
Let H denote the space of real L2 functions u : S —* R^x-periodic
in time. With respect to the usual scalar product
(u\v) = / da: / u(xj)v(x,t)dt
o o
H is a Hilbert space. We will set ||u||2 = (u\u).
We also indicate by C%„ the space of functions u e Ck(S) that are
27r-periodic with respect to t; we will set C^ for C^.
i
Definition 5.1 Let g e H. By a generalized solution of
Au = g, u(0, t) = u{tt, /,) = 0 for all < e R (5.3)
we mean a u e H satisfying
(u\A<j>) = (g\4>) for all <j> e C^, <f> = 0 on dS. (5.4)
Let us notice that, since the x-partial derivatives of the test function
<j> take arbitrary values on dSt any u e C2^ satisfying (5.4) is zero on
dS. In other words, the integral relationship (5.4) translates in the
generalized sense also the boundary condition u(0,t) = u(iv, t) = 0.
Here too, we will use complex notation for the sake of simplicity in
computation. For v € H we will let uWi„ denote the Fourier coefficients
of v with respect to the orthonormal basis {<pmin = sinmx e'nl},m =
1,2,...,n = 0,±l±2,... :
^=y^ vmns

132
6 Bifurcation problems
The following lemma shows the relationship between a generalized
solution of (5.3) and its Fourier coefficients.
Lemma 5.2 A function u e H is a generalized solution of (5.3) if and
only if there results
(m2 — n2)umtn = g-mn for all integers m, nt with m > 1 (5.5)
Proof Let it be a generalized solution of (5.3). Taking as test functions
<i> ~ <Pm,ni wc find readily the relationships (5.5); conversely, if (5.5)
holds, then by easy calculations one checks that u = £um,ti^m,Ti satisfies
(5.4).
As an immediate consequence of Lemma 5.2 we find the following.
(a) If we denote by V the set of generalized solutions of Lu = 0, then V
is a closed subspace of H and there results
V = {v € H ; vm>u = 0 for all m2 ^ n2}.
The projection onto V will be denoted by P.
(b) If W denotes the orthogonal complement of V in H, then there
results
W= {w€ H : wmjn = 0 for all m2 = n2}.
The projection onto W will be denoted by Q. Any u € H can be written
in the form
u = v + w, with v = Pu and w = Qu.
From (5.5) it follows immediately that (5.3) has a solution if and only
if g € W. Moreover, if g e W then there is a unique w e W such that
Aw = g. In other words, A is invertiblu on W with linear continuous
inverse G : W —► W satisfying
G(g) ~w& Aw = g (gtw e W).
Finally, if f satisfies the condition
(f4) f(x,t,s) is square-integrable inSfor all seR, and2n:-periodie in t
then f induces a continuous Nemitski operator f ; H —► H in the usual
way,
f(u)(x,t) = /(ar,<,u(ar,<)).
With this notation the generalized forrnulatiou of problem (5.1) becomes
Au + ef(u) = 0, u € H. (5.6)

6.5 Periodic solutions of a semilinear hyperbolic equation 133
Let us substitute u = v + w into (5.6) and consider the equation
F(e7 v,w) = Aw + ef(v + w) = 0.
To frame this problem as one of bifurcation, it is convenient to take v €V
as the "bifurcation parameter" and (e, w) € R x W as the "unknowns".
Indeed, one has F{0, u,0) = 0, for all v € V and hence the trivial
solutions are now given by (e, w) = (0( 0). So the problem turns out to be
that of seeking the possible ii0 e V such that F = 0 has solutions (etv, w)
near to (0,uo,0) with (etw) ^ (0,0). Any such solution corresponds to
a "non- trivial" solution of (5.6).
The main result of this section is contained in the following theorem.
Theorem 5.3 In addition to (f4) we suppose
(f5) there exist constants k > h > 0 such that h{s2 — si) < f{x, t, .¾) —
fix.t.s^) < k{s2-Hi)t for all (xj.) € S, for all Si < s2 € R. Then
for all e ^0 STna// enough (5.1) ftos a unique generalized solution.
Proof. If we let u = v + ?«(= Pu + Q") (5.G) becomes
Aw +ef(v +w) = 0.
Applying the projections P and Q we find the equivalent system
eP[f(v +w)]=0,
■4w+eQ[/(i;+«;)] =0.
Hence, for e ^ 0 one is led to
P[f(v +»;)] = (),
™ + eCQ[/<i> + «>)] =0
We claim as follows.
Lemma 5.4 For all w € W f/te equation P[f{v+w)] = 0 A,a,s* a unique
solution v = K(w). Moreover there results
||*(t»j)-tf(t»i)||<£|ki--t»,||. (5.8)
Prao/ o/ the lemma. Fixing w € W let us consider the map M,„ : V —* V
(5.7)

134
6 Bifurcation problems
defined by Mw(v) = P[f(v + w)]. For any v\v" € V we have
{Mw{v")-Mw{v')\v"-v')
= (P[f(v" + w)} - P[f(v' + w)]\v" - v')
= {fiv" +w) ~~ /(u' + ^)|f" " f')
= dx [f(x,t,v"(x,t)+w(xj))
0 0
- /(z, (, t/(ar, t) + w(x, t))][v"(x71) - v'(x, t)]dt.
Using (f5) we readily obtain
(Mw(v") - Mw(v')\v" - v') > /i||u" - u'||2
From a well-known lemma by G. Minty [Mi] on monotone operators it
follows that Mw{v) = 0 has a unique solution v = K{w). To prove that
K is Lipschitz-eontinuous with constant = k/h, we let u,- = K{wi), i =
1,2. From the definition of K one has P[f(vi + Wi)] = 0 (i = 1,2), and
hence
0=|(P[/(U2+^)]-P[/K+^J)] |B2-B,)|
> |(/(V2+1U2)-/(t>i+tU2)|*2-Vi)| (5.9)
- |(/(«l +W3)- f(Vi + Wi)\v2 - Vi)\.
Now, as in the preceding step, (/5) implies
|(/(V2+«M-/(*1 +t«2)|w2-*l)| >^||^-»lf (5.10)
as well as
|(/(«i +t«2)-/(«l + W\)\V2 ~ Vi)\ < k\\w2 -Wl|| ||«2 -«l||- (5-11)
From (5-9)-(5.11) we deduce
h||«2 - «lf <fc||W2-1«l|| ||«2-«x ||,
whence (5.8).
Proof of Theorem 5.3 completed. Syst#«i (5.7) is equivalent to
w + eGQ[f(K(w)+w)]*=Q.f
The operator w —► GQ[f{K{w) + w)] is obtained by composition of
Lipschitz maps and is therefore Lipschitzian itself. Then for |e| (> 0)
small enough such an operator is a contraction, and the equation w +
eGQ[f(Kw + w)] = 0 has a unique solution. This proves the theorem.
Remark 5.5 The proof of Theorem 5.3 shows that the solution of
(5-1) depends continuously on e(^ 0). As e —► 0 one has that w —► 0
while v tends to the unique solution of P(f(v)) ~ 0, namely 1¾ = A"(0).

G.5 Periodic solutions of a scmilincar hyperbolic equation 135
According to the discussion preceding the statement of Theorem 5.3, this
means that the "bifurcation set" for (5.6) branches off from 1¾ = K{0).
Remark 5.6 We have used a generalized formulation also for the
boundary conditions on dS, It is easy to see that, if u is a
generalized solution of (5.1), in the sense of Definition 5.1, then the function
u(67.) tends to zero in L2{0,2tt) as 6 -» 0 or tt. For details, see [DST].

7
Bifurcation of periodic solutions
This chapter is devoted to studying the bifurcation of periodic solutions
of first-order autonomous systems. We shall see that the natural setting
for tins kind of problems i« the so-called "Hopf bifurcation" which is
concerned with mappings depending on a two-dimensional parameter.
We will show that a suitable use of the Lyapuuov-Schmidt reduction
allows us to obtain the Hopf bifurcation in a rather straight way.
Applications to the classical Lyapunov Centre Theorem and to the restricted
three-body problem will also be given.
1 The Hopf bifurcation
Motivation
In this section we will investigate the existence of bifurcations for
mappings depending on a two-dimensional parameter. One of the
motivations of this study is the search for periodic solutions of autonomous
systems such as
|f = /<*«) (S«)
depending on a real parameter ^. Since (S^) is autonomous, the period
of the solutions we are looking for is a priori unknown, while in order
to apply the methods of nonlinear functional analysis it is convenient to
work in a space of functions with a fixed period. For this one makes the

