POLSKA AKADEMIA NAUK
MONOGRAFIE MATEMATYCZNE
KOMITET REDAKCYJNY
KAROL BORSUK, BRONISLAW KNASTER, KAZIMIERZ KURATOWSKI Redaktor,
STANISLAW MAZUR, WACLAW SIERPINSKL HUGO STEINHAUS,
WLADYSLAW SLEBODZINSKL ANTONI ZYGMUND
TOM 46
PWN-POLISH SCIENTIFIC PUBLISHERS

MAREK KUCZMA
FUNCTIONAL EQUATIONS
IN A SINGLE VARIABLE
WARSZAWA 1968

COPYRIGHT, 1968, by
PANSTWOWE WYDAWNICTWO NAUKOWE
(PWN - POLISH SCIENTIFIC PUBLISHERS)
WARSZAWA (Poland), ul. Miodowa 10
All Rights Reserved
No part of this book may be translated or reproduced in any
form, by mimeograph or any other means, without permission
in writing from the publishers.
PRINTED IN POLAND
DRUKARNIA UNIWERSYTETU IM. ADAMA MICKIEWICZA W POZNANIU

PREFACE
Although the greatest mathematicians, such as N. H. Abel, A. Cauchy, L. Euler,
B. Riemann and others, have concerned themselves with functional equations,
there existed — until recently—no exhaustive monograph of the subject. The articles
by S. Pincherle in Enzyklopadie der Mathematischen Wissenschaften (Pincherle
[3], [4]) and the booklet by E. Picard [10], in which only a few particular equations
were dealt with, could hardly be regarded as such. The non-existence of a general
theory of functional equations was, no doubt, one of the reasons of this situation.
In recent times, however, we observe great progress in this branch of mathematics.
This progress is manifested also in two newly published monographs^): one by
M. Ghermanescu [22], and the other by J. Aczel [5]. Ghermanescu's book, however,
is written in the Roumanian language and thus is inaccessible to a great number of
the world's mathematicians. Moreover, it concentrates on the contributions of
Roumanian mathematicians and leaves out many important results. AczeTs work,
on the other hand, has gained great popularity and an English translation of it, much
enlarged, has just appeared (Aczel [7]). However, it deals exclusively with functional
equations in several variables. It is by all means justifiable to separate the two topics:
equations in a single variable and in several variables, since there is considerable
difference between them, at least as great as between ordinary and partial differential
equations, both in methods and in the kind of results. Nevertheless, the fact is that
a satisfactory work on functional equations in a single variable has still been lacking.
It is the purpose of the present book to fill this gap at least partially.
This book is exclusively concerned with functional equations in which only
one independent variable occurs. But the difference equations and recurrences are
not included, although they belong to the same type. They may be regarded as
independent subjects, which since 1860 B) (Boole [1]) have again and again been
(') For the completeness of the account we mention here also J. Aczel and S. Gol^b's book
devoted to functional equations occurring in the theory of geometric objects (Aczel, Gofab [1]),
two expository articles by the author of the present book (Kuczma [13], [24]), lecture notes by
A. Sklar [1J, and a bibliography of functional equations by Gy. I. Targonski [3], unfortunately
very incomplete.
B) Nearly a hundred years before Bcole finite differences were dealt with by Euler [1]. It
would be difficult to establish when functional equations appeared in mathematics for the first
time. In the form of recurrent sequences they go back as far as Archimedes (Pincherle [4]).

6 Preface
presented in book form (e.g. Norlund [1], [2], T. Fort [1], Gelfond [1], Montel [13],
Goldberg [1], H. Levy, Lessman [1], Meschkowski [1], to mention only the most im-
important examples). A few exceptions from this principle are questions which are not
usually found in books on differences, such as the problem of monotonic or convex
solutions of difference equations, etc. It must be stressed, however, that, though
difference equations are not separately discussed, many of the results contained
in the present book apply also to difference equations.
Of the various aspects of functional equations we have chosen to present those
which are most likely to form a foundation of a theory. Thus the main stress has
been laid on questions of existence and uniqueness. This is the line developed in
recent years in the mathematical circles of Krakow and Katowice.
Starting to write this book we planned to give an almost complete presentation
of the above-mentioned topics of the theory of functional equations. Soon, however,
the scheme proved impracticable: the work would have to be far too voluminous.
Therefore some selection was necessary. Accordingly, the subject of equations of
higher orders and that of equations with superpositions of the unknown function —
which were intended to form separate parts of the book—have finally been reduced
to two chapters each. Also, in the first ten chapters the real variable is favoured
in comparison with the complex variable. This is partially due to the fact that the
complex variable requires more advanced tools and the proofs involved are usually
considerably longer; maybe the author's own preference has also found expression
here.
We have endeavored to keep the book on an elementary level, and advanced
methods have been avoided throughout. The use of fixed point theorems in function
spaces is an exception; but it is doubtful whether these methods can nowadays be
regarded as advanced. The Banach fixed point theorem for contraction maps is
taught at junior courses in most universities and undergraduate students are quite
familiar with it. Apart from this, no special knowledge is required from the reader,
except the elements of the calculus and some basic facts from the theory of complex
functions. Only the section on Riemann's Zeta function requires in addition the
knowledge of Dirichlet series, but it may be omitted in reading without loss.
In a few instances longer proofs have been omitted. Except for those few cases
(where the reader is referred to the original papers) all the-results given in this book
are proved and there is no need to consult other sources. The end of each proof
is marked by the sign ¦.
The book contains an introduction and fifteen chapters. The formulae, theorems,
lemmas and hypotheses are numbered doublewise: the first number refers to chapter
@ referring to the introduction) and the second indicates the successive item within
the given chapter. E.g. G.12) refers to the 12th formula in the 7th chapter. The
numbering of formulae is independent of the numbering of theorems, lemmas or
hypotheses. Thus there exist: formula B.3), theorem 2.3, lemma 2.3 and hypothesis
2.3 independent of each other (all of them in Chapter II of course). References to the
bibliography consist of a name and a number in brackets. E.g. Aczel [5] refers to

Acknowledgements 7
the fifth item under the heading J. Aczel. Where a number of references are given
in the text, the order of quotations is chronological.
The bibliography is placed at the end of the book. We have tried to collect all
the items relevant to the subject treated in the book. Thus papers on finite differences
are in general not included, but a number of exceptions have been made regarding
papers more closely connected with the topics dealt with in the book. The line of
division may be objected to as quite arbitrary, but this has, unfortunately, been
unavoidable. Most papers on iteration have been included, since iteration is so
closely related to functional equations.
We are aware that in spite of our endeavours many relevant items have been
omitted. We have not been able to trace all the papers in which functional equations
(in a single variable) are mentioned. We shall be obliged to the readers for further
bibliographical hints and also for any comments on the contents and form of this
book. It seems to be the first more serious attempt at producing a work on functional
equations in a single variable, and so we believe that it can still be improved.
Marek Kuczma
Katowice, in June 1965.
ACKNOWLEDGEMENTS
The author is indebted to a number of persons and institutions for their extremely
valuable help, which made possible the preparation of the present work.
I thank the editors of the Polish Scientific Publishers in Warszawa for the care
which they devoted to the publishing of this book in the best possible form and for
their understanding for my own difficulties. I offer special thanks to Professor
M. Stark, on whose initiative this book has been written, and to Mr. W. Muszynski,
who showed great patience when the work over this book was getting delayed.
I also thank the Editorial Board of "Monografie Matematyczne" and especially
Professor K. Kuratowski for including this book into the series "Monografie
Matematyczne".
I am also indebted to Uniwersytet Jagiellonski in Krakow, and especially to the
Rector, Professor M. Klimaszewski for his extraordinary kindness to me and multi-
multiform assistance while I was engaged in writing this book.
A number of persons have given me advice and bibliographical hints which
were particularly valuable. They were Professor N. G. de Bruijn, Professor L. E. Evtu-
sik (Л. Е. Евтушик), Professor Cz. Olech, Professor B. Schweizer, Professor
Gy. I. Targonski, Professor A. Turowicz, Professor A. Valeiras, Professor P. M. Vasic,
Professor A. Zajtz, Dr. I. N. Baker, Dr. D. Brydak, Dr. H.-J. Glaeske, Dr. W.
Holsztynski, Dr. A. Lundberg, Dr. A. Ma.kowski and Dr. E. Seneta, to all of whom
I offer my sincere thanks.

8 Acknowledgements
I am partcularly obliged to Dr. B. Choczewski and to my brother Marcin
Kuczma, who have helped me in the difficult task of collecting the bibliography.
I thank Dr. B. Choczewski also for bis help in proofreading and in preparing
the indices.
I am grateful to my mother and my wife for their help in preparing the manuscript.
There are also three persons to whom my indebtedness is particularly great and
whom I wish to thank for more general reasons.
Professor S. Lojasiewicz stirred up my interest in functional equations. He helped
me several times during my research. He also encouraged me to write a book.
Professor J. Aczel has taught me a great deal about functional equations and
has shown me that they may become an independent branch of mathematics,
worth devoting one's time and effort to its investigation. His book was an example
for me how to write a book on functional equations.
Professor St. Gola.b, my teacher and principal since I made my first steps in a
scientific career, did his best to train me for mathematical research. His kindness,
his readiness to help me whenever I needed it, were really overwhelming. Were it not
for his encouragement, help and approval, this book would have never been written.
Marek Kuczma

CONTENTS
Preface 5
Acknowledgements 7
INTRODUCTION
§ 1. Iterates and orbits 13
§ 2. Attractive and repulsive fixed points 17
§ 3. A real variable 19
§ 4. A complex variable 24
§ 5. Functional equations 25
CHAPTER I
GENERAL SOLUTION
§ 1. Formulation of the problem and preliminaries 29
§ 2. Construction of the general solution 30
§ 3. Non-invertible/ 35
§ 4. Automorphic functions 41
§ 5. Abel's equation and Schroder's equation 43
§ 6. General remarks 44
CHAPTER II
LINEAR EQUATION
§ 1. Solution depending on an arbitrary function 46
§ 2. Homogeneous equation 47
§ 3. Some criteria 51
§ 4. Non-homogeneous equation 52
§ 5. Case|<7(O|>l 53
§6. CaselsKftKl 54
§7. Casesr(x)=±l 57
§ 8. Examples 65
CHAPTER III
CONTINUOUS SOLUTIONS
§ 1. Solution depending on an arbitrary function 67
§ 2. Extensions of solutions 70
§ 3. Unique solution 72

10 Contents
§ 4. Lack of uniqueness 75
§ 5. The function fix) decreasing 77
§ 6. Examples 81
CHAPTER IV
DIFFERENTIABLE SOLUTIONS
§ 1. Preliminaries 84
§ 2. Solution depending on an arbitrary function 87
§ 3. Existence theorem 90
§ 4. Uniqueness theorem 94
§ 5. Regularity of solutions 98
§ 6. Lack of uniqueness 100
§ 7. An application: the Goursat problem for a hyperbolic equation 101
§ 8. Other examples 103
chapter v
MONOTONIC AND CONVEX SOLUTIONS
§ 1. Fundamental theorem 106
§ 2. Monotonic solutions 107
§ 3. Lack of uniqueness 110
§ 4. Further uniqueness theorems 113
§ 5. A finite difference equation 114
§ 6. Generalization of the previous result 118
§ 7. Recurrent sequences 121
§ 8. General linear recurrence 123
§ 9. Consequences for functional equations 126
§10. Euler's Gamma function 127
§11. Application to branching processes 131
CHAPTER VI
SCHRODER'S EQUATION
§ 1. Preliminaries 135
§ 2. Regular case 137
§ 3. Koenigs existence theorem 139
§ 4. Convex solutions 141
§ 5. Principal solution 143
§ 6. Singular case. Multiplier zero ' 145
§ 7. Case \s\ = \. A review of the results 147
§ 8. Case of a root of unity 148
§ 9. Divergence case 149
§10. Convergence case 149
§11. Conjugacy problem 156
§12. Exponential and logarithmic functions 159

Contents 11
CHAPTER VII
ABEL'S EQUATION
§ 1. General 163
§ 2. Asymptotic conditions at a finite fixed point 167
§ 3. Convex solutions . 171
§ 4. A condition for the effectiveness of the Levy algorithm 173
§ 5. Exponentially growing functions 174
CHAPTER VIII
ANALYTIC SOLUTIONS
§ 1. Special homogeneous equation 180
§ 2. Special inhomogeneous equation 183
§ 3. Entire solutions of the homogeneous equation 185
§ 4. General equation 187
§ 5. The Gamma function in a complex variable 191
§ 6. Riemann's Zeta function 193
CHAPTER IX
ITERATION
§ 1. Iteration groups 197
§ 2. Regular iteration 199
§ 3. Multiplier zero 202
§ 4. Levy iterates 204
§ 5. Regular iteration at infinity 206
§ 6. Analytic iteration 209
CHAPTER X
COMMUTING FUNCTIONS
§ 1. The real case 213
§ 2. Semipermutable polynomials 215
§ 3. Permutable entire functions 218
§ 4. Exponential function 222
CHAPTER XI
SIMULTANEOUS EQUATIONS
1. Biperiodic functions 227
2. Periodic solutions of functional equations 230
3. Further properties of the Gamma function 232
4. A continuous curve filling a square 236
5. Cantor's singular function 241

12 Contents
CHAPTER XII
EQUATIONS OF HIGHER ORDERS AND SYSTEMS OF EQUATIONS
§ 1. Systems of equations 244
§ 2. Equations and systems of equations of order m 246
§ 3. Uniqueness theorems 248
§ 4. Lack of uniqueness 252
§ 5. Gaussian normal distribution 254
CHAPTER XIII
LINEAR EQUATIONS OF HIGHER ORDERS
§ 1. Reduction of order 259
§ 2. Equation with constant coefficients 262
§ 3. Finite groups of substitutions 267
§ 4. Characterization of polynomials 271
CHAPTER XIV
INVARIANT CURVES
§ 1. Unique invariant curve 274
§ 2. Lack of uniqueness 278
§ 3. A problem of continuation 282
§ 4. Euler's equation 286
CHAPTER XV
FRACTIONAL ITERATES
§ 1. The Babbage equation 288
§ 2. Fractional iteiates 293
§ 3. Continuous increasing solutions 297
§ 4. Continuous decreasing solutions for decreasing g 299
§ 5. Continuous decreasing solutions for increasing g 300
§ 6. Regular solutions 303
§ 7. A generalization 305
Bibliography .... * 308
Index of symbols 373
Subject index 375
Index of names 377

INTRODUCTION
§ 1. Iterates and orbits. The theory of functional equations in a single variable
is inseparably connected with the theory of iteration. On one hand, iterated functions
usually occur in the solutions of such functional equations, and often even in
the equations themselves. On the other hand, functional equations are one of the
main investigation tools in the iteration theory. Consequently, before we can discuss
functional equations, we must introduce certain fundamental notions from the
iteration theory.
Suppose that we are given a function/(x), defined in a set E (for the present
we do not assume anything about the nature of the set E) and suppose that
@.1) /(?)<=?.
Definition. A set E for which @.1) holds will be called a submodulus set for
the function f{x). If we have
@.2) /(?)=?,
then E will be called a modulus set for the function/(x).
The iterates f\x) of the function f(x) are defined by
@.3) f°(x)=x, /и+1(*) =/(/"(*)), xeE, « = 0,1,2,...
According to @.1) all the functions f(x)(n=0, 1, 2, ...) are defined in the whole
of E. If E is a modulus set (i.e. if @.2) holds) and the function / is invertible, then we
can define the iterates of f{x) also for negative iteration indices w:
/->(*)=/-'(/»(*)). xeE, « = 0,-1,-2,...,
where/ denotes (') the inverse function to f(x). In the case where @.1) holds,
the negative iterates of/(x) are defined only in suitable subsets of E (f~\x) in
/"(?¦)). In each instance we have the relations
@.4) /'(*)=/(*),
C1) In the whole of this book upper indices at the sign of a function will denote iterations.
Exponents of a power of a function will be written after a bracket containing the whole expression
for the function. Indices in parentheses denote derivation. Thus/2 (x) denotes/(/(x)), while [/(x)]2
denotes fix)-fix); /(r)(*) means the rth derivative of f(x).

14 Introduction
@.5) /"(/m(*))=/"+m(*)'
whenever all the terms occurring in @.5) are defined.
In the set E the following equivalence relation can be defined ('): two points
x, у е Е are said to be equivalent under f (or shortly equivalent) if and only if there
exist non-negative integers m and n such that f(x)=fm(y). If x and у are equivalent,
we shall write xiy. The equivalence relation i is reflexive, symmetric and transitive;
consequently the set E can be split into disjoint sets of equivalent elements. The set
of all points which are equivalent to a given x0 e E will be called the orbit of x0
under /. The orbit of an x will be denoted by Cf(x) or, if there is no doubt what
function is involved, by C(x).
If the function / is invertible in a submodulus set E, then the relation i becomes
simpler: xiy means that there exists an integer к (positive, negative or zero) such
th&tf\x)=y. Thus the orbit C(x) consists of points of the formy=fk(x) for those
values of к for which f\x) is defined.
Let/(x) be a function defined in a submodulus set E. We shall classify the points
of E according to the structure of their orbits under/.
We define Q?k[f] (or shortly (gt), where A: is a positive integer, as the set of
those x e E for which there exists an integer y>0 such that
@.6) fJ+\x)=f\x)
(here j may depend on x) and @.6) does not hold for a smaller k> 0 with any integer/
The smallest integer у>0 such that @.6) holds will be denoted by Jk(x). Thus Jk(x)
is an integral-valued function defined on (St and characterized by the following
property:
Lemma 0.1. Ifxe<?k, k^l, then
@.7) ¦ /l+*(*) =/'(*) for i>Jk(x),
@.8) Г+кШГ(х) for i<Jk(x).
Proof. @.8) results from the definition of Jk(x), @.7) follows from the fact
that for i>J=Jk(x) we have by @.5) /l+*(«)=/'"/(/'+*(*))=/'"/(/'(*)) =/'(«)• ¦
Definition. If x e <&k[f], k^l, and Jk(x)=0, then x is called a. fixed point
of order к of the function f(x). Fixed points of order 1 are called shortly fixed
points. They are the points x e E which fulfil'
@.9) /(*)=*•
The remainder of the set E, i.e. the set of x e E which do not fulfil @.6) for any/
and k, will be denoted by <?0[f]. Thus we have
@.10) (?0[f]=E- U
k-l
@ First introduced by K. Kuratowski. Cf. Tambs Lyche [1], Isaacs [1], Kuczma [30], La-
sota, Pebzar [1], Reghis, Vuc [1], Sklar [1].