7.1 The Hopf bifurcation
137
time rescaling t —* t/cj (u> > 0), looking for 2?r-periodic solutions of
<^ = /(*«)- (S^)
If it is a 27r-periodic solution of {S^,^) then u{t) = u{u>t) is a (27r/u>-
periodic solution of (S^). Solutions of (S;VJ) correspond to solutions
of a functional equation F = 0, depending upon the parameter pair
<u>,M)eR2.
Studying the bifurcation of periodic solutions for (S;1) we will deal with
the case when /(^, 0) = 0; then (S^,^) has the trivial solution u = 0 and
we are interested in finding the possible bifurcation values (u>o,Mo) where
nontrivial periodic solutions originate.
The Abstract Bifurcation Theorem
Motivated by the preceding considerations, which will be discussed in
more detail in the following sections, we consider here from the abstract
point of view a Banach space X and a map F : R2 Y. X —>Y such that
FeC2<R2 xXtY) and F<u>,M,0) = 0 for all <u>,M) <=R2, (1.1)
in such a way that the equation F{u>,^, u) — 0 has the trivial solution
u = 0 for all (w,/i) € R2. Extending in a natural way the definition
given in Section 5.1, we will say that
(o>o,/io) is a bifurcation point for F = 0 if there exists a sequence
(w»,/*«)-* (^o,Mo)
and a sequence Un ^- Q such that ^(^,,,^,1½) = 0.
It is rather natural to expect that the situation corresponding to the
"bifurcation from the simple eigenvalue" will now be concerned with the
case when the derivative Fu (u>, ^, 0) has a two-dimensional kernel.
To be precise, suppose F satisfies (1.1) and corresponding to (u>o, /to) €
R2, let us set
L=Fu<o*,/2o,0), V = Ker{L) and R=R(L).
We will say that L (or F) satisfies assumption (II) if
(Il-i) dim(V) = 2,
(Il-ii) R is closed and codim(ii) = 2.
If W, (respectively Z) denotes a complementary subspace of V in X
(resp. of R in K), one has
X=V®W7 Y=Z®R,
with dim[2] = 2. Let P and Q denote the conjugate projections onto

138
7 Bifurcation of periodic solutions
Z and R> respectively. We also set M = PUJ1(o>o,^o,0) and N =
Fu_w(w0, no>°) (See Remark 1.4.4).
Theorem 1.1 (Abstract Hopf Bifurcation Theorem) Suppose F
satisfies (1.1) and let (o>o,^o) be such that L = Fu(a^),^o,0) satisfies
(II). Moreover we assume that there exists v €V such that
PMv and PNv are linearly independent. (1-2)
Then (u^),^o) w a bifurcation point for F.
Remark 1.2 Assumption (1.2) does not depend on the choice of the
projector P (on the choice of the complementary Subspace Z of R)
Indeed, (1.2) is obviously equivalent to require that Mv and Nv are linearly
independent in Y/R.
Proof of Theorem 1.1. We use the Lyapunov-Schmidt procedure. Let
Q — i _. p and set u = u -(- w, with v € V and w e W. The equation
P(o>, fi, u) = 0 splits into the system
QF(w,/*,« + «>) = 0,)
PF(u>, /i, 1.+^) = O.J
As usual, the Implicit Function Theorem allows us to solve uniquely the
first of (1.3) yielding
w — ^(u>, ^, u),
where ^ is a C2 function, defined in a neighbourhood of (u^),^o,0) in
R2 x V, with values in W, satisfying (see Section 5.3)
' ^(oj,^,0)s0,
^v{^0,M0,0) = °-
Substituting into the second of (1.3), we find
PF(u>,p,v + tf>(w,/*,«))- 0. (1.4)
Taking advantage of (1.2) let us seek solutions of (1.4) in the formu= sv,
with s € R. So we are led to
/i(u>, ^, s) := PF(u>, ^, 8¾ -(- ^(u>, ^, sv)) = 0.
Plainly one has /i(u>, ^, 0) = PP(o>, ^, ^(u>, ^, 0)) = PF(u>, ^, 0) = 0. Let
us set
h(w,/i,«) = x<w,M,s)s
and note that X is a function with values in Z, of class C1 because /i
is of class C2 and /i(u>, ^, 0) = 0. With straight forward calculation one
has
X(w, /*, 0) = — (w, /*, 0) = PP„(w, M, 0)[t> + tfw(w, ^,0)t?].

7-2 Nonlinear oscillations of autonomous systems 139
Since ^v(tJo,/2o,0) = 0, it follows that
X(wq, ^0,0) = Pi?u{oj0,Mo,0)[v] = PLv = 0.
Moreover there results
^p(wD,/*o,0) -PFIlt|1(wo/io,0)[v + Vt.(wb,/*o,0)v]
+^^((^,/40,0)^.(^0,/40,0)¾].
Using again the fact that ^>u(uo,/to,0) = 0, and since PFu(u>o,/io,0)i4 =
PLu = 0 for all u, we find that
^(wo, Mo,0) = PFu,„(u>0,/*o,0)[S] = Afv.
Similarly, one finds
^(wo,/(0,0) = PF^fm, /io,0)[v] = Nv.
From assumption (1.2) we deduce that the Jacobian \dx/d(ojtfi)\
evaluated at (c4),/io) is different from 0. Therefore the Implicit Function
Theorem applied to x allows us to solve (locally) the equation x = 0
with respect to (w,/t) in functions of s. More precisely we can find C1
functions a>(s) and fi(s) (defined in a neighborhood of 5 = 0) such that
u>(0) = wtj,/i(0) =/4o, and x(oj(s),/4(5),5) = 0. It follows that
PF(Lj(s)t /4(5), 5v + ^(oj(s), /4(5), av)) = 0
and hence the branch u = us = sv + ^(07(5),/4(5),5^), gives rise to a
family of nontrivial solutions of F = 0. Since u8 ^ 0 for 5 7^ 0 and
ua —> ip(ujotno,0) = 0 as 5 —► 0, it follows that (ub,/to) ^ a bifurcation
point.
Remark 1.3 Let us note for future reference that in the cartesian
representation of the bifurcating branch, ua = 5^ + ^(07(5),/4(5), 5v), the
remainder term tp satisfies ^(o>,/4,0) = 0 and ^v(^o,/io,0) = 0.
2 Nonlinear oscillations of autonomous systems
Consider the autonomous system
^ = /0^) <s,)
under the assumption that /et^fRx R", Rn) and
/(^,0) = 0 (2.1)
As a consequence of (2.1), (S^) possesses for any /4 € R the trivial
solution u(t) = 0 and we want to find the possible values of fi where

140
7 Bifurcation of periodic solutions
there is a branching off of nonconstant periodic solutions of (S^)- In
order to employ the abstract Hopf Bifurcation Theorem, we introduce,
as anticipated in Section 1, an additional real parameter o> > 0 and
consider the auxiiary system
<"^T = /<".«)■ (S^)
If u is a 27r-periodic solutions of (Sli^)1 then u(t) = u{wt) is a {2tt/u>)-
periodic solution of (S^).
An application of Theorem 1.1 prompts our discussion of the system
Let
X = {u€ Cl <R, R") : u(t + 2tt) = u(t)},
Y={y€ C*<R, R") : y(t + 2tt) = y(t)}
and define F : R2 x X —* Y by setting
F{oj,^,u) =uu' - fi^u). (2.2)
where, here and hereafter, the prime ' stands for d/dt. Note that F e
C2<R2 x X,Y) and F<u>,M,0) = 0 for all <u>,M) e R2.
It is perhaps worthwiie to make precise the relationship between the
bifurcation of solutions of (2.2) and the branching of periodic solutions
of (S^). Let (a>o,^o) € R2, 0¾ > 0, be a bifurcation point for F,
according to the definition given in the preceding section. Then there
exist sequences u>n —* u>o, ^n —► /¾ and un € X such that un -t 0, % ^
0, satisfying F{u>n,^n,Un) = 0. It follows that un is a sequence of 2tt-
periodic solutions of {Slin_^n)- Moreover setting un{t) = un{u>n2),un
turns out to be a sequence of periodic solutions of {S/Ja), whose period
Tn = 27r/ojn tends to To = 27r/ufc. Of course, the amplitude of the
orbit tin, namely sup(eR |un{2)|{= sup(eR |un{2)|), tends to 0 as s —► 0.
Hereafter, if the pair (u>o,^o) is a bifurcation point for F(u>,^,«) = am'—
fiHi u), we will say that from {^0,0) there bifurcate periodic solutions of
(Sfi) with period close to 2^/0¾.
In order to find bifurcations for F, we have to consider the linearized
system Fu{u>,^,0)u = 0, namely
uu' — ./4,,14 = 0, A^ := fx(ji,0).
We shall assume there exists ^0 such that, setting AQ = A ^, we get
(Ao-1) Aq is non-singular and has a pair of simple purely imaginary
eigenvalues iiit'o, uo > 0;
(Ao-2) Aq has no other eigenvalues of the form iifcufc, fc e K, k ^ 1.
Keeping the notation of Section 1, we let L = Fu(ljq>imj,Q), V =
Ker(L) and R = R(L), and start by proving a lemma.