1. Iterates and orbits 15
If the function / is invertible, then condition @.6) implies f\x) = x and thus in
this case Jk(x)=0 and every element of Gt[/] is necessarily a fixed point of order k.
The orbit C(x) then consists of exactly к elements. Such an orbit will be called
00
closed. For x e |J Gt[/] the iterates f\x) are defined for all n. In fact, if x e <Sk,
then @.6) holds whenever f\x) is defined. Assuming/"(x) defined for an w<0 we
have by @.3) and @.6)
/(Г+к-\х))=Г+к(х)=Г(х),
which means that fn+k~\x)=f-1{f{x))=f~\x), i.e. f"~\x) is defined. Con-
Consequently/"^) is defined for all n.
For x e e0 [/] this is no longer true. If x e Gon П /"(?)> then evidently/"(x) is
n=l
defined for all n. The set of all such x will be denoted by GOo[/]:
eoo[/]=®on n f"(E)= П /"(?)- U gn
n= 1 n= 1 n= 1
The other part of <?0[f] will be denoted by <?01[f]:
n=l
For every x e GOi there exists exactly one'x0 e C(x) such tha.tf~1(x0) is not defined.
This x0 may be represented as xo = C(x)n(E-f(E)). Then C(x) = {/"(xo)}n=o>i,2...-
Thus we have decomposed the set E into disjoint sets Gt:
@.11)
n=l
Each of the sets GOo, ©i, 62, ••• is a modulus set for f{x), and so is every orbit
contained (*) in these sets. The set G01 is a submodulus set for/(x), and so is every
orbit contained in it. GOi is empty if and only if/ fulfils @.2). Every orbit contained
in ®и, и>1, contains exactly n points; orbits contained in Go are denumerably
infinite B).
In the case where / is not invertible the structure of orbits can be much more
complicated. The orbit of a point may be non-denumerable, as in the case of the
hat-function (cf. Baayen, Kuyk, Maurice [1], Barna [2])
f(x) = l2x for xe<°>i>>
П B-2* for xe(i,l>,
where for every у e <0, 1) there exist exactly two values x e <0, 1> such thatf(x)=y.
C1) Cf. theorem 0.1 and the corollary.
B) The decomposition formula @.11) and the subsequent remarks concern only the case of
an invertible /.

16 Introduction
The orbit of an x e <?k, Jfc> 1, now contains also elements which are not fixed
points of f(x), and in general is not finite. Roughly speaking, the orbit of such an x
consists of a closed part, containing exactly к elements, and of a tail. The tail con-
consists of the predecessors of x (i.e. of points z such that f\z)=x for some />0) and
may be much larger than the closed part of the orbit. E. g. for the constant function
f(x) = a for all xeE
the orbit of any point x e E is the whole set E. Here a is the closed part (k = 1) and
the rest of E is the tail.
For invertible as well as non-invertible / we have the following
Theorem 0.1. Ifxe&k,k=0Q,01,0, 1,2,..., then C(x) <= gfc.
Proof. Suppose that x e ®t, k> 1. Thus forj=Jk(x) @.6) holds. Let у е C(x).
Consequently there exist integers w>0, w>0such that f(x)=fm{y). Hence we have
by lemma 0.1
Неге к cannot be replaced by a smaller one, for in that case a similar argument
would show that к in @.6) is not minimal, either. Thus ye&k, which proves that
C(x)<=Gt.
If x e Go and у e C(x), then we must have у e (?0, for otherwise we would
have by @.10) у e <&k for some fc> 1 and, on account of what has already been proved,
x e<Sk, since x e C(y) = C(x). In order to establish the validity of our theorem also
00
for к=00 and A:=01 we must prove that (for invertible/) xef)f(E) and xiy
00 00 П= 1
implies у e (~\ f\E). Let us suppose that x e(~)f(E) and x г у. Since / is invertible,
n=l n=l
there exists an m ^ 0 such that у =fm(x). If m > 0, then we have by @.1) fm+\E) <=f(E),
which implies that у e C\f\E). If m <0, then/"^) =fm+n(x) ef(E), since x ef-m(E),
00
which again means that у e P| f"(E). ш
Corollary. ^йсЛ Gt, A:=00,01, 0,1,2, ... is a submodulus set for f.
This follows from the above theorem in view of the fact that every orbit is
obviously a submodulus set.
We shall also prove the following
Lemma 0.2. If xt, x2e<&k, k^0, and if for some non-negative integers n, m,
p, q we have
Г(Х1)=Г(х2) and ПХ1)=
then there exists an integer r such that
@.12) (n-m)-(p-q) = rk.

1. Iterates and orbits 17
Proof. For argument's sake let us suppose thaXp^n. Then we have
i.e.
@.13) /p~"
Now, if x2e<?0, then @.13) implies p-n+m = q, i.e. @.12) holds with any r. If
A:>1, we may write
p — n + m— q = rk + s,
where r is an integer (positive, negative or zero) and
@.14) 0^s<k.
Thus @.13) becomes
fq+rk+s(x2)=f%x2),
whence we get for j=Jk(x2)
fJ+q+rk+s(x2)=fJ+%x2).
By lemma 0.1 we obtain hence
which in view of @.14) and of the minimality of A: implies $=0. Hence @.12) results. ¦
Finally we shall prove a theorem concerning the case where E is endowed with a
topology (e.g., E may be a set in the space of real or complex numbers).
Theorem 0.2. If the function /(x) is continuous in a subset E of a topological
space and if for an xoe E limf(xo)=x* e E, then x* is a fixed point of f.
П-Ю0
Proof. Letting w->oo in the relation (cf. @.3))
/и+1(*о)=/(/и0со)),
we obtain @.9) for x=x* in view of the continuity of/ (*).¦
§ 2. Attractive and repulsive fixed points. Now let us suppose that E is a subset
of the space of real or complex numbers. All the topological notions, however, will
C1) The properties and, in particular, the convergence and asymptotic behaviour of sequences
of iterates have been investigated by Lemeray [1], [3], [4], [8], [9], Podetti [1], Isenkrahe [1], Lon-
London [1], Bottcher [5], [8], [11], Niccoletti [1], Andreoli [1], [2], [4], Tricomi [1], [2], Fatou [3],
[15], Julia [l]-[6], Lattes [16], [17], Pincherle [5]-[9], Ritt [5], Schauffler [1], [2], Wolff [l]-[3], [10],
Denjoy [1], Lindner [1], Campagne [1], Valiron [5], [6], Steckel [1], Cartan [2], [3], Montel [4],
[7], Fortet [1], Topfer [1], [4], Heins [1], Ferrand [1], Ghabbour, Winn [1], Hamilton [2], Matsu-
moto [1], Herve [1], Thijsen [1], Barna [1], Bajraktarevic [1], Rosenbloom [2], Radstrom [1], Tala-
nov [1], [2], Karamata [1], Ostrowski [3], de Bruijn [3], Myrberg [8], [9], [15], [16], [17], Baayen,
Kuyk, Maurice [1], Obrechkoff [1], Thron [2], Sarkovskil [1], [2], [4]-[13], Kenzegulov [1], Cherry
[1], Brolin [1], Nishino, Yoshioka [1], Maharam [1], Kenzegulov, Sarkovskil [1], Baker [7], [8],
Gumowski, Mira [3]-[5], Mira [1], [2], Macintyre [1], Kuczma, Smajdor [1], Kuczma [39],
and others.
2 Functional equations

18 Introduction
be understood with respect to the set E. In particular, a neighbourhood of a point
x0 e E is to be understood as the intersection of an open interval or an open disc
centred at x0 with the set E.
We assume that E is a submodulus set for a function/(x). Let x0 be a fixed
point of the function/.
Definition. x0 is called an attractive fixed point of fix) if there exists a neigh-
neighbourhood U of x0 such that lim.f\x)=x0 for exery xe U. If we have, moreover,
n->oo
@.15) |/(*)-*o|<S|x-xo|. 0<S<l,
for xe U, then x0 is called strongly attractive. Similarly, x0 is called a repulsive
fixed point of fix) if there exists a neighbourhood [/ of x0 such that the sequence
/"(x) does not tend to x0 for any x e [/— C(x0). If we have, moreover,
@.16) |/(лс)-лсо|>в|*-*о|. в>1,
for x e [/— C(x0), then x0 is called strongly repulsive.
The notions of an attractive and a repulsive fixed point are here relative to
the set E.
Concerning the above notions we have the following (*)
Lemma 0.3. If fix) is differentiable at a fixed point x0, and if
@.17)
resp.
@.18)
then x0 is a strongly attractive or, respectively, strongly repulsive fixed point of fix).
Proof. Since
fix)-x0
|/'M=Hm
x-x0
@.15) and @.16) result from @.17) and @.18), respectively. ¦
If x0 is a fixed point of/and/is differentiable at x0, then the number/'(x0)
is called the multiplier of x0.
Lemma 0.4. If E is a modulus set and fix) is a homeomorphism of E onto itself,
then strongly attractive fixed points of fix) are strongly repulsive fixed points of
f~1ix), and conversely.
Proof. Suppose that x0 is a strongly attractive fixed point of fix), i.e. that
@.15) holds in a neighbourhood U of x0. Then, putting in @.16) x=f~1(j), we
(') In earlier works attractive and repulsive fixed points were defined by means of relations
@.17) and-@.18) respectively and so they corresponded to what has been called here strongly attrac-
attractive and strongly repulsive fixed points. However, since we consider also non-diflerentiable func-
functions, the definition had to be modified. (Cf. also Barna [2], Sarkovskil [10].)

2. Attractive and repulsive fixed points 19
obtain for у ef{U)
i.e.
1
Since /(x) is a homeomorphism, f{U) is a neighbourhood of/(хо) = хо.и
Let us note that a similar lemma for fixed points which are not strongly at-
attractive or repulsive is not true. This is immediately seen from the example of the
function/(x) = x, where every point x e E is a repulsive fixed point. The following
example is less trivial. Let E= {x: |x| = l} be the unit circle on the complex plane
and define /by
/(x) is a homeomorphism of E onto E and has a unique fixed point xo= I, which
is attractive both for / and for /-1.
Definition. If x0 is a fixed point of/(x), we denote by Af{x0), or shortly by
A{x0), and call the attractive domain or domain of attraction of x0, the set of those
x e E for which lim /"(x) = x0. (Let us note that this notion is again realtive to E.)
n->oo
Evidently x0 e A (x0). The condition that x0 should be an attractive fixed point
°f / (•*) is equivalent to the statement that x0 is an inner point of A (x0).
Lemma 0.5. If xeA(x0), then С(х)<=Л(х0).
Proof, v e C(.v) implies the existence of non-negative integers m, к such
that fm(y)=f\x). Hence lim f\y) = lim /"+m0;)=lim /"+*(x) = lim /"(x) = xo, i.e.
П-ЮО П-ЮО П-ЮО П-Ю0
yeA(xo).m
We shall also prove the following
Theorem 0.3. (Fatou [5], Barna [2].) Let /(x) be a continuous function in a sub-
modulus set E and let xoe E be an attractive fixed point of f. Then Af(x0) is an open
set, submodulus for /(x).
Proof. Let x e A(x0) and let U be a neighbourhood of x0 which is contained
in Л(х0). Since lim/"(x) = x0, there exists an N>0 such that/^x) e U. The function
n->oo
fN is continuous in E, just like /, and consequently there exists a neighbourhood
V of x such that / (F)<=[/<= Л(х0). By lemma 0.5, VclA(x0), which means that
A(x0) is open. It follows from lemma 0.5 that x e A{x0) implies/(x) e A(x0), which
means that Л(х0) is submodulus for/. ¦
If x0 e ©*[/], A; > 1, then x0e6i [/*]. The point x0 is called an attractive, repulsive,
strongly attractive or strongly repulsive fixed point of order к of/(x) if it is such a
fixed point (of order 1) of /*(x).
§ 3. A real variable. Let / be an interval, by which we mean an open interval
(fl, b), or a semi-closed interval <a, b) or {a, by, or a closed interval (a, b}, a being
always <b. One or both endpoints of / may be infinite. I denotes the closure of /.

20
Introduction
By C[l\ (or shortly C") we denote the class of functions/(x) which have con-
continuous derivatives up to order n in /. C°[I] is the class of continuous functions
in /; C°°[/] is the class of infinitely differentiable functions in /. If an endpoint x0
of / belongs to /, then continuity or differentiability at x0 means one-sided con-
continuity or differentiability. We shall sometimes write "/(*) is of class C" instead
of'/еГ".
Suppose that ? e /. By 5"[7] (or shortly 5|) we denote the class of functions
f{x) which belong to C[I] and fulfil the conditions:
@.19)
@.20)
(/(x)-*)(?-*)> 0 for xel,
for xel,
Further, we denote by R"[I] (or shortly R") the class of functions/(x) which belong
to 5|[7] and are strictly increasing in I. (Fig. 1 represents a function belonging to
5?; fig. 2 represents a function belonging to R".)
Fig. 1
a,
Fig. 2
If ? = + oo, then in the definitions of classes 5„ and R"x condition @.20) is
superfluous and @.19) is replaced by/(*)>* for xel. Similarly, if ?= - oo, @.19)
is replaced by/(*)<* for xel.
Regarding the function classes just introduced we have the following obvious
inclusions:
,@.23)
<0.24)
and
whenever I2 <= It, ? e ii n J2.
We have the following further properties.

3. A real variable 21
Lemma 0.6. If /eS"[/], then I is a submodulus interval for f
This results from @.19) and @.20).
Lemma 0.7. IffeS"[I], then ? is a fixed point off
This results from @.19), @.20) and @.22). Moreover, ? is then an attractive
fixed point of /. But we can prove something more (*).
Theorem 0.4. IffeS\[I], then for every x0 e / the sequence f (x0) is monotonic
(and strictly monotonic whenever x0 Ф ?) and
@.25) lim/"(xo) = ?.
n-*co
Proof. For xo = <i; @.25) is trivial. Suppose that хо>? (if xo<? the argument
is similar). It follows from the definition of the class 5" that ? </"(x0) < x0 and
f(xo)<f~1(xo) for «=1,2,3,... Consequently the sequence f(x0) is strictly
decreasing and convergent. Since, according to @.19) and @.24), f(x)^x for
x e (<l;,Xo)<=/, we obtain @.25) in view of theorem 0.2. ¦
Lemma 0.8. If g e R°[I] andheR°4 [I] and, moreover, (g(x)-h(x))(?-x)>0 for
хе1,хф?, thenf(x)=g[h~\x)] belongs to R°4[h(I)l
Proof. The function f(x) is continuous and strictly increasing in h{I) as a
superposition of continuous and strictly increasing functions. In order to prove
@.19) suppose that x e h(J) and, say, ?<x. Then h~1(x)el, h~1(x)>?, and con-
consequently д[h~x(x)]<h[h~г(х)] and
Inequality @.20) is obvious. ¦
We shall establish also the following
Theorem 0.5. Suppose that fe R\[I] and ? is an endpoint of I, ?,ф I. Then for
any xoel the interval <xo,/(xo)), resp. (/(.v0), xo>, contains exactly one point of
every orbit contained in I.
Proof. Let ? be the left endpoint of /, and let x be an arbitrary point of /.
We shall prove that the set C(x)n(/(x0), xo> consists of a single point.
We may assume that x</(x0), for otherwise x could be replaced by а у е C(x)
n(?,/(x0)), which exists in view of theorem 0.4. Now, since (?,/(xo)><=/(/),
f~\x) is defined. If/~1(x)</(x0), this argument can be repeated. Thus we have
two possibilities: either there exists an N>0 such that/~iy+1(x)</(x0) and/'^x)
>/(x0), or for every и>0 /""(a-) is defined and/~"(x)</(x0). The latter case, how-
however, is impossible, since the sequence /~"(x) would then be increasing and would
con verge to an n e (x,/ (x0) > n © i[f] = 0. Consequently, for an N > 0 we have/ ~ N +' (x)
</(x0) and f~N(x)>f(x0). Hence it follows that y1=f~N(x)e(f(x0),x0} and, of
course, y^eCix). If there were а у2фу\ belonging to (/(x0), xo>nC(x), there
(i) Cf. e.g. Schauffler [1], [2], Andreoli [4], Hamilton [2]. Thijsen [1], Bajraktarevic [1],
Kuczma [2].

22 Introduction
would exist an integer к such that y2 =/\у\)- We may assume that к >0, for otherwise
it is enough to interchange yr and y2. But then y2 =fk(yi) ^fk(x0), and consequently
Уг Ф (f(xo)> -*o>> contrary to the assumption.
If ? is the right endpoint of /, the argument is similar. ¦
We shall introduce some more function classes. The divided differences [х1г ...,
xn+1;/] of a function/(x) for distinct points xt, ..., xn+1e I axe denned by
[xi;/]=/(xi)>
[xl9 ...,xn+1 ;/] = , и = 1,2,...
(cf. Norlund [1], Popoviciu [1]).
We denote by M" [/] resp. Ml [I] the class of functions/(x) such that
@.26) [х1,...,хв+
resp.
@.27) [», xl+
for any distinct x,, ..., xn+2 e/. We put M"[/] = M
Functions of class M"[I] will be called convex of order n in /. Functions of
class МХ[Г\ will also be called shortly convex^) whereas M°[I] is the class of mono-
tonic functions. A function/(x) which fulfils @.26) or @.27) with a strict inequality
> or < will be called strictly convex of order n {strictly convex, strictly monotonic).
Convex functions can alternatively be defined by the inequality
@.28) /(Ax1+(l-A)x2)<A/(x1) + (l-A)/(x2), xt, x2el, Лб@, 1),
or
@.29) /(Ax1 + (l-A)x2)>A/(x1) + (l-A)/(x2), xt, x2el, Ae@, 1).
Relation @.28) defines the class M+[/]; relation @.29) defines the class М1[Г\.
The most important properties of functions convex of order n may be found
in Popoviciu [1]. Here we mention only the following:
Lemma 0.9. If I is an open interval, then M"[/]cC"-1[/] for n=\, 2, ...
Lemma 0.10. ///eM^nC1!/], n>\, then f еМ'-'Д and conversely, if
fe CY[I] and f e M"'^!], then fe Mn[I].
Lemma 0.11. The class M"+[I]nM"_[I] consists of polynomials of degree ^n.
Let A"hf(x) denote the nth difference of the function/(x) with the span h>0:
@.30) A°J{x)=f{x), A"h+1f(x) = A"J(x + h)-A"hf(x), « = 0,1,...
and let/й = {x: x,x + hel}.
Lemma 0.12. IffeM"[I], n>\, then A\fe М"-1[Д]-
(L) We shall thus make no distinction in the terminology between convex and concave func-
functions. All those functions will be called convex.

3. A real variable 23
Most of the elementary properties of functions of class M" can be derived
from the well-known properties of convex functions by the use of lemmas' 0.9
and 0.10.
A function/(x) is called completely monotonic in an (open) interval /if/e C°°[/]
and ( — l)r~1/(r)(x)>0 in / for r=\, 2, ... A completely monotonic function in /
is analytic in / (cf. Widder [1], Chapter IV).
Apparently similar to classes M" are classes M"{f} of functions {J"}-convex
of order n. Let/(x) be a function of class R$[I], ?$ I. We define the {/}-differences
of a function F(x) on / by the relations
@.31) A°{f)F(x) = F(x), A"{;IF(x) = r{
For f(x) = x + h the {/}-differences A^F reduce to the usual differences A\F.
We denote by M"+ {/} [/] the class of functions/(x) such that
for xel,
and by Ml{/} [/] the class of functions F(x) such that
^^ for xe/.
We put M"{/} [7]=M?{/} [7]uMl{/}[/], and we call the functions of class
M"{f}[I] {f}-convex of order n in /. Functions of class M1 {/}[/] will also be
called shortly {f}-convex, functions of class M°{f} [/] will also be called shortly
{f}-monotonic (*). It is obvious that
@.32) M°McM°{/}H
for every function /e R^[f\, ? $ I. Moreover, if ? is the right endpoint of /, then
M?[/]cM?{/} [/], Ml[I]cMl{f} [/], if ? is the left endpoint of /, then M° [/]
cMl{f} [I], M°[/]cM°{/} [/]. For и>1 there is no such relation between the
classes М"[Г\ and M"{f) [/] (Brydak [1], [3]), except for the case where f(x) = x + h,
h>0. Namely, we have
(a33) M"+ l(a, oo)] с M\ {x + ft} [(a, oo)] ,
M"_ {{a, oo)]cM". {x + h} {{a, oo)] .
To end this section we define the class H as the smallest family of functions
which contains the constant functions/(x) and the identity function/(x) = x, and
which is closed under the rational operations and the application of exp ( ) and
log | |. The class H is called Hardy's logarithmico-exponential scale (Hardy [2], [4])
and its members are called shortly L-functions.
Let us write e(x) = ex—\. G. H. Hardy [2] proved the following result.
Lemma 0.13. To every L-function fix) which tends to infinity as x-»oo there are a
uniquely determined positive real number ц and non-negative integers r, s such that
Anastassiadis [1], Montel [14], Kuczma [6], Brydak [1], [3].

24 Introduction
for every 5,0<8<ц,
@-34) e'[{e-\x)y-^f{x)^eri(e
where f(x)-^g(x) means lim/(x)/g(x) = 0.
Х-юо
An L-function/(x) fulfilling @.34) is said to be of type (r, s, /*)•
Finally, let us note that every /.-function is monotonic for x sufficiently large
(Hardy [4]).
§ 4. A complex variable. Now x is a complex variable and we consider entire
functions/(x). As is well known, there may exist at most one value ?, which is as-
assumed by/only at a finite number of points {Picard's exceptional value), fix) may
then be written in the form f(x)=? + P(x)eeM, where g(x) is an entire function
and P(x) is a polynomial. If, in particular, P(x) = (x — ?)v, v>0 being an integer, i.e.
@.35) /(x) = ?+(x-0Vw,
then ? is called Fatou's exceptional value.
The iterates/"(x) are also entire functions for и>0. We define $[f] as the
set of those points x of the complex plane about which the sequence {/"(x)} is not a
normal family. The point at infinity always belongs to 5- The following theorem
is proved in Fatou [15]. For the proof the reader is referred to the original paper.
Theorem 0.6. Let f(x) be an entire transcendental function. Then:
(i) 5 [/] is a non-empty perfect set containing all strongly repulsive fixed points
of f and fixed points of multiplier + 1 (of any order), but it does not contain any strongly
attractive fixed points (of any order) of f.
(ii) 5 If] is contained in the closure of the set of all fixed points (of all orders) off.
(iii) Let A be a bounded domain which does not contain the Fatou's exceptional
value of f (if it exists) either inside or on the boundary, and let x0 e «?[/]• Then for
every neighbourhood U ofx0 there exists an N such that A <=/"(?/) for n^N.
(iv) For every xoeg [/] and every point y0 different from the Fatou's exceptional
value off there exist a sequence of points xm and a sequence of positive integers nm
such that
lim xm = x0 and lim f"m (xj = y0 .
(v) 5[/] is a modulus set for f
We shall also prove a lemma concerning the maximum modulus functions:
Mf(r)= max |/(x)|= max |/(x)| .
|*|=r |x|<r
By the maximum principle Mf(r) is an increasing function on <0, oo).
Lemma 0.14. (Baker [2].) Let f(x) be an entire function and suppose that there
exist positive constants c, d and an integer n > 1 such that
@.36) Mf(cr")<dlMf(r)Y.
Thenf(x) is a polynomial.

4. A complex variable 25
Proof. Put t^logr, V(t)=logMf(et), a=logc, h = \ogd. By a theorem of
Hadamard V(t) is a convex function. Now, @.36) becomes
@.37) V(nt+a)<nV{t) + b.
We shall show that there exist constants А, В such that
@.38) V(t)<At+B
for t sufficiently large. Let us fix a fo>max@, —aj{n—Yj) and numbers А, В
so that
@.39) Ato + B>V(to),
@.40) Aa-{n-l)B>b .
Such numbers exist since a + (n— 1) t0>0, and consequently the straight lines Ato + B
= V{to)> Aa — (n—l)B=b divide the (A, B)-plane into four parts in one of which
inequalities @.39) and @.40) hold.
We define a sequence tk recursively by tk+l=ntk+a, к=0, 1, 2, ... Since for
t> —aftn—l) we have nt+a>t, the sequence tk is strictly increasing and tends to
infinity. We shall prove that
@.41) V(tk)<Atk+B
for k = 0, 1,2, ... For k = 0 @.41) is identical with @.39). Assuming the validity
of @.41) for a certain k, we have by @.37) and @.40)
V(tk+i)<nV(tk) + b<nAtk + nB + b = A(ntk + a) + nB-Aa + b<Atk+i + B.
Because of the convexity of V{t) @.38) holds for all t^t0. This means that Mf(r)
<eBrA for r sufficiently large. By a theorem of Liouville/(x) must be a polynomial. ¦
Finally, let us record the following results of J. Clunie [1] (cf. also Polya [1]):
Lemma 0.15. Let f{x) and q>{x) be entire functions and puth {x) = q>[f{x)]. Then
there exists a number 0, O<0< 1, such that
holds for r sufficiently large.
For the proof the reader is referred to the original paper by J. Clunie.
§ 5. Functional equations. Strictly speaking, a functional equation is an equa-
equation in which the unknowns are functions. Nowadays, however, this expression is
seldom used in this sense. Usually it is understood that, say, differential or integral
equations form independent branches of mathematics and are not included in the
notion of a functional equation. As regards differential equations with a retarded
argument or operator equations, mathematicians are not unanimous; such equations
are often referred to as functional equations. In the present book, however, we shall
adopt a modern definition of a functional equation (Aczel [5], [7], Kuczma [13],
[24], [28]) which excludes all the above mentioned types of equations.

26 Introduction
Roughly speaking, a functional equation is an equation between two expres-
expressions which are built from a finite number of functions (known and unknown) and
variables by a finite number of superpositions. To be able to give a precise definition,
we must define the concept of a term.
Definition. A term is defined by the following conditions:
(i) Independent variables are terms.
(ii) If fi,..., tp are terms and/(x!, ..., xp) is a function of p variables, then
f(t1,..., tp) is also a term.
(iii) There exist no other terms.
A functional equation may now be defined as follows (Aczel [5], [7]).
Definition. A functional equation is an equation between two terms which con-
contain at least one unknown function and a finite number of independent variables.
This equation should be satisfied identically with respect to all the variables oc-
occurring in it in a certain set (of any sort).
The notion of a functional equation as defined above does not comprise equa-
equations in which infinitesimal operations are performed on the unknown functions.
Thus, for instance,
<p"(x)=-<p(x)
(the differential equation of the sine and cosine),
<p'(x) = eax+b<p(x-l)
(the differential-difference equation considered by N. G. de Bruijn [2]),
(p{x)= max [g (y) + h(x- y) + y> (ax + b(x-y))~]
0<y<x
(the equation occurring in the theory of dynamic programming; cf. Bellman [4])
are not functional equations in the sense of the above definition. The following
are typical examples of functional equations: Cauchy's equation (cf. Aczel [5], § 2.1)
@.42)
the equation of associativity (cf. Aczel [5], § 6.2)
@.43) q>\x,(p{y,z)']=(p\(p(x,y),z],
the equation of the Gamma function (cf. Chapt. V, § 10)
@.44) <p(x + l) = x<p(x),
the equation of involutory functions (cf. Chapt. XV, § 1)
@.45)
It should always be distinctly stated in what set a given functional equation
is postulated, for the solution may depend quite essentially on that set. E.g. the
only solution of equation @.42) in (-oo, oo) is q>{x) = 0, whereas in (-oo, 0)u@, oo)

5. Functional equations 27
equation @.42) has also the solutions
@.46) p(x) = clog|x|.
Similarly, it should always be distinctly stated in what function class the solution
is sought. The number and behaviour of solutions depends very strongly on that
class. E.g. @.46) is the general continuous solution of @.42) in ( — oo,0)u@, oo),
but equation @.42) has also discontinuous (and only non-measurable) solutions
in this set. This is one of the characteristic features of functional equations and we
shall be able to observe it throughout the whole of this book. No similar phenomenon
appears in the theory of differential equations, where the function class in which
the solution is sought is usually determined by differentiability conditions of the
unknown function.
In the present book we shall be concerned with functional equations in a single
variable A). This, however, requires an explanation. Equations @.44) and @.45)
are undoubtedly in a single variable. But the equation
apparently containing two variables, can be put in a form involving only one variable,
by setting z=(x, y). Such equations also belong to the subject matter of this book.
It can easily be seen that no substitution allows one to write equations @.42) and
@.43) as equations in a single variable.
There is an important difference between equation @.44) and equation @.45).
The former does not contain superpositions of the unknown function, while the
latter does. The most general equation (in a single variable) that does not contain
superpositions of the unknown function can be written in the form
@.47) F(x,9>(x)
An important special case of @.47) is that where the functions ft{x) are iterates of
the same function/(x): /;(x) =/'(x). The equation then has the form
@.48) F(x,q>(x),.q>[/(x)], <p[/2(x)] ,...,? [/"(x)]) = 0 .
For equations of form @.47) or @.48) we define the order B) as the number и occur-
occurring there (cf. Ghermanescu [5], [22]). This definition is not precise and not quite
unequivocal; however, our use of it will involve no ambiguities.
Thus equation @.44) is of order 1. For equation @.45) we do not define an
order, though it is possible (cf. Kuczma [24], [28]). An equation of order zero has
(>) Equations in several variables, like @.42) and @.43), have been extensively discussed in
Aczel's monographs [5], [7].
B) Since in the present book we are concerned with a particular class of functional equa-
equations, viz. equations in a single variable, we do not think it necessary to introduce a fuller classifica-
classification, the more so as none of the existing ones seems satisfactory (cf. Kuczma [13], [24], 28]).

28 Introduction
the form
F(x,p(x)) = 0,
i.e. is an equation of implicit functions. Such equations will not be dealt with in
the present book, we shall confine ourselves to equations of positive order A).
Most of the book will be devoted to equations without superpositions of the
unknown function. In the first ten chapters we shall study the equation of order 1
@.49) F(x,(p(x),(p\J(xj\)=0
and its particular cases. In Chapter XI we shall be concerned with functions which
satisfy simultaneously several equations. Equations of higher orders and systems
of equations will be dealt with in Chapters XII and XIII. The last two chapters are
devoted to the study of equations containing superpositions of the unknown function.
B) Some special equations of order zero are treated in Euler [3], Schwering [1], Luxenberg
[1], Flechsenhaar [1], Schimmack [1], Caresse [1], Beardon [1] (the equation x^x) = [(p(x)]x);
Wavre [1], [2], Homma [1], Sikorski, Zarankiewicz [1], Lipinski [1], Charzynski [1] (problems of
uniformisation); Valiron [3], Myrberg [5], Polya [2], Ganapathy Iyer [1], Gross [1], Baker [10]
(some equations for analytic functions, related to the Pythagorean theorem); Sierpinski [5],
Malchair [1], Braun [1], Mioduszewski [1].

CHAPTER I
GENERAL SOLUTION
§ 1. Formulation of the problem and preliminaries. In this and the next few
chapters we shall be concerned with the general functional equation of the first
order @.49) and with its special cases. In the sequel we shall assume that equation
@.49) can be written in one of the following two forms:
A.1) ?[/(*)] = Sf(x, ?(*)),
A.2) (p{x) = h(x,(p\_f(xj\).
In this chapter we describe the construction of the general solution of equation
A.1) in the class Ф of functions <p(x) defined in a set E and assuming values in a
set E. We make no restrictions concerning the nature of the sets E and E. However,
we make the following assumptions regarding the functions/and g.
Hypothesis 1.1. The function f(x) is defined and invertible in E, furthermore
@.1) holds.
Hypothesis 1.2. The function g{x,y) is defined in a set QcExS and assumes
values in S. Moreover, for every fixed x в Got/] the function g{x,y) considered as a
function gx(y) of у alone is invertible in the set
A.3) Qx={y: (x,y)sQ}.
Let us denote by Fx=gx{Qx) the set of values of the function g(x, y) for у в Qx
and let us write
Г= U {х}хГх.
xe®o
According to hypothesis 1.2 there exists a unique function g ~ 1(x, y) inverse to g (x, y)
with respect to y, defined in Г. Now we define a sequence of functions gn(x,y)
by the relations
A.4)
9o(x,y)=y,
= g(f"(x), д„(х, y)), n = 0, 1,2, ... ,
= д-\Г1(х),дп(х,у)), п = 0,-1,-2, ...

30 CHAPTER I. General solution
The functions gn(x,y) are defined in the sets U(x}x^" f°r n^O, and in
xeE
\J {x} xQ"forn<0, where
xef—(E)
A.5)
и=-1, -2,...,
In formulae A.5) Sfn-i,xO)=Sfn-i(x, ;0, 9n+uJj)=9n+i(x, y), and йг"-1!,^ ),
9n+i,x( ) denote the counter-images of the corresponding sets; so these symbols
have a meaning whether the functions #„_! and gn+1 are invertible or not. It may
happen, of course, that the domain of definition of the functions gn is empty from
some n onwards.
Lemma 1.1. Let hypotheses 1.1 and 1.2 be fulfilled. Ifn^O, m^O, or n^O, m^O,
then
A-6) gn+m(x,y)=9nlfm(x),9m(x,y)~],
provided that one of the above expressions is defined A).
Proof. We shall prove A.6) for и>0,т>0; the proof in the other case is
similar. For и=0 relation A.6) is obvious. Suppose that A.6) holds for an и>0
and all m>0 for which at least one of the expressions is defined. Suppose further
that gn+m+1(x, y) is defined. By definition
gn+m+1(x, y)=g (fn+m(x), gn+m(x, y)) ,
where gn+m(x, y) must be defined. Consequently,
A.7) 6fn+m+1(x, y)=g(f"Um ML gJLT W. 9m(*,y)D,
and since the right-hand side of A.7) is defined, we obtain by definition
A-8) gn+m+i(x,y)=gn+1(fm(x),gm(x,y)).
If we suppose that gn+i(fm(x),gm(x,y)) is defined, then arguing backwards we con-
conclude that gn+m+1(x, y) is defined and A.8) holds.B
§ 2. Construction of the general solutionB). Now we assign to every xeE a set
V[x] cS. If x e CEfc, ?> 1, then V[x] is the set of у е Е fulfilling the equation
9k(x,y) = y.
If x e Goo, then
A.9) V[x\= П^-
(J) That is: if one side od A.6) is defined, then the other is also defined and the two are equaL
B) Kuczma [9]; cf. also Lecornu [1], Andreoli [3], Popovici [2], [3], [4], [10], [11], Tambs
Lyche [8], Pompeiu [1], [2], Baghi [1], Ghermanescu [4], [5], [7], [17], [22], Gersevanov [1], [3],
[4], Kitamura [1], Valeiras [1], Pellegrino [1], Hadwiger [1], Novotny [1], [2], van der Berg [1],
Aczel [4], Isaacson [1], Kordylewski, Kuczma [1], Kuczma [4], [30], [32], Weaver [1], Sarkovskii
[3], Mitrinovic [1], Reghis, Vuc [1], Lasota, Pelczar [1].

2. Construction of the general solution 31
Lastly, if x e ©o l > then there exists a unique integer / > 0 such that x efl{E) —f+* (E).
For such an x we put
., + °°
A.10) V[x]^ flfll.
Lemma 1.2. Z-e? hypotheses 1.1 аис? 1.2 be fulfilled. If a function q> еФ satisfies
equation (l.l) in E, then for every x0 e E
<p(xo)eV[xo].
Proof. Suppose that for a certain integer n f"(x0) is defined. From A.1) and
A.4) it follows (by induction) that
A.11) 9>[Г(хо)]=дя(хо,фо)).
Now we must distinguish three cases.
1. x0 e(?k,k>l. Then/*(xo)=xo and we obtain by A.11)
о, <p(x0)) ,
i.e. <p(x0) 6 V[x0].
2. x0 e ©oo- Then/"(x0) is defined and belongs to E for every integer n. Thus
<p[f\x0)] is defined, and consequently also the right-hand side of A.11) must be
defined for all n. Thus q>(x0) must belong to the domain of definition of д„(х0, у)
for every n, i.e. <р(хо)е(^] Qnxo=V[xo].
— GO
3. x0 e g01 and x0 efl(E)-fl+1(E). Then /"(x0) is defined and belongs to E
for every integer и> — / and we argue further as in case 2. ¦
Corollary. The relation
A.12) . K[x]^0 /or xeE
is a necessary condition of the existence of a solution of equation A.1) in E.
Further we shall prove (theorem 1.1) that condition A.12) is also sufficient.
Lemma 1.3. Under hypotheses 1.1 and 1.2 the set
A.13) ?* = {x: K[x]^0}
fa a submodulus set for the function f(x).
Proof. It is enough to prove that if x0 eE*, then also/(x0) sE*. We must
distinguish three cases.
1. xoe(Sb^l. There exists a y0 e J2*o cfiJo such that
A-14) вЛхо,Уо) = Уо-

32 CHAPTER I. General solution
Putting у1=д(хо,Уо) and making use of A.14), A.4) and of lemma 1.1 we get
Ух =9 Oo, 0ft(*o. УоХ\=9 [/*(*<>)' 9k(xo, УоУ] = 9к+1(хо, yo)
o) ,g(x0, yoy]=gk\_f(xo), у Л ,
which proves that y± e V[f(x0)]. Thus/(x0) e E*.
2. x0 e ©oo ¦ There exists a y0 such that gn(x0, y0) is defined for every n. Put-
Putting yi=g(xo,yo) we find on account of lemma 1.1 and of the invertibility of the
function g that the function gn-i(f(x0), y^) is defined for every n. This means that
3. x0e<S0l,x0ef\E) -f1+1(E). There exists a y0 such that gn(x0,y0) is
defined for every «>-/. Consequently, 0B-i(/(xo)» J>i) where y1=g(x0,y0), is
defined for every и> —/, i.e.
A-15) *e П ^(xo)-
But/(x0) e//+ \E)-f1+2(E), and so A.15) means that j^ e V[f(x0)] and/(x0) e ?*.¦
Lemma 1.2 says that seeking a solution of equation A.1) we must restrict our-
ourselves to the set E* in which V[x]^0. Lemma 1.3 guarantees that if hypotheses
1.1 and 1.2 are fulfilled in the set E, then they are also fulfilled in E*. Therefore
in the sequel we may assume that У[х]ф0 for xe E.
We denote by V[Et] the class of functions q>{x) which are defined in E^<^E
and are such that for every x0 e E1 <p(xo)e V[x0]. By lemma 1.2 every solution
q>(x) of equation A.1) in E belongs to the class V[E\ Further, (p{x) is uniquely
determined by its value ^(x0) on the whole orbit C(x0). But there are no further
restrictions, i.e. any value y0 e V[x0] determines a unique solution <p{x) on C(x0)
such that <р(хо)=Уо- Namely, we have the following
Theorem 1.1. Suppose that hypotheses 1.1 and 1.2 are fulfilled and V[x]^0
for x e E. Let A be an arbitrary set which has exactly one point in common with every
orbit contained in E (x) and define the function a(x) by
A.16) a(x)=AnC(x).
Then for every function q>0{x) belonging to the class V[A] there exists exactly one
function q>{x) which belongs to the class Ф, satisfies equation A.1) in E and fulfils
the condition
A.17) pW=?oW far xeA.
This function is given by the formula
A.18) <p(x) = gn(a(x),<Pola(x)]),
С1) The existence of such a set results from the axiom of choice.