7.2 Nonlinear oscillations of autonomous systems 141
Lemma 2.1 If (A0-l-2) hold, then L satisifies assumption (II) of
Section 1.
Proof. We follow a procedure similar to that used in the proof of Lemma
6.3.3. It is convenient to work with complex notation, However, let us
point out that all the quantities we will consider turn out to be real.
We begin with Ker(L). The equation Lu = 0, u € X, leads us to find
27r-periodie solutions of
dit
u>0— - AQu = 0. (2.3)
Using the Fourier method, let us put u =■ ]T Ufcelfct, where u^ e C"
and u-k = u£ (here aiid below £* denotes the complex conjugate of £).
Substituting into (2.3) then, formally, we find
(ifcofc/ - Ao)uk = 0, fceZ. (2.4.fc)
From (Ao-1-2) it follows that the matrix (ifcufc/ — AQ) is invertible for
any k ^ ±1 and hence we find Uk — 0 for all k ^ ±1. For k s= ±1 (2.4)
gives rise to
(±iwb/ - AQ)u±i = 0. (2.5)
By assumption both ±'iwq are simple eigenvalues of Aq. If £ € <C" is
a vector spanning Ker(iu>o^ — A$) then Ker(—'kjjqI- Aq) is spanned by
its conjugate £*. It follows that Ker(L) is spanned by the functions £ert
and £*e-lt. Plainly, to obtained real-valued functions we shall take
u = a£e'1 + a*f*e~'\ a € €,
and hence the real dimension of V is 2.
Next, to study R = R(L), let us consider the equation Lu = h, with
ix e X and h € Y. Using the Fourier method again, we are led to the
system of infinitely many equations
(ikw0I - AQ)uk = hk (fceZ), (2.5.fc)
where hk denotes the Fourier coefficient of h. As before, (Ao-1-2) imply
that (2.5.fc) is uniquely solvable for any k ^ ±1. Moreover, since
the same arguments used in Lemma 6^3.3 show that
J2 Vkeiht € X, (2.G)
k*±i
where r/k = (ifcu>0/ — A0)~lhk (k ^ ±1).
Let us denote by n (resp. II*) the spectral projectors associated to £
and £*, respectively. Then the equation
(±iu>0/- Ao)u±l =h±i

142
7 Bifurcation of periodic solutions
has a solution if and only if IT(/ii) = IT(/i_i) = 0. Using (2.6) also we
infer that y e W if and only if U(hi) = IT*(/i_j) = 0. This proves that
codim(#) = 2.
It is worth noting explicitly that from the preceding proof it follows
that the projection P onto Z is nothing but
Pu - n(u!)e" + ITfu-Oe"" (2.7)
In order to show that F satisfies (1.2) of Section 1, we take v =
£e;t + £*e_it and evaluate -P-Fu,^(w0[/j0,0)iJ and PFuu,(u0tno,Q)v. For
this, let us start by noticing that
Fu(^^Q)v - w^ - AJ> = iw(€e" - Ce~lt)- ^(£ew + Ce_it). (2.8)
Taking the derivative with respect to u> one finds
PNv = PFu^(oj0, Mo,0)iJ = i£e" - ire"". (2.9)
In order to evaluate Afu = ^,^(^0,^0,0)^, some preliminares are in
order. Let us denote by A(^) = a(fi) + i/?(/i) the branch of eigenvalues
of Ap such that A(^o) = iu?0 (namely a(^o) = 0 and 0(no) = wo)- Since
/ is C2 then a and /? are CJ functions of ^. Moreover, taking into
account that 10¾ is simple, it is easy to verify that one can associate to
Ap a family £^ of eigenvectors such that
(i) the map /i —► f^ is CJ,
(ii) ^&] = A(M)^,
Let i4j,0 = (cM^/d/i)^,,.
Lemma 2.3 There result
n*^[r] = (V(w,))*r./
Proo/. We shall prove the former equality. The latter follows similarly.
From (ii) it follows that
Then, setting £' = (d^/d^)^^ we get
4«,* = 4*K " U) - ^o£' + A'(M0)U + A(/zoK'.
Recalling that ^„ = £ and A(^o) = ^, we infer
4,0£-AVoK + Owb/-*>)€'.
If we apply the projector fl, since II = 0 on the range of iu>0^ — ^th the
lemma follows.

7.2 Nonlinear oscillations of autonomous systems 143
"i/?
iwp.
— iufc
Figure 7.1
The preceding lemma allows us to evaluate PMv — -P-Fu,M(aj0,^o,0)v.
In fact, from (2.8) we infer that
and thus, recalling (2.7)(
pmv = -n^u^c" - nMyf]c-ie
= -(A'C/io^e" + Art(/io)rc-u). (2.10)
Setting A'(^o) = «'(/*o) + 1/^(/10)1 we find that (2.10) becomes
PMv = -a'(/io)(€e" + re"") - i/?Vo)(£c" - S'c"").
Then the vectors (2.9) and (2.10) are linearly dependent, (over R) if and
only if tliere is a constant, c € R such that
ic = -a'(/io)-i/?Vo)
which yields <*'(/io) = 0. In conclusion we can state the following.
Lemma 2.4 Le2 us suppose, in addition to (Ao-1-2), that
(Ao-3) a'M^O.
TTiera F satisfies assumption (1.2) of Section 1.
Remark 2.5 Condition (A0-3) is a "transversaUty" condition. From
the geometrical point of view, it means that the branch of eigenvalues
X(n) crosses tlie imaginary axis transversaiiy for /*. = //« (Figure 7.1).
Lemmas 2.3 and 2.4 allow us to apply Theorem 1.1 to F given by (2.2),

144
7 Bifurcation of periodic solutions
yielding the following result, usually referred to as the Hopf Bifurcation
theorem [Ho],
Theorem 2.6 Let f e (^(R x R*\R") be such that /(^,0) = 0 for
all n € R, and suppose that for p = ^o, (Ao-1-2-3) hold.
Then from ^0 there bifurcates a branch of periodic solutions of (S^),
witii period close to 27r/u>0.
More precisely, there exist a neighbourhood J of s = 0, junctions
u>(s), n(s) € Cl{J), and a family us of non-constant, periodic solutions
°/(S^(s)) such that
(i) u>{s) -» u>0, fi(s) -> no as s -> sn,
(ii) us has period Ta = 27r/u>(s),
(iii) the amplitude of the orbit us tends to 0 as s —► 0.
Remark 2.7 One immediately verifies that the same conclusion holds
taking v — £eit(t+0f) + £*e~'<t+"> for any a € R. This fact is related to
the fact that (S^) being autonomous and therefore invariant under tile
time translation 6 —► u(. + 9), the bifurcation set is also.
Example 2.8 Let us consider the Van der Pol equation
g-(M-3z^+z = 0, (2.11)
where n is a real parameter. Equation (2.11) is equivalent to the first-
order systom ,
dx
The linearizr-d system is
d_,_ ' (212)
dt X'
Ax
—■ = y + fix,
dt
In this case one has ^ — _, whose eigenvalues are A(^) =
^[/j ± y/(n2 - 4)]. For ^ = 0 there results A(0) = ±i. Plainly, (A0-
1-2-3) hold true and Theorem 2.6 applies with /¾ = 0 and a*j = 1.
We note that, in this specific case, taking advantage of dealing with
a system in R2, it is possible to perform a direct analysis of the
characteristics of (2.12) in the "phase" plane (x, y). One finds easily that

7.3 The Lyapvnov Centre Theorem
145
(1) for any ^, < 0 (2.12) has no periodic trajectories but the trivial one
x = 0, y — 0,
(2) for any ^ > 0 (2.12) has a unique periodic solution, which is
asymptotically stable. See Figures 7.2 and 7.3.
Note also that the trivial solution x = 0, y = 0 is stable for all ^ < 0
and unstable for ^ > 0.
Figure 7.2 Phase portrait of (2.12)
Figure 7.3 The closed orbit of (2.12)
3 The Lyapunov Centre Theorem
Consider the first-order system
£-/(»). (S)
where / € C2(R",R"). A singular point of (S) is a p € Rn such that
/(p) = 0. In this section we will be concerned with the existence of small

146 7 Bifurcation of periodic solutions
oscillations of (S) near an equilibrium p, namely periodic solutions of (S)
with orbits confined near p. Let us suppose that p = 0 is a singular point
of (S) and let A = /'(0). If no point of the spectrum of A belongs to the
imaginary axis, then the behaviour of the solutions of (S) near p = 0 is
completely understood. It is possible to show (see, for example [P]) that
there are two invariant manifolds M and TV, with dim(M)+dim(7V) = n
and M n N = {0} such that for all q £ M (resp, N) the solution of the
Cauchy problem
du
u(0) = ,,
tends to 0 as t —► +oo (as t —► —oo, respectively). The behaviour of the
solutions near p = 0 is represented in Figure 7.4.
Figure 7-4
As a consequence, a necessary condition for (S) to have closed orbits near
a singular point, say p = 0, is that A have a pair of purely imaginary
eigenvalues ±iu>o- However this condition is not sufficient, in general.
For example, consider the two-dimensional system
x' - ~y~x(x2 +y2),
(3.1)
y = x - y(xz + /). '
The eigenvalues of A are ±i; on the other hand, if (x(t), y(t)) is a solution
of (3.1), one has
d I"1, 2 2J >
^ 2^ +V ) -as +yy = ■
■(x

7.3 The Lyapunov Centre Theorem
Figure 7-5
It follows that
x'+y*
2t+c
and hence (3.1) has no periodic solutions (see Figure 7.5),
Lyapunov has shown in a celebrated theorem [Ly] that (S) does
possess periodic solutions near 0 provided it has a "non-singular" first
integral b. Recall that a first integral of (S) is a non-constant real-valued
function b € C^R^R) such that b(u(t)) = constant for any solution
u(t) of (S). We point out that, in view of the uniqueness of the solutions
of the Cauchy problem
du
d*
= f(u), u(0) = p,
b is a first integral of (S) if and only if
/(P) ■ V6(p) = 0, for all p € R". (3.2)
Examples of such systems are the second-order cojiservative (or gra-^
dient) systems, namely systems like
^+VU(u) = 0,
(3.3)
(HS)
or the Hamiltonian systems
x' = -Hy(x,y),
y' = Hx(x,y),
where (x, y) e R*1 x R". In fact, the Hamiltonian itself H = H(x, y) is
a first integral of (HS). Note that (3.3) is a particular case of (HS): it
suffices to take H(x, y) ~ ||y|2 + U(x).
The following lemma shows the role played by the first integral.

148
7 Bifurcation of periodic solutions
Lemma 3.1 Suppose b is a first integral of (S) and consider the
modified system
^ = /(u)+MV6(U),M£R. «S)„)
Ifu~ u(t) is a T-periodic solution of (S^) then u is in fact a T-periodic
solution of (S).
Proof. Let u(t) be any solution of (SM) for some p ^ 0. Setting 0{t) ~
b(u(t)) one has
/?(*) = ^*(«(0) = V6(«(*)) ■ «'(*) = V6(«(*))" /(«(*)) + m|V6(u(*))!2
Using (3.2) we get that
/7(<) = m|v»(u«)Ij-
If, for example, ^. > 0, then ,5(() is non-decreasing. In the other hand,
since x is T-periodic, we deduce 0(0) = b(u{0)) = b(u(T)) = /?(T).
Hence
ff(t) = mIv»(»(«))I5 = o
and u solves (S).
Lemma 3.1 suggests we seek small oscillations of a system with a first
integral as periodic solutions of (5M) above, bifurcating from ^o = (0> 0)-
The following theorem gives conditions under which such a bifurcation
occurs.
Theorem 3.2 (Lyapunov Centre Theorem) Suppose that f €
C2(K",Rn) is such tiiat /(0) = 0. Letting A = /'(0), we suppose that
(A-l) A is nonsingular and has a pair of simple eigenvalues ±iu>o;
(A-2) for all k eZ, k ^ ±1, ifcu>o is not an eigenvalue of A.
Moreover, let us assume that (S) has a first integral b e C^fR^R)
such that b"(Q) is non-singular.
Then (S) possesses small oscillations near p = 0.
More precisely, there exist a neighbourhood J of s ~ 0, a Junction
oj(s) € Cl( J), and a family us of non-constant, periodic solutions of (S)
such that
(i) lj(s) —► uq, as s —► 0;
(ii) ua has period Ta = 2tt/uj(s);
(iii) the amplitude of the orbit u8 tends to 0 as s —*Q.
Proof. According to Lemma 3.1, we can replace (S) with (SM) which

7.3 The Lyapunov Centre Theorem
149
can be studied by means of Theorem 2.6, with /(^, it) = f(u)+fiVb(u).
First of all we note that from (3.2), and recalling that b is C2 here, it
follows that
!'{i)y ■ V&(0 + f(0 • Vb"(Ov = O, for all £, y € R". (3.4)
Putting £ = 0, one has
Ay ■ V6(0) + /(0) • b"(0)y = 0, for all y € R".
Since /(0) = 0 and A is non-singular, it follows that V6(0) = 0. As
a consequence, we infer that f(n,0) = /(0) + ^V&(0) = 0. Moreover,
setting B = b"(0), one has
^M = /*(/*> 0) «= i4 + ^B.
We shall apply Theorem 2.6 with ^o ~ 0. Since A0 — A^ — A, (A0-l-2)
follow from (A-l-2). It remains to verify that (A()-3) holds.
First, let us consider (3.4). Since / is continuously differentiate,
/(0) = 0 and b" is continuous, then it is easy to verify that the map
£ -* /(£) ■ b"{Qy is differentiate at £ = 0 with derivative /'(0)[.] •
b"(0)y — A[.] ■ By. Hence we can differentiate (3.4) at, £ = 0, yielding
/"(°)[y»zl ■ Vb(Q) + Ay ■'Bz + Az ■ By = 0 for all y, z € R".
Since V6(0) = 0 it follows that
Ay -Bz + Az-By = 0 for all y, z € R",
that is (note that B is symmetric),
ArB + BA=Q. (3.5)
Then, up to a change of coordinates, the matrix A has the form
A \S 01
with
L<"0 0 J
and R does not contain ±iu>0 in its spectrum, because ±M} are simple
eigenvalues of A. Let us write
" U Ml
B=|_Mr Cj
where U (resp. C) is a symmetric 2 x 2 (resp. (n —2) x (n - 2)) matrix.
From (3.5) it follows readily that
SU = US (3.6')
and
SM = M.K. (3.6")

150
7 Bifurcation of periodic solutions
Recalling that o>o ^ 0, from (3.6') one deduces with elementary
calculations that there is 6 € R such that
U [0 6\
From (3.6") and using the fact that wq ^- 0 and ±"kjo are not eigenvalues
of R, one infers that the 2 x (n — 2) matrix M is the 0 matrix, *
To see this, let X,Y e R"-2 denote the two rows of M.
"•[y]
Then from (3.6") it follows X,Y satisfy the system:
:' XR + u0Y = 0
L YR - oj0X = 0
One finds X — Uq1YR and hence Y(R2 + (JqI) = 0. Since ±iufc are not
eigenvalues of R, then one infers that Y = X = 0.
From the preceding arguments we deduce that B has, with respect to
the same basis used for (3.5), the form
-6 0 .,
0 6 L
0 C
where 6 ^ 0, because B is non-singular.
Consequently there results
ti8 — a>o
—OJo 1^
0 R + pC_
and hence the branch of eigenvalues X(fi) such that A(0) =
by
A(/i) = ^ + ioj0-
This proves that (Ao-3) holds true. An application of Theorem 2.6,
jointly with Lemma 3.1, yields the existence of a C1 function oj(,*>) —*
0¾ and of family us of non-constant solutions of (S) with period TB =
27r/oj(s), such that the amplitude of us tends to 0 as s —► 0.
B =
A + nB =
0
- 10¾ is given
Remarks 3.3
(i) The above result being local in nature, it would be sufficient to
consider in Theorem 3.2 a vector field f and a first integral b defined in
a neighbourhood of 0 in R".
(ii) If A has several purely imaginary eigenvalues ±io>fc,o>fc > 0, k —
' More generally, it is possible to show that if R and S are square matrices having
disjoint spectra and if M is a matrix such that. SM = MR, then M = 0.

7.3 TTie Lyapunov Centre TheoTem
151
1,.,., m, the non-resonance condition (A2) is always satisfied at ±Iu>*,
where u>* = max{u>fc, 1 < k < n}.
(iii) It has been proved by J. Moser [Mo] that non-resonance
conditions (A 1-2) can be eliminated at the expense of the existence of a first
integral b e C2(Rn,R) such that b"(Q)is positive-definite. The
following example (see [Mo]; see also [MW]) shows that, in this more general
form, if b"(0) is merely nondegenerate, (S) may have no periodic
solutions at all. Let x,y € R2, x — (3:1,3:2), y — (yi,y2) and consider the
Hamiltonian system (HS) with Hamiltonian
H(z, y) = \(x\-x\+y\- vl) + (M2 + \y?)B(x, y),
where
B(x,y) = (i/i!/2 - xix2).
Here the matrix A has the form
A =
0 0-10
0 0 0 1
10 0 0
0-1 0 0
and has double eigenvalues ±i. If x = x(t) and y = y(t) is a solution of
(HS), there results
^(xm+vix,) = -*[B(x,yf - (M2 + |j,|2)2,
and therefore (HS) has the trivial solution x = 0, y = 0 only.
(iv) Since b" is non-singular, the arguments of Lemma 3.1 show that
here the auxiliary parameter \i — 0.
(v) According to Remark 1.3, the family of periodic solutions u,, has
the property that
► £e + f e as s —► 0
s
where £ is such that A£ = iu>o£.
The Lyapunov Centre Theorem applies both to second-order
gradient systems like (3.3) and to Hamiltonian Systems (HS). Let us state
explicitly this kind of result.
We consider (HS) with H e C2(R" x R",R). Set z = (x,y) e
R2", H(z) = H(x,y) and VH{z) = (Hx(z),Hy(z)). If J denotes the
symplectie matrix (i.e. J : (x,y) —► (—y,x)), then (HS) can be written
in the more compact form
Since in (HS) the Hamiltonian H is a constant of the motion, namely