2. Construction of the general solution 33
where n is chosen in such a manner that
A.19) /" [>(*)]=*
00
and, moreover, n^Ofor x e \J &k.
k = l
Proof. We must prove that relation A.18) actually defines a function (p{x)
in the whole of E and that this function satisfies equation A.1) in E. Let us fix an
xeE. Since a(x)eC(x), there exists an integer n such that A.19) holds. We shall
distinguish two cases.
1. xe<Sk,k^l. Suppose that A.19) is fulfilled by an integer n. Then the rela-
relation fp[a(x)]=x holds for all p=n+rk, r=0,±\,±2, ..., and in view of lemma
0.2 only for these. Thus we may assume that Q^n^k— 1. From the definition of
the sets V[u] it follows that F[d(x)]cfflJ(j), since by theorem 0.1 а(х)е61. In
view of A.5) we obtain hence
K[e(x)]c=fl^(x) for j=O,...,k,
and in particular
K[e(x)]<=fl^x).
Consequently q>0[a{x)] efijw, i.e. the expression gn(a{x), q>o[a{x)\) has a meaning.
Let us note also that, since n^k-l, F[a(x)]cO"(+I andbyA.5) 9o[a(x)]eg~J(x)(Qx),
i.e.
A.20) <p(x) = gn(a(x), <po[a(x)-])eQx .
Now, for p=n + rk and r>0 in view of lemma 1.1 we have
gn(a(x), <PoLa(x)J)=gn(fkLa (x)],?0 (/" [>(x)]))
= 9n+rk(a(x), (po\_a(x)\)=gp(a(x), <po[a(x)]),
sinpe a(x) eSt and (po[a{x)] e V[a(x)]. Thus function A.18) is unambiguously de-
defined, mdependently of the choice of an и^0 fulfilling A.19). Now we shall prove
that A.1) holds.
We have/(x)=/"+1[a(x)], whence (*)
V [/(*)] = 9n+ i(a (x), 4>0\_a (x)]) ,
i.e. according to A.4), A.18) and A.19)
?[/(*)] = 9(ГИ*)], ф{х))<ро1а{х)\) = д{х, <p(x)) .
2. xeS0. Then, by lemma 0.2, n fulfilling A.19) is unique (positive, negative
or zero). If x e Goo, then by theorem 0.1 a(x) e ©Oo and by A.9)
A.21) K[fl(x)]cfi;w.
C1) The previous argument shows that formula A.18) is meaningful and defines the func-
function q>(x) for any x sQjj.. Since for x e Qjj. also f(x) e Qjj., this shows that p [/(*)] is defined.
3 Functional equations

34 CHAPTER I. General solution
If x e ©oi» then a(x)eGOi and there exists a unique integer />0 such that a(x)
=/~"(x) ef\E)-fl+\E). It follows that -n^l, i.e. и>-/and in view of A.10)
relation A.21) holds as well. Consequently, q>0[a{x)] e Q"a(x) and the expression
9n(a(x)> <Po[a(x)]) has a meaning. Further let us note that by A.9) and A.10) V[a(x)] с
сйЦЛ and by A.5) (po[a{x)] eg~tl(x)(Qx) and A.20) holds also in this case.
To prove that A.1) holds, let us note at first that the relation
implies the relation
A-22) 9n+1(x,y)=g(fn(x),gn(x,y)),
which shows that the latter is valid provided that both sides of A.22) are defined
(whether n is positive or not).
In view of A.19) we have/(x)=/"+1[a(x)] and by A.18)
q>{x) = gn{a{x),(p0[_a(x)§,
<P [/(*)] = gn +1 {a (*), <Po\? (*)]) •
Hence, according to A.22) and A.19)
g(x,<p(x))=g(x, gn(a(x), (po[_a(x)]))
since both these expressions are defined (g(x, q>(x)) in view of A.20) and (p[f(x)]
in view of the earlier part of the proof, which is valid for any x e CEq , and of theorem
0.1).
The solution obtained fulfils condition A.17), since for xeA we have a(x)
= x, i.e. in A.19) we may take n=0. To prove the uniqueness, let us suppose that
q>(x) is an arbitrary function belonging to the class Ф, satisfying equation A.1)
in E and fulfilling condition A.17). Let us fix an arbitrary x e E and an integer n
such that A.19) holds. Then by A.11) and A.17) we have
q>(x) = <p{fn[a(x)]) = gn(a(x), q>[a(x)]) = gn(a(x), <po\_a(x)]),
which means that (p{x) coincides with the function A.18).и
Remark. In order to obtain the set A occurring in theorem 1.1 we had to
resort to the axiom of choice. In most cases, however, such a set can be constructed
effectively (cf. e.g. theorem 0.5).
The conditions of theorem 1.1 may be slightly weakened by dropping the
condition of the invertibility of g (x, y) for x e (?oi • Thus hypothesis 1.2 is replaced
by the following

2. Construction of the general solution 35
Hypothesis 1.3. The function g{x,y) is defined in a set QcExS and assumes
values in E. Moreover, for every fixed xe&00[f] the function g(x,y) considered
as a function gx(y) of у alone is invertible in set A.3).
Then also the sets V[x] must be replaced by modified sets Fmod[x] defined
as follows:
V[x] for
00 ^
П*? for xe(S01.
n = 0
Evidently V[x]cVmod[x] and consequently lemma 1.2 holds with Fmod[x] in place
of V[x]. The function class Vmou[Ex], Ex<=lE, may be defined just like ?[EX] with
the sets V[x] replaced by Fmod[x].
Let В be an arbitrary set containing exactly one point of every orbit contained
in E and put
A.23) A=(?-/(?)) и (В п П /"(?)) •
n=0
It is easily seen that the set A thus defined also has exactly one element in common
with every orbit contained in E. Consequently, formula A.16) defines a function.
Theorem 1.2. Suppose that hypotheses 1.1 and 1.3 are fulfilled and Fmod[x]^0
for x e E. Let В be an arbitrary set which has exactly one point in common with every
orbit contained in E, and define the set A and the function a(x) by A.23) and A.16),
respectively. Then for every function (po{x) belonging to the class Vmou[A\ there exists
exactly one function q>{x) which belongs to the class Ф, satisfies equation A.1) in
E and fulfils condition A.17). This function is given by formula A.18), where n is chosen
00
so as to fulfil A.19) and, moreover, n>Ofor xe (J <5k.
k=l
The proof of the above theorem does not differ from that of theorem 1.1. The
essential point is that, according to A.23), for xef01 we have a(x)e E—f(E)r
i.e. /=0.
§ 3. Non-invertible / (x). Since by lemma 1.2 every solution of equation A.1)
in E restricted to the set A belongs to the class VIA], theorem 1.1 gives the general
solution of equation A.1) in E, essentially under the following three assumptions:
1. E is a submodulus set for f(x).
2. The function f(x) is invertible in E.
3. The function g(x,y) is invertible with respect to у for every x e (?0[/].
The first of the above conditions seems quite natural, the remaining two are
restrictive. If the function g(x,y) is not invertible, the general solution of A.1)
can be obtained in a similar way (cf. Kuczma [9]). The case where /(x) is not invert-
invertible is more complicated. In the present section we shall give an analogue of theorem
C1) Kuczma [30]. Cf. also Tambs Lyche [8], Reghis, Vuc [1].
3»

36 CHAPTER I. General solution
1.1 without the assumption of the invertibility of/(x), but with hypothesis 1.2 re-
replaced by a stronger one. Thus in what follows we shall make the following as-
assumptions regarding the functions / and g.
Hypothesis 1.4. The function /(x) is defined in a submodulus set E.
Hypothesis 1.5. The function g(x, y) is defined in a set ExE and assumes values
in E. Moreover, for every fixed x e E the function g(x,y) considered as a function
9x(y) °f У alone maps E onto itself in a one-to-one manner.
Accordingly, there exists a unique function g~l(x,y) inverse to g(x,y) with
respect to y, defined in ExS. The function f(x) has no unique inverse function.
Let {f^ix)}, X e A, be the family of all functions inverse to fix), i.e. the family
of all functions that fulfil the condition
A.24) Я/Г'(*)]-* for xe/(?).
The condition/^ 1[f(x)]=x in general is not fulfilled.
We definefx"(x), n>0, as the nth iterate of the function/J1(x)./7"(x) is de-
defined in a set E\clE, which may be empty ('). For the sake of uniformity, we shall
sometimes write also /"(x) instead of/"(x). For non-negative n, /"(x) is indepen-
independent of X.
If /"(x!) = x2, n>0, there always exists an index X e Л (in general not unique)
such that fx"(x2) = xl. It is enough to put fxl[ft(xl)]=fl~1(x1) for i=\,...,n,
and then to extend/71 onto f(E) to a function fulfilling A.24).
Now we may define functions хд„(х,у) by formulae A.4), where/" and/"
are replaced by/" and/", respectively. For non-negative n, xgn{x,y) is independent
of X and is defined in ExE. For n<0, xgn(x,y) depends on X and is defined in the
л
set П E\xE, which may be empty as well.
;=i
Lemma 1.4. Let hypotheses 1.4 and 1.5 be fulfilled. If for an x0 e E and for an
n>0 fx-"(x0)=f-"(x0), then /Л*о)=/Л*о) for all /<«.
Proof. This results from the fact Ла1/[/я";(х)]=/я";+1(л:)(с^^гти1аA.24)).и
Lemma 1.5. Under hypotheses 1.4 and 1.5, if for an xoeE and for an n f"(xo) =
ffao), then xgn(x0, y)=llgn(x0, y) for у е Е.
Proof. This follows from lemma 1.4 and from the fact that for и>0 кда does
not depend on Хм
Similarly to lemma 1.1, also the following lemmas can be proved.
Lemma 1.6. Let hypotheses 1.4 and 1.5 be fulfilled. If n^O, m>0, or n^O, m<0,
then
*9n + m(x> У) = лдп[/т(х),х9т(х, У)] ,
provided that one of the above expressions is defined.
(i) E.g.if?=(-oo,0)u@, oD),f(x) = x2,tbenfJ1(x)=ex(x)y/7,xe@, oo), where [ex(x)P=l.
~ (
(The family of functions ex(x), and consequently also that of functions f~x (x), has cardinality 2C.)
For eXo(x)= -1, fi20(x) is not denned for any x e (- oo, 0)u@, oo), i.e. E\o=0.

3. Non-invertible / 37
Lemma 1.7. Let hypotheses 1.4 and 1.5 be fulfilled. If n^O and for an xoeE
and a X e Л we have fx"[fn(x0)] = x0, then
x9-n[f"(x0),gn(x0,y)]=y for yeE.
Now we can define the sets V[x]. If x e <&k ,k > 1, then V[x] is the set of у e Е
fulfilling the equation
where j=Jk(x) is the integer occurring in lemma 0.1. For хе&0 we put V[x] = E.
(If/is invertible, this definition agrees with that given in § 2 since then 7=0.) Rel-
Relatively to the sets V[x] so defined, the function class ^[Et], EtcE, may be de-
defined as in § 2.
Lemma 1.8. Let hypotheses 1.4 and 1.5 be fulfilled and suppose that for an x0 e E,
for some non-negative integers n, m, p, q, and for some parameters X, ц е Л we have
A-25) A"m[/Vo)]=/,"9[/PW]-
Then for every ye V[x0]
A.26) xg_m[/"(x0), д„(х0, y)]^^-,[/"(*<>), 0P(*o, JO] •
Proof. Let us suppose that we have
A.27) n — m = p — q.
Without loss of generality we may assume that />>и. Then we have also
and
/Г[/'(*„)]<"[/,"й"в)(/РЫ)] •
By A.25) we get hence
d-28) f-(q-m)UP(xo)-]=r(xo),
i.e.
/;![/p(xo)]=/;
and
On account of lemma 1.5 we obtain
A-29) x9-mU"(x0), дАхо,уУ\=„д_п1Г(хо), дп(х0,у)\ .
Now by lemma 1.6 and by A.28) we have
»9-qU"(xo), gP(x0, У)] =„0-*,(/„"("-m) [/"(*<>)], „0-(*-m) [/"(*<>), gP(x0,
= !ig-m(fn(x0), „g-^-m) U"(XO), 9p(Xo , УI) •
Applying lemma 1.6 again we obtain in view of A.27)
9P(x0, y)=gp-n[f\x0), gn{x0, y)] = gq-m\f(x0), gn(x0, y)].

38 CHAPTER I. General solution
Further we obtain from A.28)
/P(*o)=/9~m [/"(*<>)]
and
/м-(9~т)[/9~т(/"(*о))] =/"(*<>)•
Hence by lemma 1.7
Mg-(e-m)[/p(x0), др(х0,УУ]
=ltg-(q-m)(fq~m[/"(*<>)], gq-n[/"(*<>), gn(x0, y)J)=gn(x0, y),
and finally
,^-i [/P0o)> 0P(*o, УI =M9-m [/"(*o)> gn(x0, УУ] .
whence in view of A.29) we obtain relation A.26).
Now let us suppose that A.27) does not hold. Then by lemma 0.2 there exist
integers к > 1 and гФО such that x0 e &k and
A.30) (n-m)-(p-q) = rk.
Without loss of generality we may assume that r>0.
Let us puty=y^(x0) and let us choose an index сое Л such that
A-31) f~q [Л*,,)] =/
Since p — q=(p+j) — (q+j), we have on account of the first part of the proof
A.32) „g-qU"(xo), gP(x0, y]\=a,9-(f?j) Up+'(xo)> gP+j{x0, y)~\ ¦
Now we shall prove that for every integer s^Owe have
A-33) gP+j+sk(x0 > y)-gP+j{x0, y).
For j = 0 relation A.33) is trivial. Suppose it true for an s>0. We have by
lemma 1.6
A.34) 9P+j+(,+ i)k(x0, y) = gp+SkUJ+k(xo), 9]+к(.Хо,уУ] ¦
But since;=Л(х0) and ye V[x0], we have/y+*(x0) =f\x0) and gJ+k(x0,y)=gk(x0,y).
Thus applying again lemma 1.6 and making use of the induction hypothesis we
obtain from A.34)
(x0, y)=gP+Sk UJ(xo)> gj(xo, x)]
y)=gP+j{xo, y),
which proves that A.33) is valid for all integers
Sincey'=.4(x0), we have according to lemma 0.1
A-35)

3. Non-invertible / 39
and hence
A.36) f
Relations A.25), A.31) and A.36) give
f-mffti/ \-\_f-(q + j)r fp + j + rk/ 4-1
Jx и \xo)l —/ю и \хо)л
Now, by A.30) we have
and in view of the first part of the proof
A0-m[/"(*o), 9n{x0, y)l = a>g-(q + j)UP + J + rk(Xo), 9P + J+rk(x0 , УI ,
whence by A.35), A.33) for s=r, and A.32) we obtain relation A.26).и
The following two lemmas are analogues of lemmas 1.2 and 1.3, but they re-
require independent proofs according to the new definition of the sets V[x].
Lemma 1.9. Let hypotheses 1.4 and 1.5 be fulfilled. If a function (p{x) belonging
to the class Ф satisfies equation A.1) in E, then for every xoeE <p(x0) e F[x0].
Proof. Suppose that for a certain integer n (positive, negative, or zero) and
a certain X e Л fl(x0) is defined. Then in view of A.1) and A.4) (induction!)
A.37) <p [Л"(х0)] = хфо, 9 (xo)) •
If x0 e <gt, fc> 1, and ;=Л(х0), then of course q>[fJ+k(x0)]=<p[fJ(x0)] and by A.37)
9J+k(xo, 9 (xo)) = 9j(xo, <P CO)],
which means that q>(x0) e V[x0]. If x0 e Go» the lemma is trivial.¦
Corollary to lemma 1.2 is also valid in this case.
Lemma 1.10. Under hypotheses 1.4 and 1.5 the set A.13) is a submodulus set for
the function fix).
Proof. Suppose that for an xoeE we have V[xo]^0. If xo e(?o> then this
means that ЕФ0, which in view of theorem 0.1 implies V[f(xo)]=3^0. If x0 е&к,
к>1, then there exists a y0 such that
A-38) д}-+к(хо,Уо) = 9Ахо>Уо)>
where j=Jk(x0). We shall prove that yi=g(xo,yo) belongs to V[f(x0)]. We shall
distinguish two cases/
(a) 7=0. Then by lemma 0.1 we have fk[f(x0)]=f(x0), which shows that
Л[/(хо)]=0. Further, according to lemma 1.6 and relation A.38) (with;=0)
9kU(xo) Ji] = 9kU(xo) ,9i(x0, УоУ] = 9k+ i(*o > Уо)

40 CHAPTER I. General solution
(b);>0. Then, by lemma 0.1 /J-1+*№o)]=/y[Axo)] and for/</-l,
fJ+k[f(xo)]^fJ[f(xo)]- Consequently, Jk[f(x0)]=j-l, and by lemma 1.6 and rela-
relation A.38)
0;-i+?/(*o)> yil=9j-i+k[f(x0), 9\(xo, УоЯ = 9]+к(хо, У о)
i.e. у у e V[f(xo)].m
Theorem 1.3. Suppose that hypotheses 1.4 and 1.5 are fulfilled and ?[х]Ф0
for x e E. Let A be an arbitrary set which has exactly one point in common with every
orbit contained in E, and define the function a(x) by A.16). Then for every function
<Po(x) belonging to the class V[A\ there exists exactly one function <p(x) which belongs
to the class Ф, satisfies equation A.1) in E and fulfils condition A.17). This function
is given by the formula
A.39) »W=rf-,(/"[e(x)],ff.[eW )
where the integers n,m^0 and the index Xe A are chosen in such a manner that
A.40) /аЛЛя (*)])=*•
Proof. Since by A.16) xia(x), there exist n,m,X fulfilling A.40). In view
of lemma 1.8 the right-hand side of A.39) is independent of the choice of n, m, X
fulfilling A.40). Consequently the function <p(x) is unambiguously defined by for-
formula A.39) in the whole of E and evidently belongs to the class Ф, We must prove
that (p{x) satisfies equation A.1) in E, fulfils A.17) and is the unique function with
these properties.
Let us fix an x e E and n,m, X such that A.40) holds. Since f(x) e C(x), we
have a[f(x)]=a(x). Hence, according to A.40)
and consequently
A.41) (
On the other hand A.39) and A.40) give
A.42) д(х,<р(х)) = д {/Гп[/"(«(*))]> я»-»(/"!>(*)]. 0-0(*). Ыв(
But by the definition of the functions яд_т A) we have
x9-m(/"l> (x)], д„[а (х), <po(a (x))])
= 9 ~ \frif\a (x)]), дя_m+ i(/"[e (x)], gB[e (x), ?0(« (x))])},
whence
This is correct if m>0. For m=0 this step is superfluous and relation A.43) is obvious.