152
7 Bifurcation of periodic solutions
H(z(t)) =const. for all solutions of (HS), it makes sense to look for
periodic solutions of (HS) on the Hamiltonian surface H(z) — h.
We suppose that
(HO) H(0) = 0, V-ff(O) = 0 and H"(0) > 0 (that is H"(0) is positive-
definite),
(HI) JH"(0) has n pairs of purely "imaginary simple eigenvalues
±iojfc, k = 1,2,... ,n,
such that Wi/wj is not an Integer for all i ^- j.
Theorem 3.4 Suppose that H e C2(Hn x R",R) satisfies (H0-1).
Then for all £ > 0 small enough (HS) has n (geometrically) distinct
periodic orbits on the surface H(z) = e. More precisely, the surface
H(z) = £ carries n distinct periodic orbits zk whose periods tend to
2ir/Ljk, k = 1,2,...,n.
Proof. For all k = 1,2,..., n, we can apply Theorem 3.2 with f =
JVH, A = JH"(0) and b — H. Indeed, (HO) implies, in particular, that
b"(0) = H"(Q) is non-singular; and (Hi) implies that (Ai-ii) hold true
for all k =1,2,...,n.
Then there exist n branches z^s, k — 1,2,..., n. of periodic solutions
of (HS) with period Tk_s converging to 2-KJWk as s —► 0; moreover,
||3fc,.!b~->° as s -+0. (3.7)
In addition, Zk,s depends in a Cl fashion on s and (see Remark 3.3(v))
!iins_0-^ = Vk := fa1"1 + t*ke~'lwt, k = 1,2,...,n (3.8)
where A£k = '^k^k-
Since H is a first integral of (HS), then H(zklS(t)) is independent of
t. We set
hk(s) = H(zk,a(Q)).
From (3.7) one immediately deduces that hk(s) —► H(0) = 0.
Moreover, since Zk,a is Cl with respect to s and H'(0) = 0, it follows
readily that hk is twice differentiable at 5 = 0 and, using also (3.8), one
finds
/4'(0) = H"(Q)wk ■ wk
where wk — £*+&*
Since H"(Q) > 0, it follows that for all e > 0 small enough and any
k =1,2,...,n, the equation hk(s) = £ has a solution s = s(k,£) and
s(k,e) -»0ase — 0 fc= 1,2,...,n.

7.4 The restricted three-body problem
153
Correspondingly we find n solutions Zk,e = Zk s{k ei (fc = 1,2,..., n) of
(HS) such that H(zkiS) = e. Finally, from (3.8) we also deduce that, for
e small, the orbit of Zk_£ is close to that of svk, that is to that of svk,
up to higher-order terms; then the Zk,E (fc = 1,2,...,n) correspond to
geometrically distinct orbits. This completes the proof of the theorem.
Remarks 3.5
(i) As in Remark 3.3 (i), H could be defined in a neighbourhood of 0
in R".
(ii) Theorem 3.4 has been extended by Weinstein [W] (see also [Mo])
who proved the following result. Suppose H satisfies (HO). Then for all
e > 0 small enough (HS) has n distinct periodic orbits on the surface
H(z) = £. In comparison with the result of Moser recalled in Remark
3.3 (iii), one has to point out tfiat in the case of a general conservative
system one can exibit examples where (S) has only one solution on each
surface b = e.
4 The restricted three-body problem
One of the most classical application of the Lyapunov Centre Theorem
is to the existence of small oscillations near the equilibrium points of the
planar restricted three-body problem. This problem deals with three-
bodies P\,P2 (called primaries) and Q, with masses Mi,M2 and M3,
respectively, under the action of the Newton Gravitational Law. To
make the problem more feasible, one considers the restricted problem,
which is concerned with the case when the mass of one particle is
negligible with respect to the others. If, say, M3 = 0 then /¾ and P-i are not
influenced by Q and they move according to the solutions of a two-body
problem. Our aim is to study the motion of Q under the attraction of
tile two primaries.
Actually, we shall make some further simplifications. First of all, we
suppose that the primaries move on circles, rather than more general
elliptical orbits, with constant angular velocity 7. Moreover we will
assume that the motion of Q occurs on the same plane as that of P|, P2.
This problem is usually called the restricted planar three-body problem.
Even with these simplifications, it is still quite interesting, because many
problems arising in celestial mechanics fit in this frame.
Let us introduce a rotating coordinate system xOy (Fig. 7.6), such
that (i) the origin O coincides witii the barccntre of Pi and P2 and (ii)
Pi and P2 are at rest on the x-axis. With a suitable choice of the units,

154
7 Bifurcation of periodic, solutions
iy jf Q = (x, y)
Figure 7.6
we can take Mi + M2 =1,7=1 and g (the gravity constant) = 1. We
also set Mi = m in such a way that Pi = (—m, 0) and P2 = (1 - m, 0)
and let (x, y) denote the coordinates of Q and
pi =v/[(z+m)2+y2],
p2 = y/[{x+m- if +y2]
the distances from Q to Pj and P2, respectively.
The third body Q is subjected to combined action of the
centrifugal and Corioiis forces and to those due to the Newtonian attraction,
corresponding to the potential
1 —m m ,
U(x,y) = +—.
Pi P2
In conclusion, we find tbe system
x" - 2y' - x- Ux(x,y),
y" + 2x' ~y~Uv(x,y),
where, here and hereafter, primes f denote d/dt
(4.1)
Equilibrium points
The possible equilibria of (4.1) can be found by solving the system
-x = Ux(x,y),
~V = Vy(x,y), J
namely the pair of equations
(x + m)(l-m) m(x + m-l)

7.4 The restricted three-body problem 155
Figure 7.7
-,-^-^. (4.3,
Pi Pi
The lattor is satisfied for y = 0. Substituting y = 0 into (4.2) we find
T - (s+m)(l-m) m(s+m-l)
[z + mP |x + m-l|3 ' * " '
Equation (4.4) has three solutions, corresponding to the so called Eulcr
points Li,L2,L3. (Figure 7.7).
It is also convenient to introduce the potential
<b(x,y)=±(x*-t-y2)+U(x,y). (4.5)
Equations (4.2) and (4>3) are nothing but ¢^=0 and $,, = 0,
respectively. Hence the Euler points arc the solutions of ¢3:(3:, ()) = t). One
checks immediately that $**(.£*) > 0, i = 1,2,3. As for $„„ one finds
^ , ~, m 1 —m
«w(3!,o)-i-[!c_m + ip-F-^p.
Since in L2 both [x-m| and \x—m+lj are < 1 we infer that $,,,,(1/2) < 0.
In Li and L3 one finds, with elementary calculations, that x^yy(x,Q) is
< 0 in L3 and > 0 in Lx. In both cases it follows that $yy < 0. In otiier
words, letting for 1 < j < 3
wc get
bj = 0, flj > 0 and c, < 0. (4.6)

156
7 Bifurcation of periodic solutions
La
*.
/ \
/ \
/ \
\
Li - mi
'.1 - m L3
V
is
Figure 7.S
Let us come back to (4.2)-(4.3) and look for solutions with y 7^ 0. Setting
h = l/ff\ and fc = 1//9¾ we find
3; = h(x + m)(l - m) 4- fcm(x + m - 1),
(4.7)
1 = /i(l - m)+km. (4.7')
Multiplying (4.7') by x and subtracting from (4.7) one lias readily ft. =
k — 1. Thus there are two more equilibria L4 and L5, the Lagrangian
points, such that Pi, P2 and L4 (or L5) are the vertices of an equilateral
triangle (Figure 7.8).
Setting a = 04.5 = $^(£4,5),6 = 64,5 = $1,1/(-^-1.5) and c = C4/, =
$yj/(L4>5), one finds readily
«4^(2,-^4
(4.8)
The configuration consisting of the two primaries and a Lagrange point
is, for example, that of the system Sun-Jupiter-Trojans (tlie last arc a
group of asteroids).