3. Non-invertible / 41
A.43) дв_т+1(/"[в(х)], ffB[e(x), ?0(e W)])
=g {/я"т(/"[« (*)]), ябг-т(/"С« (*)], e-[e (*), Ф W)])} •
From A.41), A.42) and A.43) we obtain q>[f(x)]=g(x,q>(x)), which proves that
p(x) satisfies equation A.1).
If xeA, then a{x) = x and A.40) holds with n=m=0. Then A.39) becomes
q>(x)=(po(x), i.e. condition A.17) is fulfilled.
Lastly, let (p{x) be an arbitrary function belonging to the class Ф, satisfying
equation A.1) and condition A.17). Let us fix an arbitrary x e E and n, m, X such
that A.40) holds. By A.37) and A.17) we have
<P [/"(a (x))] = gn[a (x), <p(a (x))] =gn\_a (x), <po(a (x))]
and
= x9-m(fla (*)]' 9„[а (x), ф (x))]),
which means that q>{x) coincides with the function A.39).и
Now the contents of lemmas 1.2 (with corollary) and 1.9 can be given a more
complete form.
Theorem 1.4. Suppose that hypotheses 1.1 and 1.2 or 1.4 and 1.5 are fulfilled.
Then relation A.12) й a necessary and sufficient condition for the existence of a solu-
solution belonging to the class Ф of equation A.1) in E.
The above theorem results directly from lemma 1.2 and theorem 1.1 or lemma
1.9 and theorem 1.3.
Theorem 1.5. Suppose that hypotheses 1.1 and 1.2 or 1.4 and 1.5 are fulfilled
and V[x]^0 for x e E. In order that there exist a solution <р е Ф of equation A.1)
in E fulfilling the condition <p(xo)=yo, x0 e E, it is necessary and sufficient that
y0 e V[x0].
Proof. The necessity follows from lemma 1.2 resp. 1.9. To obtain the suffi-
sufficiency choose the set A in such a manner that x0 e A and a function (p0 e ?[A]
in such a manner that (ро(хо)=Уо and then apply theorem 1.1 or 1.3.И
Remark. One may replace hypothesis 1.2 by hypothesis 1.3 and the set V[x]
by Fmod[x] in theorem 1.4, but not in theorem 1.5.
§ 4. Automorphic functions A). Following M. Ghermanescu, we shall call the
solutions of the equation
C1) Rawson [1], Pincherle [1], [2], [3], [4], Appell [1], [2], Rausenberger [l]-[3], Hurwitz [2],
Jaggi [2]-[4], Fatou [1], [4], [5], [12], Schottky [1], Popovici [2], [4], Starke [1], Myrberg [3], [6],
[12], Chajot [1], Hardy, Titchmarsh [1], Marty [1], [2], Shimizu [1], [2], Ganguillet [1], Chayoth
[1], Volterra, Peres [1], Ferrar [1], Birkhoff [6], Rademacher [2], Montel [7], af Hallstrom [1],
Pastides [1], Foures' [1], [2], Zaharcuk [1], Ghermanescu [18], [22], Koljagin [2], Pre§ic [1],
Browder, Werner [1], Kuczma [30].

42 CHAPTER I. General solution
A.44) ?> [/(*)] = ?>(*)
automorphic functions. If/(x) fulfils @.1), we may apply to A.44) theorem 1.3. In
this case g(x,y)=y, and so the function kgn{x,y) equals у for л=0, + 1,+2, ...
and is independent of A, x and n. Formula A.39) becomes
A.45) ^(x)
Since the function a(x) (defined by A.16), where A is an arbitrary set having exactly
one point in common with every orbit contained in E) is evidently constant on
every orbit, we obtain hence the following result.
Theorem 1.6. Suppose that the function /(x) fulfils condition @.1). A function
cp(x) defined in E satisfies equation A.44) in E if and only if <p{x) is constant on every
orbit contained in E.
It is interesting to note that this theorem may be proved without the use of
the axiom of choice. If <p(x) is constant on every orbit contained in E, then evidently
A.44) holds for every x e E. Conversely, let us suppose that a function <p(x) satisfies
equation A.44) in E and let C<=E be an orbit under/. Let xt and x2 be two arbi-
arbitrary points of С Then there exist non-negative integers n, m such that/"^) =/m(x2).
It is easily shown by induction that <p[f\x)] = (P{x) for every г>0; thus
pU\d\ pi) and
Hence
which shows that <p(x) is constant on С
Formula A.45) can be given another form, which does not require the axiom
of choice. Let Eli be the quotient space of the set E by the relation i, i.e. the set
of all orbits contained in E. Then the formula
A.46) 9>(х) = Ф[С(х)],
where Ф is an arbitrary function defined on Eli, gives the general solution of equa-
equation A.44) in E. Both formulae, A.45) and A.46), involve an arbitrary function;
but the former contains a function cpQ defined on a set A, whose existence is guar-
guaranteed by the axiom of choice, while the latter contains a function Ф defined on
the well-defined quotient space E/i.
From formula A.46) one may easily deduce the following.
Theorem 1.7. Suppose that the function f(x) fulfils condition @.1) and т(х) is a
function which maps E onto a set F=t(E) and т(х1)=т(х2) if and only if xtix2.
Then the general solution of equation A.44) in E is given by the formula
where W(x) is an arbitrary function defined on F.
Solutions of equation A.44) have been written by various authors in various
forms. In particular, if/(x) is invertible in a modulus set E, then the general number-

4. Automorphic functions 43
valued solution of equation A.44) may be written in the form
(i.47) «»(*)= X> LA*)],
— oo
where н(х) is an arbitrary function defined in E and such that series A.47) con-
converges (*).
§ 5. Abel's equation and Schroder's equation. We shall now apply the results of
§ 3 to the Abel equation
A.48)
and the Schroder equation
A.49)
These equations will be treated in more detail in Chapters VII and VI.
Let/(x) fulfil hypothesis 1.4 and define the function D(xt, x2) for xl5 x2 e (?0,
Xi i x2, by the formula
A.50) D(x1,x2)=n-m,
where n,m are such that/n(x1)=/m(x2). According to lemma 0.2 the function
D(xt, x2) is unambiguously defined by A.50).
Hypothesis 1.6. E is an Abelian group with respect to the operation +, and
for any integer пфО we have псфО.
E.g., S may be the set of real or complex numbers, сфО.
Equation A.48) is a particular case of A.1), where g(x,y)=y+c. Hence
!,д„(х,у)=у+пс are independent of x and A. By hypothesis 1.6 we have
F[x]=0 for xe\J(?k,
(c=l
whereas V[x] = E for x e (?0. Thus theorem 1.4 implies the following
Theorem 1.8. B) Let hypotheses 1.4 and 1.6 be fulfilled. Then equation A.48)
has a solution ере Ф in E if and only if
A.51) /*(x)^x for every x ? Eand every integer k>0.
00
In fact, condition A.51) is equivalent to (J <?k = 0. If f\xo)=xo, k>0, then
*=i
xoe(?fc. Conversely, if xo?©fc,A:>O, then for j=Jk(x0) and x*=/'(x0) we have
/*(x*) = x*.
@ Appell [1], [2], Popovici [2], [4], Badescu [1], Volterra, Peres [1], Ghermunescu [22].
B) Tambs Lyche [1], [3]. Cf. also Kuczma [30].

44 CHAPTER I. General solution
Now suppose that A.51) holds and let A be a set containing exactly one element
of every orbit contained in E. Define the function a(x) by A.16) and d(x) by
A.52) d(x)=D(a(x),x), xeE.
The following theorem is an immediate consequence of theorem 1.3.
Theorem 1.9. О Let hypotheses 1.4 and 1.5 and condition A.51) be fulfilled,
and let A be a set having exactly one point in common with every orbit contained in
E. Then for every function (po(x) defined in A and taking values in E there exists exactly
one solution q> e Ф of equation A.48) in E, fulfilling condition A.17). This solution
is given by
<p(x) = <po[a(xj]+d(x)c,
where a{x) is defined by A.16) and d{x) by A.52) and A.50).
Now we turn to equation A.49).
Hypothesis 1.7. E is a vector space over the field IF of real or of complex numbers
and s^O is a number from ^, which is not a root of unity.
In particular we may have E=&.
Now g{x, y)=sy and ^gn(.x,y)=s"y. Since 51 is not a root of unity, we have
V [x~\ = {0} for x ? I J (?
where в is the zero of E, and of course V[x]=E for x e (?0.
Theorem 1.10. (Kuczma [20], [30].) Let hypotheses 1.4 and 1.7 be fulfilled and
let Abe a set having exactly one point in common with every orbit contained in (So •
Then for every function q>Q(x) defined on A and taking values in E there exists exactly
one solution срвФ of equation A.49) in E fulfilling condition A.17). This solution
is given by
00
(в for xe{J<?k,
where a(x) is defined by A.16) and d(x) by A.52) and A.50).
The above theorem also results directly from theorem 1.3.
§ 6. General remarks. The theorems contained in the present chapter show that
equation A.1) usually has a very large family of solutions. The general solution
depends on an arbitrary function. This situation is disadvantageous. If one is led
by a concrete problem to a functional equation, such a vast family of solutions
is awkward and worthless. One would like to know which solution of the equation
should be taken as representing the solution of the original problem. So now the
following question appears: whether among all solutions of equation A.1) there exists
О Tambs Lyche [1], [3]. Cf. also Kuczma [30].

6. General remarks 45
a unique solution (or a unique finite-parameter family of solutions) characterized by
a certain particular property! Such a solution might then be called the principal
solution and it would be reasonable to regard this solution as the solution of the
problem considered. In the theory of functional equations in a single variable the
most important problem is to decide under what additional assumptions an equa-
equation has a unique solution (a unique finite-parameter family of solutions).
Thus instead of considering equation A.1) in the class Ф of all functions defined
in E and taking values in S, we shall seek solutions in a narrower function class
Ф* с Ф. As Ф* we shall take the classes of continuous, differentiable, analytic,
monotonic, convex functions, etc. Depending on the hypotheses assumed and the
form of the equation in question, various function classes will turn out to be best
suited to furnish a unique (principal) solutions of the equation. In the next chapters
we shall deal with this problem, and we shall establish a number of results to the
effect that the equation considered has in a function class Ф* a solution depending
on an arbitrary function (such a result may be regarded as negative) and also to
the effect that the equation has in Ф* a unique solution or a unique finite-parameter
family of solutions. As we shall see, a particular role will be played here by the points
of the set ©i[/] (the fixed points of the function/). The conditions that ensure the
uniqueness of a solution cp(x) of the equation A.1) will always concern the behaviour
of <p(x) at, or in a neighbourhood of, a point «^ e ®i[/]-
However, the expression "equation A.1) has in a function class a solution
depending on an arbitrary function" must be given a precise meaning. E.g. if the
set A consists of a single point, then "a solution depending on an arbitrary function
on A" means simply a one-parameter family of solutions. Since in the sequel the
set E will always be a subset of the space of real or complex numbers, we shall adopt
the following definition.
Definition. Let Ф be the class of all functions defined in a set E and assuming
values in a set 3, and let Ф* <= Ф be a subclass of Ф and Ф*[Е±], E± <= E, the class
of functions cp б Ф* restricted to the set E±. We shall say that equation (I A) has in
the function class Ф* a solution depending on an arbitrary function, if there exists an
open set A с E such that every function q>0 e Ф*[А] can be extended (not necessarily
uniquely) to a solution <p e Ф* of equation A.1) in E.

CHAPTER П
LINEAR EQUATION
§ 1. Solution depending on an arbitrary function. In this chapter we shall de-
develop the theory of continuous solutions of the linear equation A)
B.1) 9[f(x)]=g(x)9(x) + F(x),
where x is a real variable. Values of <p lie in the field E of real or of complex numbers.
The class of functions defined in an interval / and taking values in S will be denoted
by ФЩ.
At first we consider the case where the following conditions are fulfilled.
Hypothesis 2.1. /e R°[I], where ?. is an endpoint of I, ?$I B).
Hypothesis 2.2. g e Ф[Г\, Fe4> [/], g{x) andF{x) are continuous in Iandg(x)j=O
for x el.
Theorem 2.1. (Kordylewski, Kuczma [4].) If hypotheses 2.1 and 2.2 are fulfilled^
then equation B.1) has in I a continuous solution cp e Ф[Т\ depending on an arbitrary
function. More precisely, for any x0 e / and an arbitrary continuous function <ро(х)
from the class Ф[/о], where /o = <Xo >/(*(>)> or Io = (f(xo)> *o>> fulfilling the condition
B.2) <PoU(xoJ] = 9 Oo) PoOo) + F Oo)»
there exists exactly one solution cp(x) of equation B.1) in I, belonging to Ф[Г\ and
such that
B.3) <p(x)=(po(x) for xelo.
This solution is continuous in I.
Proof. For definiteness let us assume that ? is the left endpoint of/; in the other
case the proof is analogous. Then /0 = </(x0), xo>. Let <po(x) be a continuous function
belonging to Ф[10] and fulfilling B.2). By theorems 0.5 and 1.1 there exists exactly
A) Choczewski, Kuczma [1], Kordylewski, Kuczma [2], [4], [5], Kuczma [2], [20], Bielecki,
Kisynski [1], Bajraktarevid [11]. Cf. also Catalan [1], Ostrowski [2], Pompeiu [1], [2], Popovici
[10], van der Berg [1], Steinhaus [1], Steinberg [1], Szmuszkowicz [1], Targonski [5], [6],
Kwapisz [2], Burek [1], Choczewski [4]-[7], Coifman, Kuczma [1].
B) One or both the endpoints of / may be infinite.

1. Solution depending on an arbitrary function 47
one function q> e Ф[Г\ satisfying equation B.1) in /and fulfilling the condition
<p(x)=<po(x) for xe(/(xo),xo>.
According to B.2) <p(x) fulfils B.3). It remains to prove that <p(x) is continuous in /.
In (f(x0), x0) the function <p(x) is continuous by B.3). The continuity of q>0 in /0
implies further that
lim q>{x)= lim foW=?'o№o)]
x->f(xo) + 0
We have also, since <p{x) satisfies B.1),
lim <p(x)= lim fl»[/(x)]= lim {g(x)<p(x)+F(x)}
f(xo) — 0 x-*xo — 0 x-*xo — 0
= lim
x-»;co— 0
Thus ?>(x) is continuous for x e </(x0), x0). Now, /= \J /n[</(x0), x0)]. Supossing
that ?>(x) is continuous for x e </n(x0),/n~1(x0))=/"~1 [</(x0), x0)], n>0, we have
forx6</n+1(xo),/n(x0))
9(x) = g[/-1(x)-]9[/-1(x)]+F[/-1(x)-], Г1(хN<Лх0),/п-1(х0))
and thus <p{x) is continuous for x e </n+1(x0),/"(x0)) =/"[</(x0), x0)]. Similarly,
we prove that <p(x) is continuous for x e/"[</(x0), x0)], n<0. ¦
§ 2. Homogeneous equation. The conclusion of theorem 2.1 is not valid if t, e /.
In this section we shall discuss this problem for the homogeneous equation
B.4)
We assume that the following conditions hold:
Hypothesis 2.3. feR^I], fe/C).
Hypothesis 2.4. g еФ[Г\, g(x) is continuous in I and д(х)фО for xel,
We consider the sequence of functions
B.5) еп(х)=П0[Л*)], Х6/, л = 1,2,3,...
i=0
Three cases may occur.
@ Also in this case { may be infinite. The continuity of a function <p at { is then understood
as the existence of a finite limit lim <p(x). This allows one to apply the results of this chapter also
to the case where f(x)=*+const.
If <J is infinite, then some changes in notation must be made. E.g. if <J= — oo, then instead of
the interval (fi, $+Sy one must take an interval (— oo, d), where d is a sufficiently small number.
These changes, however, do not alter the argument. For the sake of simplicity, we proceed in the
sequel as if ? were finite.

48 CHAPTER П. Linear equation
(i) The limit
B.6) G(x)=limGn(x)
n-»oo
exists in I. Moreover, G(x) is continuous in I and G(x)?=0 in I.
(ii) There exists an interval J <= I such that
B.7) limGB(x)=0
n-»oo
uniformly in J.
(iii) Neither of the cases (i) and (ii) occurs.
We shall prove the following
Theorem 2.2. (Choczewski, Kuczma [1], Kuczma [20].) Let hypotheses 2.3 and
2.4 be fulfilled.
In case (i) equation B.4) has a one-parameter family of continuous solutions
in I: for every ne S there exists exactly one continuous function q> e Ф [/] satisfying
equation B.4) and fulfilling the condition
B.8)
This solution is given by
B.9)
In case (ii) equation B.4) has in the class Ф [/] a continuous solution depending on
an arbitrary function. In this case every continuous solution (pipe) of equation B.4) in I
fulfils the condition
B.10) lim?>(x)=0.
In case (iii) the function q>{x) = 0 is the only continuous solution of equation B.4)
in the function class Ф[Г\.
Proof. We may assume that ? is the left endpoint of/. If «^ is the right endpoint,
the argument is similar. If ? is an inner point of /, we consider separately each of
the two parts into which / is divided by ? (with ? included in both). We may do
so in view of @.24).
Case (i). In this case д(х)фО in /. In fact, for хф? д(х)Ф0 by hypothesis 2.3,
and if we had g(?)=0, then Gn(?)=0 for all n, and consequently also
contrary to the assumption. Thus we have by B.5)
B.11) G Г/(хI = G (x),
whence
а г t(v\~\ = q (x)
0(x)

2. Homogeneous equation 49
Thus functions B.9) actually satisfy equation B.4) in /. By (i) they are continuous in /,
and it follows from lemma 0.7 that #(?) = 1 (otherwise Gn (?) = [#(?)]" would not
converge to a limit ^0), whence G(g) = \ and B.8) is fulfilled.
On the other hand, let q> e Ф [/] be a continuous solution of equation B.4) in /,
fulfilling B.8). Then induction yields
B.12)
I* $
By theorem 0.4, lim/"(x)=? and thus, since <p is continuous in / and ?, e /, B.9)
follows from B.6), B.8) and B.13).
Case (ii). Suppose that B.7) holds uniformly in an interval / <= /. We choose
an x0 ? /, хоф?, and we choose a<b such that <a, by <= / n </(x0), xo>. Let q>0(x)
be an arbitrary function defined and continuous in </(x0), xo> and fulfilling the
conditions
B.14)
po(x)=O for xe</(xo),xo>-(a,6).
It follows from theorem 2.1 that <po(x) can be uniquely extended onto / — {?} to
a continuous solution <p{x) of equation B.4). Moreover, we put q>(^)=0. The function
<p(x) thus defined satisfies equation B.4) in / (cf. lemma 0.7) and is continuous
in / — {?}. Thus it is enough to show that
B.15) lim
x->$ + 0
Given an e>0, we can find a positive integer TV such that
|Gn(x)[<e/M for n>N, xe(a,b),
where
M= sup |?>(x)|.
<A*o),*o>
For an arbitrary x e (?, fN(x0)) we can find an index m^N and x* e </(x0), xo>
such that x=/m(x*). Hence by B.12) we get
Now, if x* ? (a, b), then
B.16) \9(x)\ = \GJi
and if x* ф(а, b), then <p(x*)=0 and B.16) holds all the same. Thus B.16) holds for
all x ? (i,fN(x0)), whence B.15) follows immediately.
4 Functional equations

50 [CHAPTER П. Linear equation
For an arbitrary continuous solution q>(x) of equation B.4) condition B.10)
follows from B.12) and B.7) in view of theorem 0.4.
Case (iii). Let us suppose that equation B.4) has a continuous solution q>(x)^0
in /. We shall show that then either case (i) or case (ii) must occur.
) = цфО, then <р{х)фО in /. In fact, if we had q>(xo)=O, x0 e /, then by B.12)
)]= Urn Gn(x0Mx0)=0,
n-*co n-* oo
contrary to the assumption. Consequently, B.12) gives
and thus limit B.6) exists, is continuous and different from zero in /, i.e. case (i) occurs.
Now let us suppose that cp(?)=0. Since cp{x)^0, there exists an interval /= <a, by
/-{{} such that
for X ? J .
Given an e>0, we can find а <5>? such that
|p(x)|<?C for
We can also find an index N such that
B.17) f\b)<5 for
B.17) implies the inequality
f\x)<5 for xeJ,
and hence we obtain according to B.12)
|Gn(*)l= i /*, <e for xeJ,
K*)|
Consequently, Gn(x) tends uniformly to zero in /, i.e. case (ii) occurs.
Remark. As follows from the proof of case (ii), if there exists an x0 e /,
such that </(x0), xo><=/, then every continuous function q>0(x) on </(x0),
fulfilling B.14) can be uniquely extended to a continuous solution of equation B.4)
in /. Thus (cf. theorem 2.1) in this case every solution of B.4) continuous in /— {{}
can be extended by putting ?>({)=0 to a continuous solution of B.4) in /. In view
of theorem 2.1 we obtain so the general continuous solution of equation B.4) in /.
But if no interval </(x0), xo>, x0 e I, x0 ф ?,, is contained in / (that this may actually
happen is shown in Choczewski, Kuczma [1]), then the procedure described above
(in the proof of case (ii)) does not furnish the general continuous solution of B.4)
in/.
If ij is an inner point of /, then the solution should be prescribed on two intervals:
and </(jc2), x2}, where xx<?<x2.

3. Some criteria 51
§ 3. Some criteria. One would like to have some simple criteria to decide which
of the cases (i), (ii), (iii) from the preceding section occurs. It can be seen that a
great deal of information can be obtained from the value of g(g).
Theorem 2.3. Iff(x) and g{x) fulfil hypotheses 2.3 and 2.4 and if, moreover,
B.18) И)|<1,
then case (ii) occurs.
Proof. Suppose that { is the left endpoint of /; in other cases the argument
is similar. Let us choose an arbitrary с в I, сф^, and put /=<{, c>. By B.18) there
exist a <5>0 and a constant 3, 0<3< 1, such that
B.19) \g(x)\<$ for xe<{,{+<5>.
Further, we can find an index N such that/"(c) 6 <?> ? + <5> for n^N. Hence
B.20) /n(x)e<{,{+<5> for
We set
JV-l
J i = 0
Then by B.19) and B.20) we have for x e / and n>N
n-l
„«| = | rig
i0
| ri | | li
i=0 i = N
which shows that lim Gn(x)=0 uniformly in /.¦
n-»oo
Theorem 2.4. Iff(x) and g{x) fulfil hypotheses 2.3 and 2.4 and if there exists a
<5>0 such that
B.21) |flf(
?/гел case (iii) occurs.
Proof. Again we may assume that t, is the left endpoint of /. We fix arbitrarily
an x0 ? /. There is an index N such that /п(х0) е <?, ? + <5> for л^М Hence and
in view of B.21) we get for n>N
|<?„(*о)| = \Т\9 [/"(xo)] 11П в [/*(xo)] |
1=0 i = N
and consequently Gn(x0) cannot approach zero. Since x0 has been chosen arbitrarily
in /, this shows that case (ii) cannot occur. Nor can (i) occur, since g [/"(x)] tends
to д(^)ф\ as л-юо. So (iii) is the case.B
Corollary. If hypotheses 2.3 and 2.4 are fulfilled and
B.22) |flf(O|>l,
then case (iii) occurs.
4*

52 CHAPTER II. Linear equation
If
B.23) 9(S) = 1,
then all the three cases (i), (ii), (iii) can actually occur (cf. Kuczma [20], Choczewski,
Kuczma [1]). Case (i) may occur only if B.23) holds (otherwise the infinite product
cannot converge), but some further assumptions are necessary to ensure that (i)
actually is the case.
Theorem 2.5. (Choczewski, Kuczma [1].) Iff(x) and g^x) fulfil hypotheses 2.3
and 2.4, ? is a strongly attractive fixed point off and, moreover, if there exist positive
constants 5, ц and M such that
B.24) |<7(x)-l|<M|x-?|" for xeln (?-&, i + S),
then case (i) occurs.
Proof. We may assume that ? is the left endpoint of /. We may also assume
that 8 in B.24) is chosen in such a manner that
B.25) |/(x)-{|sS3|x-{| for xe(i,i+5},
where 0<$< 1. Let us fix а с e /, сф^. We can find an index N such that f(c) e
<{, { + <5> for n^N. This and the monotonicity of/ imply that
B.26) f"(x)e(Z,Z + Sy for
Now by B.26), B.24) and B.25) we have for jc e <{, c> and
whence it follows that the infinite product fj g [/"(*)] absolutely and uniformly
n=0
converges in <{, c>. Since с has been chosen arbitrarily in /, limit B.6) exists and
is continuous and different from zero in /, i.e. case (i) occurs. ¦
§ 4. Non-homogeneous equation. Since the difference of two solutions of equa-
equation B.1) must satisfy the homogeneous equation B.4), the results of §§ 2-3 allow
us to draw some conclusions concerning the number of continuous solutions
of B.1).
For the non-homogeneous equation B.1), we can also form the sequence Gn(x)
defined by B.5), and distinguish cases (i), (ii) and (iii) according to the behaviour
of Gn(x) (cf. § 2). Theorem 2.2 implies the following
Theorem 2.6. Let hypotheses 2.3 and 2.4 be fulfilled. In case (i) equation B.1)
has in I either one-parameter family of continuous solutions or none. If B.1) has in I a
continuous solution <po(x), then the general continuous solution in I is given by
G(x)
where t] eE is an arbitrary constant and G(x) is given by B.6).

4. Non-homogeneous equation 53
In case (ii) equation B.1) has in I a continuous solution depending on an arbitrary
function, or has no continuous solution in I.
In case (iii) equation B.1) has in I either exactly one continuous solution or none.
As we see, the problem remains to decide whether equation B.1) has in / at
least one continuous solution. In the next two sections we shall show that a con-
continuous solution of B.1) in / actually exists provided |gf({)|^l. If |fif(?)| = l> then
equation B.1) may happen to have no continuous solution in / (cf. § 7). This may
be compared with theorem 2.1, which asserts that B.1) has in /— {{} a continuous
solution depending on an arbitrary function. Here we see the prominent role played
in the theory by the point ?, characterized by the property { e G^I/] 0). In fact,
all the conclusions of sections 2-7 will remain valid if we replace (in the hypotheses
as well as in the assertions) the continuity in / by the continuity at ?.
§ 5. Case |fif(?)|>l. Let us supplement hypotheses 2.3 and 2.4 by the following
Hypothesis 2.5. F{x) belongs to Ф[Г\ and is continuous in I.
We have from B.1)
<2.27> ^
(Note that in virtue of hypothesis 2.4 and condition B.22) we have д(х)фО in /.)
Using B.27) one can prove by induction the formula
B.28) <p(x)=——— ?——ГТ > xe/, л=0,1,2,...,
Gn{x) i,o Gt+1(x)
where Gn(x) are given by B.5). Suppose that <p{x) is a continuous solution of B.1)
in / and let л-юо in B.28); in view of the fact that \Gn{x)\^>co (resulting easily
from B.22)), while <p[f"+1(x)]->p(f), we obtain
B.29)
Thus B.29) is the only possible form of a continuous solution of equation B.1) in /.
Now we shall prove that formula B.29) actually defines the unique continuous
solution of B.1).
Theorem 2.7. (Kordylewski, Kuczma [4].) If hypotheses 2.3, 2.4 and 2.5 and
condition B.22) are fulfilled, then equation B.1) has a unique continuous solution
q> ? Ф [I] in I. This solution is given by formula B.29).
Proof. The uniqueness follows from theorems 2.6 and 2.4 (cf. the corollary)B).
C) According to theorem 0.4 the other points of / belong to (?0 [/]•
B) The above considerations yield another proof of the uniqueness of the continuous solution
of B.1) in case B.22).

54 CHAPTER II. Linear equation
Thus we need only prove that B.29) actually defines a continuous solution of B.2)
in/.
We may assume that ? is the left endpoint of /. We can find a <5>0 and a &> 1
such that
B.30) |<7(x)]>6> for хб<{, i+Sy.
Next we choose an arbitrary eel, c?=?, and find an index N such that
B.31) f(c)e{i, i + 8y for n^N.
B.31) and the monotonicity of/imply that
B.32) /n(x)e<{,{ + <5> for n>N and xe<{, c>.
Let us write
B.33) L= inf \g(x)\, M= sup \F(x)\.
We have by B.32), B.30) and B.33)
M M
n-JV+l
for nl>iV and хе<{, с>.
Consequently, the series on the right-hand side of B.29) converges uniformly in
<?, c> for every с el, с^{, and so cp(x) is a continuous function in /. Further we
have by B.29)
whence by B.11)
и,- ?
by B.5) and B.39) we obtain hence B.1). Thus tp(x) satisfies equation B.1).¦
§ 6. Case [fif({)[<l. Now we shall suppose that condition B.18) is fulfilled.
We start with establishing some lemmas.
Lemma 2.1. Let /e 5°[/] (*) and let functions g еФ [I] and F еФ[Г[ be bounded
in I. Further, let q> e Ф [/] be a solution of equation B.1) in I. Then
B.34) |р[Пх)]|<М(хI—L-+C\(p(x)\ for xel and n = \,2,...
where
B.35) M(x) = sup|F@|, L = sup|<7@|,
and /, = ({, x> or <x, Q.
Proof. Since for every xel we have Ifix)<=Ix, the inequality
B.36) M[/(x)]*SM(x)
Of may belong to / or not.

6. Case |<К?>|<1 55
holds for every x e /and for every positive integer /. We have by B.1)
B.37) ^[/(^I
for every xel. Replacing in B.37) x by f(x) (cf. lemma 0.6) and making use of
B.36) and B.37) we obtain
x)]+L \<p [/(*)] |
Induction yields
i.e. B.34). ¦
Lemma 2.2. Let feR°[I] (') and let functions g еФ[/] and Fe0[I] be bounded
in I. Moreover, suppose that there exist numbers <5>0 and S, 0<i9<l, such that
B.38) \g(x)\<9 for xe(?-<5,? + <5)n/.
Then every solution среФ[Г\ of equation B.1) in I which is bounded in an interval
(f(xo),xoy resp. (xo,f(xo)y, xoel, хоф^, is also bounded in ({, xo> resp. (x0, ?)¦
Proof. We may assume that xo>?. Let <p e Ф[Г\ be a solution of equation
B.1) bounded in </(*„), *o>:
B.39) \9(x)\<C for xe(f(xo),xoy.
There exists an index N such that fN~ \x0) e ({, ? + 8). Consequently,
/n(x)e({,{+<5) for xe(i,xoy and n>N-l
We define L and M(x) by B.35) and put M=M(x0), M=M[fN~\x0)]. Let us fix
an arbitrary хе(?,хоу. If x e </(x0), ^oX then \(p(x)\^C by B.39). If x e
(fN(x0), f(x0)), then there exist an n^N and an x* e </(x0), xo> such that
x=f"(x*). So we have by B.39) and lemma 2.1
1 r^
df
Lastly, if xe(?,fN(x0)), then there exist an n^\ and an x* e </"(*<>)>/N"'(*<>)>
such that x=/"(x*). We now use lemma 2.1 applied to the interval ({,/*
In this interval we have \g(t)\<8<\, and so
Thus in every case |?>(x)| ^max (C, Cx, C2), i.e. ?>(x) is bounded in ({, л:0>
(') f may belong to / or not.

56 CHAPTER II. Linear equation
Theorem 2.8. (Kordylewski, Kuczma [4].) LetfeS%[I] (') and suppose that the
function F ? Ф [/] fulfils the condition
B.40) \imF(x) = 0.
Suppose further that g e Ф [/] and that there exist numbers <5>0 and S,O<S<1, such
that B.38) holds. Then every solution q> e Ф [/] of equation B.1) ш / w/г/с/г й bounded
in a neighbourhood of ? fulfils the condition
B.41) \im.<p(x) = 0.
Proof. We may assume that ? is the left endpoint of/. Further we may assume
that 5 in B.38) is chosen in such a manner that { + 6 e /and F(x) and ?>(x) are bounded
in ({, { + <5) (according to B.40) F(x) is bounded in a neighbourhood of {). Thus
B.42) |?>(*)|*SC for xe({,{+<5).
We have by B.40) for the function B.35)
KmM(x) = 0.
Consequently, given an e>0,' we can find a Si, 0<Sl <S, such that
B.43) M(x)<i(l-S)e for xe(i
Further we can find an index N such that
B.44) SN<— ¦
1С
We put
(«,*>
Then m(x) belongs to S°[I] and is monotonic (not necessarily strictly). Set
<52 = mN(i + <5,) - i. Since for every n /"(({, { + <5,)) = (Z, m\S, + <?,)), we have in par-
particular /*(({, { + Sj) = ({,{+<52). Consequently, for every x e ({, ? + S2) there exists
an x* ? ({, ? + Si) such that/)V(x*)=x. Hence by lemma 2.1 and by B.42), B.43)
and B.44) we have for x e ({, { + <52)
)
1 — of
which proves relation B.41). ¦
(') ? may belong to / or not.

6. Case |sr(OI<l 57
Theorem 2.9. (Kordylewski, Kuczma [4].) If hypotheses 2.3, 2.4 and 2.5 and
condition B.18) are fulfilled, then every solution <p e Ф[Г\ of equation B.1) in I that
is continuous in I— {{} is continuous in the whole of I.
Proof. We may assume that ? is the left endpoint of /.
Since (cf. lemma 0.7) ? e ©i[/], the set V[x] consists of the roots rj e 3 of the
equation
B.45) r\=g(S)r\+F(?).
According to B.18) equation B.45) has the unique root
7
Let <p(x) be a solution of equation B.1) in /, continuous in /—{{}. In view of
lemma 1.2
Since <p{x) is continuous in /—{{}, it is bounded in </(xo)> xoy for any x0 el,
хоф^. By lemma 2.2, q>(x) is bounded in a neighbourhood of ?. We put
B.47) () ()@ H()
On account of B.47) and B.46) we obtain
V [/(*)] = <7(*)V(*) +Я (x).
The function ^ (x) is, just like ?> (x), bounded in a neighbourhood of ? and the function
/f(x) fulfils the condition 1шЯ(х)=0. By theorem 2.8 Нт^(л:)=0, i.e. lim?>(x)
= ?>@. This proves that q>(x) is continuous at ?, and thus it is continuous in the
whole of/.¦
Theorems 2.9 and 2.1 imply the following
Theorem 2.10. If hypotheses 2.3, 2.4, 2.5 and condition B.18) are fulfilled, then
equation B.1) has in la continuous solution q> e Ф [/] depending on an arbitrary function.
More precisely, for any xoel and an arbitrary continuous function <ровФ [70], where
Io = (xo,f(xo)y or (J\xo),xo}, fulfilling condition B.2), there exists exactly one
solution <p ? Ф [/] of equation B.1) in I such that B.3) /го/Л. 77гй solution is continuous
in I.
§ 7. Case fif(x) = ±b The case where [#({)!= 1 (it is called indeterminate case)
is much more difficult, as it can be seen from the example of the homogeneous
equation (§ 3). Contrary to the situation when \д(?)\ф1 (cf. theorems 2.7 and 2.10),
in the present case it may happen that a continuous solution of equation B.1) in I
will not exist at all. In the present book we shall confine ourselves to the important
case g(x)=±l. Some further information about the case where |fif({)| = l may be
found in Choczewski, Kuczma [1].

58 CHAPTER II. Linear equation
We shall thus be concerned with the equations (')
B.48) vlf(x)}+(p(x)=F(x)
and
B.49) q>U(x)\-<p(x)=F(x).
For equation B.48) g{x)= — 1, whence Gn(x) = (— 1)", and consequently, we have
case (iii). For equation B.49) g(x) = l and Gn(x) = l, and consequently, we have
case (i). This may also be seen from the following
Theorem 2.11. (Bielecki, Kisynski [1], Kuczma [2].) Suppose thatfeS%[I], { e/,
and F б Ф [/] is continuous in I. Ifq> e Ф [/] is a continuous solution of equation B.48)
in I, then
B.50) 9 (x) = i F @ + ? (- l)n{F [/"(jc)] - F ({)} •
n = 0
Similarly, 1/<реФ[Г\ is a continuous solution of equation B.49) in I, then
B.51) *»(*) = *-I *¦[/"(*)],
7 e S1 & a constant.
Proof. Let <pe Ф[Ц be a continuous solution of equation B.48) in /. As in
the proof of theorem 2.9 (cf. in particular formula B.46)) we obtain
B.52) H0 = $F@.
We put
B.53) () ()@ H()
It follows that y/(x) is a continuous solution of the equation
B.54) V [/(*)]+ V (*)=#(*),
and, moreover,
B.55) lim
Now, we have by B.54)
B.56) 4,{x) = H{
whence, replacing x by f{x), we obtain
B.57)
B.56) and B.57) yield
B-58)
@ Cayley [3], Hilb [1], Belardinelli [1], Raclis [1], Popovici [4], Picard [10], Gilman [1],
Hardy [5], Gersevanov [4], Steinhaus [1], McKiernan [1], Szmuszkowicz [1], Kuczma [2], [6], [8],
Kordylewski, Kuczma [2], [5], Schweizer [1], Bielecki, Kisynski [1], Ghermanescu [22], Bajrak-
tarevic [11], [13], Coifman, Kuczma [1].