7.4 The restricted tliree-body problem
157
(4.1')
(4.1")
Small oscillations
Let us refer to the system {4.1) which will be written in the form
x" -2j/ = ^x(xty)t]
y" + 2x' = $,,(:£, y),
where $ is given by {4.5). In order to apply the Lyapunov Centre
Theorem. {4.1') has to he transformed into a first-order system. If we set
p ■= x' and g = y', {4.1') becomes
x' — P,
y' = 9,
p' = 2g + $*,
q'= -2p + $y, J
which is of the form u' - f(u), where u = (x,y,p,g) € R4 and / has
components
f(x,y,p,g) = (p,g,2g + $r,-2p + $y).
I11 terms of the new coordinates, the equilibria are given by
v,j ~ (xj,y3,0,0), 1 < j < 5, where (xj,Vj) = Lj.
It iw inimediately verifiable the system {4.1") has a first integral {the
Jacobi integral) given by
•/(■'•> y, p,q)=^(p2+ q2) - $(x, y).
The Hessian J"(uj) is given by (wc keep the notation introdueed before)
J"M =
■-aj -bj 0 0
-bj -Cj 0 0
I) 0 10
. 0 0 0 1.
Consequently
dct[J"(uj)] =
-M.,
Taking into account (4.6) and (4.9) we find
dct[J"(u,)] < 0, fori = 1,2,3,
27
drt[./"(i;,)] = -m(l - m) > 0, for j = 4,5,
and in any case J" is nonsinguiar at each equilibrium point.
It remains to evaluate t!ie matrix Aj = /'(zj,J/j,0,0). We obtain
■0 0 10"
. _ 0 0 0 1
' ~ aj b,i 0 2
.bj c, -2 0.

158 7 Bifurcation of periodic solutions
The eigenvalues A of Aj satisfy the equation
X4 - (aj + cj - 4)A2 + Dj = 0, (4.9)
with
3 \ ~bi ~c3
On the Euler points Lj, j = 1,2,3, we have bj = 0, and (4.9) becomes
A:| - (a, +Cj - 4)A2 + ojCj = 0. (4.9')
Since in addition a,c,- < 0, then (4.9') has a unique pair of purely
imaginary roots A = iiWj, j = 1,2,3, and the Lyapunov Centre
Theorem applies without any further restriction yielding the following.
Theorem 4.1 In a neighbourhood of the Euler points Lj, j = 1,2,3,
the restricted planar three-body problem has a family of periodic solutions
whose periods tend to 2ir/ljj.
As for the Lagrangian point L4 (the same holds for the symmetric one
L5) we liave (see (4.8))
3 , 3V2~ ,. 9
<z=-, &=—(2m-l), c«-.
The (4.9) becomes
A"+A2 + —m(l -m)=0,
4
which possesses two pairs of imaginary roots ±io/, ±iu>" , with, say,
0 < lj' < lj", provided ^m(l - m) < \, namely for all 0 < m < mo
(or 1 - mo < m < 1) where mo as 0.0385... is the smallest roots of
27m(l - m) = 1. Let us consider the range 0 < m < m0 (for example,
in the crt.se when the primaries are Sun and Jupiter, the mass ratio i«
m ss 1/1000, a value which is widely in the range (0, mo «* 0.0385); the
Name for the: Earth-Moon system where m « 1/82 as 0.012).
Taking a>o = u>" we can apply the Lyapunov Centre Theorem directly,
while when we consider the pulsation u/ we have to require the non-
resonance condition lj"'/lj' $. N, namely that lj" ^- fcu>' for all k € N.
This leads to excluding the solutions of
um = fcV2, 1
27 n \
= —mil — m).
4
This system has solutions whenever m satisfies
7m(1-m>=(TW (4-10)

7.4 The. restricted three-body problem
159
In conclusion, if rtik denotes the sequence of solutions of (4.10) such that
m-k —* 0, we can still apply, for m ^ rtik, the Lyapunov Centre Theorem
yielding the following.
Theorem 4.2 Suppose that 0 < m < mo; then in a neighbourhood
of the Lagrange points L4[5 the restricted planar three-body problem has
a family of periodic solutions whose periods tend to 2ix/u)"; if further,
m ^ /fta-, then there exists a second family of periodic solutions v)hoac
periods tend to 1-k /lj''.
Note that these periodic solutions correspond to bounded trajectories
in the inertia! frame of reference. For other results on the restricted
three-body problem, see for example [SiM].
Remark 4.3 The stability of the linearized system u' = A3u, namely
of
x"-2y' = a,x+b3yA
y" + 2x' ^bjX + cjy, J
at the equilibria Lj,j= 1,2,3,4,5, can be easily discussed.
At the Euler points Li,L2l Ai *nc matrix Aj has a real positive and
a real negative eigenvalue. Thus L3(j = 1,2,3) are unstable equilibria
for (4.11) and are said to be linearly unstable.
Unlike the preceding case, the matrices A$, AA have, for 0 < m < m(),
two pairs of purely imaginary eigenvalues. Therefore, at £/4,//5, (4.11)
has bounded orbits only and the Lagrangian points are said to be linearly
stable.
The question of the (nonlinear) stability of L4,L5 is much more
delicate. It lias been shown there are tiireo exceptional values iiii(i =
1,2,3),0 < mi < m2 < m3 < m0, such that for all m € (0,m0),m ^
vi; (t = 1,2,3), the Lagrangian points are stable in the sen.se of
Lyapunov. For more details, see [Molj.

Problems
1. Let Inv(X, Y) denote the set of all A e L<X, Y) thai, are invertiblc
with inverses ^1 € L(Y,X). Show that
(i) lnv(X,Y) is open,
(ii) the map (-)^1 :lnv(X,Y) ->lnv(Y,X) defined by (-)~l(A) = ^1
is differentiable and
d(-)-l(A)xH -^~A~lHA~l.
2. Tf / satisfies tiie Caratiieodory condition (C) and
\f(x,s)\<a(x) + b\s\°
with a = p/q, a G Lq and b > 0, show that / is continuous from Lp to
L".
3. Let K : Cl x Cl -» R be sucli that K(x, y) = K(y,x) for all (x,y) €
A xfl and
/ |^{x,y)|pdxdy < oo.
If f satisfies (C) and {2.1) show tHat the Hammerstein operator
H : u(x) -» I K(x,y)f(y,u(y))dy
is continuous from Lv into itself.
4. Tf, in addition to the conditions of problem 3, / has partial derivative
f„ satisfying (C) and {2.7) of Section 1.2, with p > 2, show that H is
F-differentiable on V and
dH{u)[v) = f K{x,y)fs{y,u{y))v{y)dy.
n
5. Consider the operator TV defined in Section 1.2, eqn {2.19) and assume
/ satisfies {2.17) therein. Show that if a < (n + 2)/(n - 2) then N is
compact.
HintUse the compactness of the Sobolev embedding of Hq(CI) into 1^ (£))
for any p < 2ra/(n - 2).

Problems
161
6. Consider F e C*(X,Y) with F(0) = 0. Let us set JV =Ker(F'(0))
and suppose that (i) JV has a complementary subspace Z in X, and (ii)
F'(0) is onto Y. Show that there are e > 0, neighbourhoods 0 (resp.
V, W) of 0 in JV (resp. in Y,Z) and a map $€C'(0xl/, W) such that
F(£ + ¢(¢,¾)) = u, for all (£, v) 60x1/.
Apply this result to the case in which X = TR.m,Y = Rn(m > n)
and the rank of the matrix A = F'(0) is n, and show that the equation
F(x) = 0 has a solution x = £ + 2, with £ = </(£) := <^(f,0).
7. Consider the boundary-value problem
P1) f-A««/(«)i.in,
and suppose that / 6 C(R), /(0) = 0 and /'(0) + \k. Prove that (PI)
has a solution for all £ € R,|e| small. This solution is unique in C2-a(Cl).
8. Prove that the boundary-value problem
Au = u3 in £),
u = .9(2) on d£), J
has a unique solution for all g 6 C0*"(d£)).
9. Let F(u) = Au + A/(u),A e R. Suppose that / e C2(R) is such
that /'(0) = 1 and uf"(u) < 0 for all u ^ 0. Setting X = C2-a(0) n
C0(f)),y = C°-a(Cl), prove that
(i) for all A < Ai, F : X —► Y is locally invertible on X,
(ii) if, in addition, /(0) = 0, then for A = Ai for the singular set of
F,E0, one has £0 = {0}.
10. Consider the boundary-value problem
|Au+A/(u) = /1(1) inJl
1 w = 0 on tW7,
where / e C2(R) is bounded and such that /(0) = 0,/'(0) = 1 and
uf"(u) < 0 for all u ^- 0. Show that (P2) has a unique solution for all
h e C°-a(f)).
/firai Use Problems 7 and 9.
11. Extend the preceding results to tiic case in winch / satisfies /(0) =
0, /'(0) = 1, uf"(u) < 0 for all «^0, but is possibly unbounded. Deduce
that for all h € C°-a(Cl) the boundary-value problem
Au + Xu — it3 = /1(2:) in £), ]
u = 0 011 d£),
has a unique solution provided A < Aj.