7. Case g(x)s + l 59
Induction then gives
n = 0
whence, by B.55) and theorem 0.4, on letting т-юэ, we obtain
п = 0
i.e. after taking into account B.52) and B.53) we get B.50).
Now let <p(x) be a continuous solution of equation B.49). Since (p(g) must
be a root of the equation t] = t]+F(l;), we must have F(?)=0. We write t]=<p(l;)
and у/(х)=<р(х) -<р(?) and the proof follows as for equation B.48).¦
Evidently solutions B.50) and B.51) need not exist. We shall exhibit here one
concrete example.
We consider the equation
B.59) <p \-—\-<p(x) = x, xe@, oo).
Here I=@, oo), <J=0, and В is the space of real numbers. The functions
/(jc) = — e R°o [<0, oo)] and F (x) = x e C°[<0, oo)]
fulfil the hypotheses of the preceding theorem. We have, moreover,
/"(Л) = ^-, /г=0,1,2,...
l+nx
But the series
n=0 n=0
(occurring in B.51)) diverges for every xe@, oo), which shows that equation B.59) has
no continuous solution in @, oo).
Therefore, some additional assumptions must be made in order to ensure the
existence of continuous solutions of equations B.48) and B.49). It becomes obvious
from the example of equation B.59) that the regularity of functions f{x) and F(x)
alone does not produce the desired effect. The functions x/(l + x) and x are analytic
in <0, oo). Consequently, conditions imposed on/and F must be of another kind.
Theorem 2.12. (') Suppose that hypotheses 2.3 and 2.5 are fulfilled and that there
exist a real-valued function H{x), bounded in I, and positive numbers 5 and S<\ such
that
\F{x)-F^)\^H{x) for
and
for
Then equation B.48) has a continuous solution in I. Moreover, if F(l;) = 0, then also
equation B.49) has a continuous solution in I.
@ Kuczma [2], Kordylewski, Kuczma [5], Bajraktarevic [11], [18], Choczewski, Kuczma [1].

60 CHAPTER II. Linear equation
Proof. We may assume that ? is the left endpoint of /. We shall show that
the series
00
B.60) ? co"{F [/"(*)] -F@},
n=0
where co= + l or —1, converges uniformly in the interval <{, c> for every eel,
сф?. Thus formulae B.50) and B.51) define continuous functions in /. It may easily
be checked that these functions satisfy equation B.48) and B.49), respectively.
Let us fix а с e I, сф {. We can find an N such that/"(с) е ({, i + S) for ,
We write
sup H(x) for
B.61) An=\
sup H(x) for n>N.
For x e </n+1(c)> f\c)y, n>N, we have
sup H(x)=SAn_1,
</"(c),/"-4c)>
whence it follows that
B.62) ЛП<6>ЛП_! for n>N.
According to B.61) and B.62) the sequence An is decreasing.
Now let us take an arbitrary x e ({, c>. Since f\x)^fn{c), there exists an integer
such that f(x) e </n+*+ \c), f+\c)}, and consequently,
Hence for x e ({, c> we have
The inequality \F[f(x)] — F(i)\^An holds also for x=?, wliich shows in view of
B.62) that series B.60) converges uniformly in <{, с>.и
Theorem 2.13. О Suppose that hypotheses 2.3 and 2.5 are fulfilled and that there
exist positive constants 8, к and С such that
\F(x)-F@\^C\x-t;\K for xe(l;-d, t; + d)nl.
Further, suppose that ? is a strongly attractive fixed point off. Then equation B.48)
has a continuous solution in I. Moreover, if F(?) = 0, then equation B.49) has a con-
continuous solution in I.
Proof. We put H(x) = C\x-?\K and apply theorem 2.12.И
Corollary. Iff{x) fulfils hypothesis 2.3 and, moreover, f{x) and F{x) are of
class C1 in I,f'(O ?, then equation B.48) has a continuous solution in I. Moreover,
if F(Q = 0, then equation B.49) has a continuous solution in I.
Bielecki, Kisynski [1], Kordylewski, Kuczma [5].

7. Case g(x)=±l 61
This results from theorem 2.13 in view of lemma 0.3.
The condition which is not fulfilled in the case of equation B.59) is/'(<!;) ?^ 1.
/ x V 1
In fact, = ^ and for x=0 equals 1.
\l+xj (l+xf
We shall give some further sufficient conditions for the existence of continuous
solutions of equation B.48). They will concern the case where 3 is the space of real
numbers.
Theorem 2.14. (') Suppose that f fulfils hypothesis 2.3 and F(x) is a continuous
real-valued function, {f}-monotonic in a neighbourhood of ?. Then equation B.48)
has a continuous solution in I.
Proof. We may assume that ? is the left endpoint of /. There exists a <5>0
such that the expression F[f(x)]—F(x) has a constant sign in <{, ?+<5>. We shall
show that series B.60), where co= — 1, converges uniformly in <{,c> for every
с в I, c^t;.
Let us fix а с e/, сф^, and an e>0. We can find a number 8X, 0<Jt <<5, such
that
\F(x)-F@\<s for jee<?, ? + *,>•
Next we can find an index N such that
Л*)е<?,? + *,> for X?<?,c> and
The alternating series
converges, since the sequence F[f\x)] — F(l;) tends monotonically to zero on account
of the {/}-monotonicity of F. Moreover, we have
|
n-N
which proves that series B.60) with со = —1 uniformly converges in <{, с>.и
Corollary. Suppose that f(x) fulfils hypothesis 2.3 and F(x)=F1(x)-F2(x),
where Flt F2 are continuous real-valued functions, {f}-monotonic in a neighbourhood
of ?. Then equation B.48) has a continuous solution in I.
Proof. Let (px{x) and (p2{x) be continuous solutions of the functional equations
and
@ Bajraktarevic [11]. Cf. also Kuczma [2], [6].

62 CHAPTER II. Linear equation
respectively, which exist by theorem 2.14. Then <p(x) = <p1(x) — <p2(x) is a continuous
solution of equation B.48). ¦
Let T=[akn], k=0, 1,2, ..., л=0, 1,2, ..., akn e S, be an infinite matrix fulfil-
fulfilling the conditions
B.63) ]imakn = 0 for every n>0,
fc-юо
B.64) Ko| + Ki| + ... + |atn|<L for every л>0,
where L is a fixed positive number (independent of n and k),
00
B.65) Takn=Ak and ]imAk=l.
n = 0 fc-»oo
We write
B.66) <Tjx) t ''
t
i = 0
and
B.67) sk(x)=
n = 0
(provided that the series converges for every k^O). We have the following
Theorem 2.15. (Bajraktarevtf [11].) Suppose that feR%[I]C) and Fe<P[7] and,
moreover, that F(x) is continuous at ?. IfUmsk(x) = <p(x) exists in I B), where sk(x)
is given by B.67) and B.66) and T is a matrix fulfilling conditions B.63)-B.65), then
<p (x) satisfies equation B.48) in I.
Proof. We have
whence
B.68) sklf(xy]=AkF(x)-sk(x)- ? (-l)n+1akn{Flfn+1(x)-]-
n = 0
Let us fix an x e I. To a given e>0 we can find an index N such that
^i for
where L is the constant occurring in B.64). Further, since the sequence F[f"(x)]
tends to zero, it is bounded:
for n = 0,l,2,...
@ ij may belong to / or not.
B) We say then that the series in B.50) is summable by the method T;cf. e.g. Hardy [5]%
Knopp [1].

7. Case g(x)=±l 63
By B.63) we can find an index К such that
i i e
\ОьЛ< for k^K and n<N.
1 ' 2MN
Thus for k^Kv/e have
п=0
JV-1
which
B.69)
shows
e
2MN
that
uf i T
lim
e
J2L
n = 0
Consequently, in view of B.65) and B.69), passing in B.68) to the limit as k-y со,
we get
vU(x)]=F(x)-9(x),
which was to be proved. ¦
The above theorem allows us to improve theorem 2.12 in the case where S is
the space of real numbers and the difference F(x)—F(Q has a constant sign in a
neighbourhood of ?.
Theorem 2.16. (*) Suppose that f(x) fulfils hypothesis 2.3 and F(x) is a real-
valued function, continuous in I. Further, suppose that there exists an x0 el, xo>?,,
and/or B) x'o ? /, x'0<?, such that
B.70) 0<шЯ(х)<4 f°r хе<
n
B.71) Я[/(*)]/Я(*)<1+- for xe<JXx>o),r+1(x'o))u(f"+\xo),r(xo)>,
n
where A and a are positive constants, H(x)=F(x)—F(Q and a>= +1 or — 1. Then
equation B.48) has a continuous solution in I.
Proof. We may assume that ? is the left endpoint of/and co= +1.
@ Bajraktarevid [11]. Cf. also Bajraktarevic [18].
B) We require the existence of xq if € is the left endpoint of /, the existence of Xq if ? is the
right endpoint of /, and the existence of both if ? is an inner point of /. If ? is an endpoint of I,
conditions B.70) and B.71) are postulated only in </"(Xo),/"+1(Xo)) or only in (/"+1(xo),/"(*<>)>.

64
CHAPTER II. Linear equation
> 1
At first we shall show that the sequence — H[f"(x)] is decreasing for every
x e <?, xo>. Let us fix an *e (?> *o>- There exists a k^O such that xe
C/*+1(*o),/W>. Then/V)e(/*+n+1(*o),/*+Vo)> and by B.71) we have
и + 1 и + 1 n+k+1
n + k+1 H[/"(x)] H[/"(x)]
и + 1 п+к
For x=?, the assertion is trival.
Next we show that the series
A72) 1
absolutely and uniformly converges in <?, xo>. Let us fix an x e (?, xo> and let
t1),/W>. Then we have by B.70)
0.73,
и и
For x=(i;, B.73) holds trivially. Our assertion results directly from B.73).
Further, we show that the sequence
B.74)
(-1)
H[fl(x
uniformly converges in (?, xo>.
Let us fix an x e ((^, xo> and let x e (/*+ 1(x0),f\x0)}. We have
i=n+l
i=o
i=n+p+l I
H[f
i+n+1(
Further, since the sequence — H[f"(x)] is monotonic, we have
n
н.+ 1»[/-
¦j + n+1
i+n+1(

7. Case g(x)=±\ 65
and finally, by B.70)
i . "^ A v A
B.75) у„+р(х)-у„(х)к I
A A
For x=(^ inequality B.75) holds all the same. This proves that у„(х) converges
uniformly in <?, xo>. Its limit
is thus a continuous function in <?, xo>.
Now, it follows(x) from the convergence of series B.72) and sequence B.74)
that the series in B.50) is summable to y(x) = limyn(x) by the method C1 = [ckn],
where B)
1
for и=0, ..., к,
k+1
0 for n>k.
Thus, in view of theorem 2.15, y(x) is a continuous solution of equation B.48) in
The solution obtained may be uniquely extended onto the whole / according
to theorem 2.1. Namely, there is a unique solution (p(x) of equation B.48) in /— {?,}
which coincides with y{x) on </(x0), xo>. This solution is continuous in /-{?}
and must coincide with y(x) on (?, x0). Extended onto / by putting
<p(x) provides a continuous solution of B.48) in the whole of /.¦
§ 8. Examples. 1. The equation
B.76) <pBx)=H<p(x)+x]
occurs in statics C). It may be written in the form
so that we can apply theorem 2.7 (S=0). We have/"(x)=x/2", Gn(x)=2n and B.29) becomes
n-0 Z / n-0
(!) Cf. e.g. Knopp [1], § 60, theorem 7.
B) This is the so-called Cesdro method, or the method of first means.
C) Poinsot [1], pp. 175-178. Cf. also Pompeiu [1], [2].
S Functional equations

66 CHAPTER II. Linear equation
Thus <p(x)=$x is the only continuous solution of equation B.76) in any interval containing
zero.
This solution could easily have been guessed. One may expect B.76) to have a solu-
solution of the form <p(x)=cx. Inserting in the equation gives c=J. It follows from theorems
2.4 and 2.6 that the solution found is the only continuous one.
2. In the theory of infinite series we meet the equation (!)
B.77) p(*2)+?(*)=*.
According to theorems 2.14 and 2.11 the function
CO
B.78) ?(*)= E(-0n*2"
n-0
is the only solution of equation B.77) continuous in @,1). After writing equation B.77)
in the form
we may apply theorems 2.14 and 2.11 with f=l. Then
CO
B.79) *(*)= X(-l)«x2—'
n-0
is the only solution of B.77) continuous in @, oo). Solutions B.78) and B.79) do not
coincide (Szmuszkowicz [1]). Consequently, equation B.77) has no continuous solution in
@, oo) nor even in <0,1).
3. In the case of the equation
B.80) <p(x+l)=—o(x)
x+1
we have ?= + oo. Here fn(x)=x+n, and
ft x+i x
Consequently, lim Gn(x)=0 uniformly in every interval (a, 6> С @, oo), i.e. case (ii) occurs.
И-Ю0
Thus equation B.80) has a continuous solution in @, oo) B) depending on an arbitrary
function.
Nevertheless it is possible to choose among solutions of equation B.80) a one-para-
one-parameter family of particular solutions characterized by better behaviour as x->co. As we shall
see in Chapter V, § 10, the functions
p(x)ij,
Г (x+1)
where Г(х) is Euler's Gamma function and r\ is an arbitrary real constant, are the only
monotonic solutions of equation B.80).
@ Hardy [5], p. 77, Steinhaus [1], Szmuszkowicz [1].
B) I.e. continuous in @,oo) and possessing a finite limit (in view of B.10) necessarily equal
to zero) as x tends to + oo.

CHAPTER III
CONTINUOUS SOLUTIONS
§ 1. Solution depending on an arbitrary function. In the present chapter we shall
obtain for equation @.49) results(x) analogous to those presented in the preceding
chapter for the linear case. However, we shall now confine ourselves to real-valued
functions. Equation @.49) will always be considered in one of the forms A.1)
and A.2).
Fig. 3
Let Q be a region (an open, simply connected set) on the real plane and let /
be a real interval. As in Chapter I, § 1, we introduce the sets (cf. fig. 3)
Qx={y: (x,y)eQ).
Throughout the whole of this chapter the region Q will be subjected to the following
condition:
(i) Bajraktarevic [8], Kordylewski, Kuczma [1], Kordylewski [2], [3], Kuczma [5], [8]. Cf. also
Kitamura [1], Chayoth [1], Sarkovskil [3], Pelczar [1], Andreoli [3], Volterra, Peres [1], Ward,
Fuller [1], Popovici [10], Bajraktarevic [4], [5], [10], van der Berg [1], de Rham [3], [4], Cooke [1],
Kuczma, Vopenka [1], Kordylewski, Kuczma [5], Kuczma [13], [24], [331.

68 CHAPTER III. Continuous solutions
Hypothesis 3.1. For every x el the set Qx is an open interval (possibly infinite).
Let f(x) be a function defined in / and fulfilling /(/)<=/. We assign to Q
another set Q* defined by the condition
/(*) if xeI>
\QX if хф1,
where Q* = {y: (x,y)e Q*}.
J being an interval, we define Ф[Т\ as the class of real-valued functions <p(x)
of a real variable which are defined in J and such that for every x e J <p(x) e Qx.
Let g(x, y) and h(x, y) be functions defined in Q and Q*, respectively. We
denote by Fx and Ax the sets of values of g(x, y) for у e Q and h(x, y) for у ей*,
respectively:
rx=gx(Qx), Ax=hx(Q*),
where gx{y)=g(x, y), hx(y)=h(x,y). The sets Qx, Гх, Лх will be supposed to be
connected by various relations.
Hypothesis 3.2. For every xel we have AxcQx.
Hypothesis 3.3. For every xel we have FxcQfix).
Hypothesis 3.4. For every xel we have Fx=Qf(x).
We shall also have some variants of the hypotheses concerning the behaviour
of functions g(x, y) and h(x, y).
Hypothesis 3.5. The function h(x, y) is defined and continuous in the strip
C.1) {(x,y):xeI,yeQt}.
Hypothesis 3.6. The function g(x,y) is defined and continuous in the strip
C.2) {(x,y): xel, yeQx}.
Hypothesis 3.7. For every fixed xel the function gx(y) = g(x,y) is invertible
in Qx.
In the present section we aim at proving a theorem analogous to theorem 2.1.
However, we start with some weaker results.
Lemma 3.1. Let fe R\[I], where ? is an endpoint of I (*), ?$I, and let hypotheses
3.1, 3.3 and 3.6 be fulfilled. Thenfor any xoe Iand for an arbitrary function <p0 e Ф[10],
where /o = <xo,/(xo)> or </(x0), xo>, fulfilling the condition
C.3) <Poif(xo)] = g{x0, <Po(xo)),
there exists exactly one function <p(x) belonging to the class <&[J0], where J0 = (x0, 0
or (?, xoy, satisfying equation A.1) in Jo and fulfilling the condition
C.4) <p{x) = <po(x) for xelo.
The function <p(x) is continuous in Jo.
A) Also in this chapter ?, may be infinite; cf. the faotnote on p. 47.