12. Consider the boundary value-problem
, . , ..K„ , „,it) = h(x) in Cl,
(P3) J v ; V '
(Au + Xku + b(u)
u = 0 on d£l.
Suppose that b € ¢^(¾.) is bounded and such that \k~i < C\ <
\k + b'(s) <c2< Afc+i. Let
fl+ = lim sup 6(s), 0" = Urn sup 6{s),
and set
n"1" = lim inf b(s), a = lim inf b(s)
S—'+00 ( 3—'-OO
A+ = a+ J<j>k + (3~ J <j>k,
n+ n-
,4-= or /V + P+ J fa,
n+ n-
5+ = /3+ f<f>k + a~ J <f>k,
n+ ti~
m n-
Prove that (P3) lias a solution provided
min(,r,>l+)< f h<t>k <max(B-,B+).
13. Consider the boundary-value problem
(P4)
(Au+ A|it -(- 6(u) = o^i in £),
w = 0 on d£l,
with b satisfying
(i) b € ^(R), and A! -(- tf(a) < A2,
(ii) ^=0,
(Hi) b(s) > 0 for all s € R.
Prove that there exists a > 0 such that (P4) has a solution if and only
if 0 < a < a; and, moreover, that if 0 < a < a then (P3) has at least
two solutions.
14. Consider (P4) with b satisfying (i)-(ii) and
(iv) sb(s) >0 for all s + 0.
Prove that there exist a' < 0 < a'' such that (P4) has a solution if
and only if a' < a < a"; and, moreover, that if a' < a < a" and a ^ 0
then (P4) has at least two solutions.

15. Let h be a 27r-periodic continuous function and consider the
problem
<P5) y" + f(y) = h(t), y(0) - y(2ir) = y'(0) - y'(2n) = 0.
Suppose that / e C^R), f'(y) < 1 and that f(y) -» f± as y -» ±oo.
Prove that (P5) has a solution provided /~ < (2tt)~1 /Q2?r /t < /+.
16. Consider
. f Au + Alu+/{.r,u) = 0iiiSl,
\ u = 0 oh d£),
witli / € C1 satisfying (i) Ai + /'{«) < A2; keeping the notation of
Section 4.1, let
) = f fity
G(t)= / /(ty,+tu(0toi-
(a) Prove that if G{**) < 0 {> 0) then u* = t"4> + w(t*) is a super-
solution (sub-solution) of (P6).
(b) Use this to show that (P6) has a solution provided it possesses a
sub-solution <p and a super-solution ip (without requiring that <p < ip).
Hint If G vanishes at some to then tufa + w(to) solves (P6). if, say
G(t) < 0 for all t, then each ttpi + w(t) is a super-solution of (P6). hi
particular, for t > 0 large enough, t.4>\ + w(t) > <p.
(c) Find a counterexample of a b.v.p which has a sub-solution <p and
a super-solution ip but does not possess any solution.
Hint Take a problem like -Au + X2u + h(x) = 0). *
17. Consider the (linear) Vol term operator A from X = C(0,1) into
itself, defined by
:'"L1) -» J u(s)<iii.
Show that the spectrum cr(A) of A contains only A = 0, which is not an
eigenvalue and hence A = 0 is not a bifureation point. See Remark 5.1.5
(a).
18. Let A be the operator from X = L2(a, b) into itself defined by
: u(t) -» / fc(s, t)u(s)ds,
where Cl = [a,b] x [a, 6] and fc(s,t) e L2(Cl), is symmetric and positive-
definite.

164 Problems
Show that (i) A = 0 is not an eigenvalue of A, (ii) A possesses a
sequence Xn of eigenvalues, with Xn —► 0, (iii) hence A = 0 is a bifurcation
point for F = XI - A,
19. Discuss the Examples 5.4.5-6 using Theorem 5.4.2 instead of 5.4.1.
20. Consider the system
n" = A sin ^, J
tp" = Xu cos ipy J
together with the boundary conditions
u'(0)=:u(l)=0, 1
^(0) = /(1) = 0,/
describing the equilibria of a rotating beam.
Prove that the solutions Xk of
l + cosv/|A| -cosh^AI = 0
are bifurcation points for the above problem.
Extend the result to a system of the form
u" = Xf(n,<p),)
ip" - \g(u,tp). J
21. Let ifi, if) : R —► R be smooth. Discuss the bifurcation of periodic
solutions for the Ltenard equation
x" — (fi(x)x' + if)(x) = 0,
in dependence on the parameter ^= ^(0)-
(Note that for <p{x) = ft - 3x2 and if)(x) — x this 13 nothing but tin;
Van der Pol equation).
22. Given ¢,^) • R -» R, smooth and such that ¢(0) = ^(0) = 0, discuss-
tlio Hbpf bifurcation for the system
*' = #y), 1
y' = if)(x) + fty. J
23. Discuss the Lyapunov Centre Theorem in the case of the Haurilto-
liian system
»' = -»„/
with H(x,y) = U(x) + V(y).

Bibliography
[Ama] Aniaiin, H., Fixed point equations and lionlincar eigenvalue
problems in ordered Banach spaces, SIAM Review 18(1976), 620-709.
[AAM] Anmnn, H., Ambrosctti, A. &: Mancini, G-, Elliptic equations
with noninvertible Fredholm linear part and bounded nonlinearities,
Math. Zeit. 158 (1978), 179-94.
[AmaH] Amaiin, H. h Hess, P., A multiplicity result for a class of
elliptic boundary value problems, Proc. Royal Soc, Edinburgh 84A(1979),
145^51.
[AMI] Ambrosotti, A. Sz Mancini, C, Existence mid multiplicity
results for nonlinear elliptic problems with linear part at resonance. The
case of the simple eigenvalue, Jour. Diff. Equal. 28 (1978), 220-45.
[AM2] Ambrosctti, A. Sz Mancini, G-, Theorems of existence and
multiplicity for nonlinear elliptic problems with noninvertible linear part,
Annali Scuola Norm. Sup. Pisa, Seme IV 5 (1978), 15-28.
[AP] Anibrosetti, A. &c Prodi, G., On the inversion of some differ-
entiable mappings with singularities between Banacb spaces, Ann. Mat.
Pura Appl. 93 (1973), 231-47.
[AmiT] Amick, C,J. &c Turner, R.E.L., A global branch of steady
vortex rings, Jour. Reine Angew. Math, 384 (1988), 1-23.
[Be] Bergcr, M.S., Nonlinearity and Functional Analysis, Academic
Press, New York, 1977.

166
Bibliography
[Bp] Berger, M.S. h Podolak, E,, On the solutions of a nonlinear
Dirichlet problem, Indiana Univ. Math. Jour. 24 (1975), 837-46.
[B6] Bohme, R., Die Losung der Verzweigungsgleichungen fur nicht-
lineare Eigenwertprobleme, Math. Zeit. 127 (1972), 105-26.
[Br] Brezis, H., Analyse fonctionelle, theorie et applications, Mas-
son Ed., Paris, 1983.
[Ca] Caccioppoli, R., Un principio di inversione per le corrispon-
denze funzionali e sue applicazioni alle equazioni alle derivate parziali,
Atti. Ace. Naz. Lincei 16 (1932), 392-400.
[CH] Courant, R. &: Hilbert, D,, Methods of Mathematical
Physics , Interscience, New York, 1962.
[ChR] Chow, S.N. & Hale, J.K„ Methods of Bifurcation Theory,
Springer-Verlag, New York, 1982.
[CrR] CrandaU, M, h Rabinowitz, P.H., Nonfinear Sturm-Liouville
eigenvalue problems and topological degree, Jour. Math. Mechanics 19
(1970), 1083-1102.
[DST] DeSiinon, L. h Torelli, G., Soluzioni pcriodiclic di
equazioni afle derivate parziali di tipo iperbolico nonlineari, Rend. Sem.
Mat. Univ. Padova 40 (1968), 380-401.
[DI] Dieudonne, J,, Elements d'analyse, Gautheir-VUlars, Paris,
1969.
[D2] "Dieudonne, J., Sur le polygone de Newton, Arkiv der math. 2
(1950), 49-55.
[Fi] Field, M.J., Differential Calculus and its Applications, Van
Nostrand Reinhold, New York, 1976.
[Fu] Fucik, S., Solvability of Nonlinear Equations and Boundary
Value Problems, Reldel, Dordrecht, 1980.
[GT] Gilbarg, D, h Trudinger, N., Elliptic Partial Differential
Equations of Second Order, Springer-Verlag, New York, 1977-
[GS] Golubitski, M, &: Schaeffer, D., A theory of imperfect
bifurcation, Comm. Pure Appl, Math, 32 (1974), 21-98,
[Ho] Hopf, E,, Abzweigung einer periodischen Losung von einer sta-
tionaren Losung eines DlfTerentialsystemes, Ber. Math. Phys. Sachsis-
che Akademie der Wissenschaften Leipzig 94 (1942), 1-22,

Bibliography
167
[KW] Kazdan, J.L. h Warner, F,W,, Remarks on some quasilinear
elHptic equations, Comm. Pure Appl. Math 28 (1975), 567-97.
[Ko] Kolodner, 1.1., Heavy rotating string, a nonlinear problem,
Comm. Pure Appl. Math. 8 (1955), 395-408.
[Krl] Krasnoselski, M.A., Topological Methods in the Theory of
Nonlinear Integral Equations, Pcrgamon, Oxford, 1965,
[Kr2] Krasnoselski, M.A., Positive Solutions of Operator Equations,
Noordlioff, 1964.
[KFS] Kufner, A., John, O. h Fucik, S., Function spaces,
Acadcmia, Prague, 1977.
[LL] Landcsman, E. Sz Lazer, A.C., Nonlinear perturbations of
linear eigenvalue problems at, resonance, Jour. Math. Mechanics 19 (1970),
609-23.
[LC] Levi-Civita, T., Determination rigourcusc des ondes
d'auipleur linie, Math. Annalen 93 (1925), 264-314.
[Le] Levy, P., Snr los fonctions de lignes hnplicites, Bull. Soc.
Math, de France 48 (1920).
[Lyl] Lyapunov, A.M., Sur les figures d'equilibre peu differents des
eflipsoidcs d'unc masse liquidc liomogeue douee d'un mouvement dc
rotation, Zap. Akad. NaukSt. Petersburg (1906), 1-225.
[Ly2] Lyapunov, A.M., Probleme general de la stabilite du mouve-
mont,, Ann. Far.. Sri. Tovlouse 2 (1907), 203-474.
[Mar] Marino, A., La biforcazione nel caso variazionale, Conf.
Sent. Mat. Univ. Bari 132 (1!)73).
[MP] Marino, A. k. Prodi, G., La teoria di Morse per gli spazi di
Hilbert, Rend. Sem. Mat. Univ. Padova 41 (1968), 43-68.
[MM] Marsden, J. &c McCracken, M., The Hopf Bifurcation and its
Applications, Springer-Vcrlag, New York, 1976.
[Maw] Mawhin, J., Topological degree methods in nonfinear boundary
value problems, CBMS Regional Conference Series Math. #40, A.M.S.,
Providence, R.I., 1977.
[MW] Mawhin, J. &c Willein, M., Critical Point Theory and Hamil-
tonian Systems, Springer-Verlag, New York, 1989,

168
Bibliography
[McS] McKean, H,P, & Scovel, J.C., Geometry of some simple
nonlinear differential operators, Annali Scuola Norm. Sup. Pisa, Serie IV
13 (1986), 299-346.
[Mi] Mlnty, G.J., Monotone (nonlinear) operators in Hilbert
spaces, Duke Math. Jour. 29 (1962), 341-6.
[Mol] Moser, J., Lectures on Hamiltonian Systems, Mem. A.M.S. 68
(1968).
[Mo2] Moser, J., Periodic orbits near an equilibrium and a theorem
of A. Weinstein, Comm. Pure Appl. Math. 29 (1976), 727-47,
[Ne] Nekrasov, A.I., Waves of stationary type (in Russian), Izv.
Ivanovo. Voznesensk pol. Inst. 6 (1922), 155-71.
[Ni] Nirenberg, L., Topics in Nonlinear Functional Analysis,
New York Univ. Lecture Notes, 1974.
[P] Perron, G., Uber Stabiiitat und asymptotischc Verhaiten der
Integraies von Differentialgleichungsystemen, Math. Zeit. 29 (1929),
748-66. -::
[Pr] Prodi, G., Probiemi di diramaziorie per equazioni funzionaii,
Boll. U.M.I. 22 (1967), 413-33.
[PW] Protter, M.H. h Weinberger, H.F. Maximum Principles in
Differential Equations, Prentice-Hall, Englewood Cliffs, N.J., 1967.
[Rl] Rabinowitz, P.H., Periodic solutions of nonlinear hyperbolic
partial differential equations, Comm, Pure Appl. Math. 20 (1967),
145-205.
[R2] Rabinowitz, P.H., Some global results for nonfinear eigenvalue
problems, Jour. Funct. Anal. 7 (1971), 487-513.
[R3] Rabinowitz, P.H., Existence and nonunlqueness of rectangular
solutions of the B^nard problem, Arch. Rat. Mech, Anal. 29 (1968),
32-57.
[Sa] Sansone, G., Equazioni differenziali nel campo reale,
Zanichelli, Bologna, 1965.
[Sche] Schechter, M., A nonlinear elliptic boundary value problem,
Ann. Scuola Norm. Sup. Pisa 27 (1973), 707-16.
[Schm] Schmidt, E,, Zur Theorie der linearen und nichtlinearen Inte-
gralgleichungen, 3 Teil, Math. Annalen 65 (1908), 370-99-

Bibliography
169
[Schw] Schwartz, J.T., Nonlinear Functional Analysis, Gordon &:
Breach, New York, 1953.
[SiM] Siegel, C.L. h Moser, J., Lectures on Celestial Mechanics,
Springer-Verlag, 1971.
[T] Turner, R.E.L., Internal waves hi fluids with rapidly varying
density, Ann. Scttola Norm. Sup. Pisa, Seriz IV, Vol. VlII-4 (1981),
513-73.
[Va] Vainberg, M.M., Variational Methods for the Study of
Nonlinear Operators, Holden-Day, San Francisco, 1964.
[VT] Vainberg, M.M. h Trenogin, V.A., The methods of Lyupanov
and Schmidt in the theory of nonlinear equations and their further
development, Uspehi Mat, Mauk 17 (1962), 13-75; English translation in
Russian Math, Surveys.
[Ve] Velte, W., Stabilitat und Verzweigung stationaren Loaungen
der Navier-Stokesschen Gleichniigen behu Taylor problem, Arch. Rat.
Mech, Anal. 22 (1966), 1-14,
[W] Weinstein, A., Normal modes for non-linear Hamiltoiiian
Systems, Inv, Math. 20 (1973), 47-57.
[Y] Yoslda, K., Functional Analysis, Springer-Verlag, New York,
1974.

Index
Bcnard problem 112-119
bifurcation
definition 80
equation 91
for a variational operator 1G6
from a multiple eigenvalue
101-104
from an odd eigenvalue 105
from a simple eigenvalue 91-101
global 105-106
Hopf 135-139
necessary condition 80
sub critical 97
supercritical 97
transcritical 97
bifurcation of periodic solutions
136-159
definition 137
bifurcation problems 107-135
buckliiig of an elastic beam 86-89
composite-map formula 11
conservative systems 147
derivative
Frechet 12
higher 23-26
partial 26-29
differential
FVechet 9-10
Gateaux 12
differentiation rules 11-12
Dirichlet problems
asymptotically linear 52-54
at resonance 62-71
bifurcation 99
boundary value 5
semilinear 61-78
with asymmetric noniinearities
71-78
cigenspace 2
eigenvalue
comparison property 6
continuity property 7
existence 6
of a linear elliptic problem 6
of a iiliear operator 2
variational characterization 7
eigenvector 2
elliptic operator 5
first integral 146-147
functional 12
Frechet derivative 12
differential 1G
Fretiiiohn alternative 2
Gateaux differential 12
global inversion 45-60
theorem 47-52
with singularity 54-59
gradient 12
gradient systems 147
Green operator 6
Haniiitoiiian systems 147, 151
Hopf bifurcation 1,36
Hopf bifurcation theorem 144
abstract 137-138
Holder spaces 3-4
hyperbolic semilinear equation
130-135
implicit function theorem 36- 38
inverse of a linear map 30-31
inversion
along a path 48
global 46-52
global with singularities 54-59
local 30-36

Index
171
LP spaces 3
local inversion 30-36
theorem 32-33
Lyapunov centre theorem 148-160
Lyapunov-Schmidt reduction 89-91
manifold of codimension one 66
maximum principle 7-8
mean-value tlieorein 13
minimal surface 36-36
multiplicity
algebraic 3
geometric 3
Neniitski operator 16-22
continuity 16-17
differentiability 17-2G
operator
compact 2
Green 6
linear 2
Nemitski 16-22
potential 12, 21-22
variational 12, 106
oscillations
of autonomous systems 139-145
small, for second order systems
119-123
small in the 3-body problem
164-167
small near an equilibrium
143-161
periodic solutions
bifurcation 137
of autonomous systems 139-146
of conservative systems 148-161
of hamiitonian systems 161-163
of hyperbolic equations 130-135
under small perturbation 38-44
Poincare inequality 4
potential operator 12, 21-22
proper map 46
range of an operator 2 /.-.'
Rettich theorem 4
Schwarz theorem 28
singular ordinary point 56
singular point 46
point for a differential system 145
Soboiev
embedding theorems 4-6
spaces 4
splitting 1
stability of closed orbits 38-44
string, rotating 1G7-112
Stlirni-Liouviiie
problems 7
bifurcation 97-98
subsoiution 76
Slipersollition 76
Taylor's formula 28-29
three-body problem (restricted)
163-169
equilibrium points 164-156
Euler points 154 ...
Lagraiigian points 155-156
linear stability 159
Hinat! oscillations 156-169
topological complement 1
traiiHversallty condition 143
water waves 123-129
weak solution 6
0565^8

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