1. Solution depending on an arbitrary function 69
Proof. We may assume that ? is the left endpoint of /, so that /0 =
and Jo = (?, xoy. By theorem 0.4 xe60[f] for every xeJ0 and, moreover, if we
assume E=J0, хе<?0ЛЛ- Consequently, according to the terminology of Chapter
I, § 2, we have
C.5) ^od|>]=n^-
n=0
Q° is the set of reals, Qlx = Qx (cf. A.5)). We shall prove that
C.6) fi" = fi* for n>l.
For this purpose let us define the functions д„(х, у), n^O, by relations A.4). C.6)
means that the function д„,х(у)=д„(х, у) is defined for у е Qx. For n=\ this is true
and, moreover, g1 X(QX) = FX. Suppose that gnjy) is defined for yeQx and gnx(Qx)
<=rfn-i(x). Then, in view of hypothesis 3.3, for arbitrary x eJ0 and у е Qx we have
an,x(Gx) <=rfn-Hx)<=QfHx),i.e. gn+1,x(y)=g(f"(x),gniX(y)) is defined and belongs to
Г/п(х). On account of the induction principle gn,x(y) are defined in Qx for every
n^l (and gnx(Qx)<= rfn-i(x)). Thus QX<=Q" for every n^l. The inclusion ?2"c??x
results from A.5). Consequently C.6) holds and C.5) becomes
PmodM=fi* for xe-V
Evidently, Jo—f(Jo) = (f(Xo)>xoy. Thus by theorems 0.5 and 1.2 there exists
exactly one function 9>(x) belonging to Ф[/о]. satisfying equation A.1) in Jo and
coinciding with <po(x) on (/(x0), xo>. By C.3) <p(x) fulfils C.4). We must prove
that <p(x) is continuous in Jo.
We have
We shall prove by induction that <p(x) is continuous at every xe Kn= /0 u
П
U </'+ 1(x0),fi(x0)) for n=0, 1,2, ...' For xe(f(xo),xoy <p(x) is continuous in
i = 0
view of C.4). Further, we have
lim <p{x)= lim <p\_f{x)~\= lim ^(x,9>(x))
jc-»/(jco) — 0 x^*xq—0 x^*xq — 0
= lim g (x, (po{xj)=g(x0, <po(xo)) = <Po [/(^o)] = <P [КхоУ]>
x->xo-0
since <po(x) is continuous in /0, ^(x, y) is continuous in strip C.2) and C.4) holds.
On the other hand, the continuity of <p0 in /0 implies that
lim <p(x}= lim <po(x)=<polf(xo)'] = <plf(xoy].
x^f(xo) + 0 x->f(xo) + 0
Consequently, <p(x) is continuous at/(x0), i.e. it is continuous at every point x ? Ko.
Supposing <p(x) continuous at every point of Kn, n>0, we have for xeKn+i

70 CHAPTER III. Continuous solutions
and the continuity of <p in Kn+1 follows from that of <p in К„ and from the continuity
of f~l and g.m
Lemma 3.2. Let fe R^[F\, where ? is an endpoint of I, ?$I, and let hypotheses
3.1,3.2,3.5 be fulfilled. Then for any x0 e Iand for an arbitrary function <p0 e Ф[/о]A)
fulfilling
C.7) 9>oC*o) = h(x0, <Polf(xo)J) ,
there exsists exactly one function <p(x) belonging to the class Ф[A— Jo)vlo], ful-
fulfilling condition C.4) and such that for every x el—Jo relation A.2) holds. The func-
function (p(x) is continuous in (I—Jo)ulo.
The required function <p(x) can be constructed successively in the intervals
<Гя(.хо),Гя~1(хоУ) or (/~"*o),/""(*o)> with the aid of relation A.2). The con-
continuity of <p is then verified as in the proof of lemma 3.1. On the whole, the proof
of lemma 3.2 is very similar to that of theorem 3.2, and therefore we do not enter
into details here.
Theorem 3.1. (Kordylewski, Kuczma [1], Kordylewski [3].) Let f e R%[I], where
? is an endpoint of I, йф I, and let hypotheses 3.1, 3.4, 3.6 and 3.7 be fulfilled. Then
equation A.1) has in I a continuous solution <p e Ф[1] depending on an arbitrary func-
function. More precisely, for any x0 e / and for an arbitrary continuous function <po(x)
belonging to Ф[10], where /o = <^o./(^o)) or </(x0), xo>, and fulfilling condition
C.3), there exists exactly one function <p(x) belonging to the class Ф[1], satisfying
equation A.1) in I and fulfilling condition C.4). The function (p(x) is continuous in I.
Proof. Let h(x,y) denote the inverse function (with respect to y) to g(x,y),
which exists by hypothesis 3.7. For every fixed xe/ the function hx(y)=h(x, y)
is defined in Гх and hx(Fx)=Qx. Thus, according to hypothesis 3.4, hjj) is defined
in Q* and hx(Q*)=Qx. Therefore, th; function h(x,y) fulfils hypotheses 3.2 (where
Ax = hx(Q*)) and 3.5.
Functional equations A.1) and A.2) are equivalent in the function class Ф[1].
Thus our theorem results from lemmas 3.1 and 3.2.¦
§ 2. Extensions of s»lati»ns. In the present section ws shall prove sams exten-
extension theorems, which will Ьг useful later. Theorem 3.1 says that if we know a solu-
solution in an interval <xo,/(xo)> or </(x0), xo>, it can be extended onto the whole
interval / considered. Now we shall show that if we know a solution in a somewhat
larger set, then its extention is possible under weaker conditions.
Theorem 3.2. (Kordylewski [3].) Suppose that /e5?[/]B), /0=/п(?-<5, й + 5),
where 5 is a positive number, and let hypotheses 3.1, 3.2 and 3.5 be fulfilled. If <p0
e Ф[10] is a continuous solution of equation A.2) in Io, then thzre exists exactly one
function <реФ[1] satisfying equation A.2) in I and fulfilling condition C.4). This
function <p(x) is continuous in I.
5 I О We preserve the notation from lemn a 3.1.
-a , B) ? may belong to / or not.

2. Extensions of solutions 71
Proof. О We may assume that ? is the left endpoint of /. We may also as-
assume that ? ? /.
We put
m(x)= sup/@-
This function is continuous and increasing in /. Moreover, we have
C.8) /(x)<m(x)
in / and by induction
/"(*)<«"(*) f°r «=0,1,2,...
We choose xo=? + S and define a sequence х„:
xn+1 = sup {x:xel,m(x)<xn}.
Let us note that on account of the monotonicity and continuity of m{x) we have
the relation
C.9) m(x)<xn for x<xn+1, n = 0,l,2, ...
Moreover, since m{x)<x, the sequence х„ is increasing and is finite or infinite.
In the former case the last existing х„ is the right endpoint of /, in the latter case
х„ tends to the right endpoint of /. Thus in any case B)
J-/o = U <*»»*»+1)-
Now, for x e <x0, Xj) we put
C.10) <p(x) = h(x,<po[f(xfl).
By C.8) and C.9) for xe^xo,Xj) we have/(x) e/0, and so <po[f(x)] is defined.
Since <poe Ф [/„], we have q>0[f(x)] e Qf(x) = Q*. In view of hypothesis 3.5 the func-
function <p(x) is defined by C.10) and is continuous in {xo.^i)- Moreover, we have
by C.10)
lim <p(x)= lim h(x (
and, since <po(x) satisfies equation A.2) in Io,
lim <po(x)= lim h(x, <polf(x)l) = h(x0, <potf(xo)']),
x->xo-0 x->xo-0
which shows that the function <p(x) defined in <?, xt) as
(!) One could obtain a shorter proof making use of theorem 1.3, but such a proof would not
have a constructive character. It is not easy to indicate a set that has exactly one point in common
with every orbit contained in /, though it may easily be shown that there exists a set with the above
property which is contained in Io. The proof presented here is constructive.
B) Here we assume that / does not contain a right endpoint, say b. If b e I, then, after having
defined <p(x) for x e (g, b), we define it additionally for x=b as <p(b)=h(b, <p[f(b)\).

72 CHAPTER III. Continuous solutions
(<po(x) for xe<<f;,x0),
<P(X) = \, , Г,, ЧПЛ С , N
?>о [/(*)]) for хе<хо,Х!)
is continuous in <?, xx). Moreover, by hypothesis 3.2 and by the assumption con-
concerning <po(x) we have <p(x) e Qx for every x e <?, xj, i.e. уеФ [<?, Xj)].
We proceed further by induction. Having defined <p e Ф[(й,х„)] in <?, х„),
we define it in <х„, х„+1) putting cp{x) = h(x, <p[f(x)]) and then verify its continuity
in <?, xn+1), and we show that <p e Ф[<<!;, xn+1)]. It follows directly from the con-
construction that <p{x) so obtained satisfies equation A.2) in /, fulfils condition C.4)
and is the only function with these properties. ¦
Let / be a submodulus interval for/(x), and suppose that hypotheses 3.1, 3.2
and 3.5 are fulfilled. Let <po(x) be an arbitrary function belonging to the class Ф[1].
We may define a sequence of functions <pn(x) by
It is evident that if the sequence <р„(х) converges in / to a function <p (x) belonging to
Ф[1], then <p(x) satisfies equation A.2) in /.
Lemma 3.3. If xoe/ and the sequence (pn{x) converges for x=/(x0) to a limit
к e Qf(xo), then (pn{xQ) converges to h{xo,X) e QXo.
. Proof. This results directly from relation C.11) and hypotheses 3.2 and 3.5.И
Theorem 3.3. (Kordylewski [3].) Suppose thatfeS%[I] 0), /0=/n(?-c5, ? + 3),
where 6 is a positive number, and let hypotheses 3.1, 3.2 and 3.5 be fulfilled. Further,
let <Po(x) be a function belonging to the class Ф [7] and define the sequence <р„(х) by
relation C.11). If cpn(x) converges in Io to a continuous function belonging to Ф[10],
then (pn{x) converges in the whole of I and its limit <p (x) provides a continuous solu-
solution of equation A.2) in I belonging to the class ФЩ.
Proof. For every x0 e I we have lim /"(xo) = ^ and thus
C.12) /"(xo)e/o for n sufficiently large.
It follows thus from lemma 3.3 that the sequence <р„(х) converges in / to a function
(p{x) belonging to Ф[Г\ and satisfying equation A.2) in /. The function <p(x) restric-
restricted to /0 by hypothesis is a continuous solution of A.2) in /0. By theorem 3.2 cp{x)
can be uniquely extended from /0 to a solution <p* e Ф [I] of equation A.2) in /
and (p*{x) is continuous in /. But the function <p (x) itself is an extension of <p (x)
restricted to /0 to a solution of A.2) in /; thus the two functions must be identical.
Consequently <p(x) is continuous in /.¦
§ 3. Unique solution. Now we are going to prove that in some cases the con-
continuous solution of equation A.2) in / may be unique if ? e I. Since ? e f&ilf], for
any solution (p of A.2) the value t\ = (p{?) must fulfil the equation
C.13)
? may belong to / or not.

3. Unique solution 73
Theorem 3.4. (Kuczma [8]. Cf. also Kordylewski [3].) Suppose that /e S° [I],
? e /, and let hypotheses 3.1, 3.2 and 3.5 be fulfilled. Further, let r\be a root of equation
C.13) such that r\ e Q^ and suppose that the inequality
C.14) \h(x,y1)-h(x,y2)\^\y1-y2\, 0<$<l,
holds in a neighbourhood of (?, rj) {}). Then there exists exactly one function <p{x)
which is continuous in I, belongs to Ф[Г\, satisfies equation A.2) in I and fulfils the
condition
C.15)
This function is given by
C.16) <p{x)= lim <pn{x),
where the sequence (pn{x) is defined by C.11) and <po(x) is an arbitrary continuous
function belonging to Ф [Ц and fulfilling condition C.15).
Proof. We may assume that ^ is the left endpomt of/. We choose ac>{ and
a d> 0 such that the set
C.17) D=(t;,cyx<r,-d,r, + dy
is contained in QnQ* and relation C.14) holds in D. Further, let с be chosen in
such a manner that
C.18) \h(x,ti)-h(Z,ti)\^(l-S)d for xe<?,c>,
which is possible, since h(x,r[) is continuous at ?.
Let J5" be the space of continuous functions <p on <?, c> fulfilling condition
C.15) and
C.19) \<p(x)-ri\^d for xe<?,c>.
It follows from C.19) that <p e ^ implies <p e Ф[(?, с>]. We introduce in J5" the
metric
C-20) p(<pt, <p2)= sup \<Pi(x)-<p2(x)\.
«¦ c>
As is well known, J^ is then a complete metric space.
We define the transform
C.21) V{x)=h{x,<pU{x)})
for <p e ,F. i//(x) is evidently continuous in <?, c>; moreover, we have for x e <?, c>
according to C.13), C.14), C.18) and C.19)
(') A neighbourhood is here meant relatively to Q, i.e. C.14) is postulated in the common
part of an open disc around (?, tf) and of ?2.

74 CHAPTER III. Continuous solutions
= \h(x,q>\J(x)$-h(?,ri)\
and by C.15) and C.13)
which means that C.21) transforms J5" into itself. Further, we have in virtue of
C.14) for \//1(x) = h(x, <pAf(M) and y/2(x) = h(x, <p2[f(x)]), fi.^e ^,
\Vi(x)-щ(х)\ = \h(x, ^[/(x)])-h(x,
whence
i.e. C.21) is a contraction map. On account of Banach's theorem there exists a
unique fixed point of C.21) in J^ given as the limit of successive approximations.
In other words, there exists a unique continuous function <ре&~сФ[(!;, c>], sat-
satisfying equation A.2) in <?,c> and fulfilling condition C.15), given by C.16).
Putting /0 = <^,c) and applying theorem 3.3 yields the existence of the desired
solution. Its uniqueness follows from the uniqueness of the continuous solution
in <?, c> and from the fact that any continuous function <p e Ф [I] fulfilling con-
condition C.15) restricted to <?, c> belongs to & provided that с had been chosen
sufficiently small. ¦
In the above theorem the condition of the continuity of the solution of A.2)
may be replaced by the condition of boundedness. Namely, we have the following
Theorem 3.5. (Bajraktarevic [8].) Let f{x) be defined in a submodulus interval I
and let Q be the strip Jx(—d, d), where IcJ and d^co. Suppose that the function
h(x, y) is defined in Q and fulfils condition C.14) in Q, and that there exists a d! <d
such that (x)
C.22) |/r(x,j)|<max(|j|,d') in Q.
Then there exists exactly one function <p (x) which is bounded in I by a constant less
than d, belongs to Ф[Г\ and satisfies equation A.2) in I. This function is given by for-
formula C.16), where the sequence <pn{x) is defined by C.11) and <po{x) is an arbitrary
function belonging to Ф [I] and bounded away from d.
The proof of the above theorem is similar to that of theorem 3.4. As the
space & we now take the set of functions <p(x) fulfilling the condition
|<p(x)|<M for xel, d'<M<d,
with metric C.20). ¦
(') Instead of C.22) one may assume fl=/x <-</,</>,</< +со, and \h(x,y)\<d in Q.

3. Unique solution 75
The above theorem may often be applied in cases where theorem 3.4 is useless.
E.g. the equation
C.23) <p(x)=jr<p(x—i)
has, by theorem 3.5, the unique bounded solution <p(x) = 0 in (—со, со), while on
account of theorem 3.1 equation C.23) has in ( — со, со) a continuous solution
•depending on an arbitrary function. Another example will be discussed in § 6.
Finally, let us note that condition C.14) is fulfilled, for example, in the case
where the function h(x, y) has a continuous derivative dhjdy in the set C.1) and
dh
Yy(i>
§ 4. Lack of uniqueness. Relation- C.14) is essential for the uniqueness of a
continuous solution of equation A.2). If it is not fulfilled, then, in general, A.2)
has in / a continuous solution depending on an arbitrary function (compare the
situation for the linear equation). We shall now investigate this very case. It
will be more convenient to consider the equation in form A.1).
Theorem 3.6. (Kuczma [5].) Suppose thatfe S° [I], ^ e I, and let hypotheses 3.1,
3.3 and 3.6 be fulfilled. Further, let ц be a root of the equation
C.24) n
such that цеп/, and suppose that the inequality
C.25) \д(х,У1)-д(х,У2)\^9\У1-У2\,
holds in a neighbourhood of (?, rf) (*). Then there exist positive numbers c, d such
that for every xoel, 0<|x0 — ?\<c, and for every solution cp(x) of equation A.1)
such that
<3.26) \<p(x)-ri\^d for xelo,
where I0 = {x0,f(x0)) or (/(x0), xo>, we have
C.27) \im<p(x)=r].
Proof. We may assume that ? is the left endpoint of/. We choose c>0, d>0
in such a manner that the set
is contained in Q, relation C.25) holds in D and, moreover,
<3.28) \g(.x,ti)-g(Z,ti)\^(l-S)d for xe<c,
If (p(x) is a solution of equation A.1) in / and \(p{x) — t\\^d for an хе(?,
then by C.24), C.25) and C.28)
Compare the footnote on p. 73.

76 CHAPTER III. Continuous solutions
^9\<p(x)-ri\ + (l-
This shows that if a solution <p{x) of A.1) in / fulfils C.26), then \<p(x)-r\\^d in
UfVol
0 °o
Now we shall show that (?, xo)c\Jf\Io). Set х„ = f\x0). By theorem 0.4
О oo
the sequence xn decreases to ?, and thus (?, xo> = U (х„+1,х„>. Since f(xn+1) = xn+2
о
and/(xn) = xn + 1, for every у е (х„+2, х„+1> there exists (by the Darboux property)
an хе(х„+1,х„> such that f(x)=y. Thus (х„+2, х„+1>с/((х„+1, х„>), whence
(х„+1, х„>с/"(/0) for every и^О. Consequently
00 00
/J» -у \ j— /J^ -у \ ___ I I /*-у* -у* \ j— I I /*"/T 
(,C ) Xo) С ^, X0/> — i^J (Xn+ j , Хп-> С {JJ (l0).
о о
Let <p{x) be a solution of equation A.1) in /fulfilling condition C.26). It follows
from what has already been proved that
C.29) \(p(x)-n\^d for хе(?,х0).
Let us write
tg{x,<p{x))-g{x,rj)
k(x)=\ (p{x)-r\
VO if (p(x) = t],
F(x)Ug(x,n)-g^,ri),
Then we have for x e I
and in view of C.29) and C.25)
Moreover, by C.29) the function xix) is bounded in (?, x0) and evidently lim F(x)
= 0. From theorem 2.8 we obtain the relation lim/(x) = 0, i.e. relation C.27). ¦
Theorem 3.7. Let feR°[I], ?el, and let hypotheses 3.1, 3.4, 3.6 and 3.7 be
fulfilled. Further, let ц be a root of equation C.24) such that rj e fi4 and suppose that

4. Lack of uniqueness 77
inequality C.25) holds in a neighbourhood of (E,,tf). Then equation A.1) has in I a
continuous solution <p e Ф[1] (*) fulfilling C.15) depending on an arbitrary function.
Proof. Results from theorem 3.1 and З.б.и
Remark. If we replace in the above theorem hypotheses 3.4 and 3.7 by hy-
hypothesis 3.3, we obtain a continuous solution of A.1) in a neighbourhood of ?.
One must then appeal to lemma 3.1 instead of theorem 3.1
Let us note that condition C.25) is fulfilled, for example, in the case where the
function д{х, у) has a continuous derivative дд/ду in the set C.2) and
It may quite well happen that equation C.24) has more roots. Then the behaviour of
the solutions of A.1) may be different in a neighbourhood of each root. E.g. in the case
of the equation
C.30)
where !—(,— со, со), J2=(—со, со)х@, со), i=Q, equation C.24) has two roots: »/i = 1 and
42 — 2. We have дд/ду = 2У~1 log2 and thus
@,1)
= |log2|<l and
да
/@,2)
by
I log 4| >1.
(log means here the natural logarithm). Consequently equation C.30) has a unique con-
continuous solution fulfilling ?>@) = 2 B) and an infinity of continuous solutions (a solution
depending on an arbitrary function) fulfilling p@) = l.
Another example will be discussed in §6.
§ 5. The function f{x) decreasing. In the investigations of equation @.49) the
case of a monotonic function/(x) is particularly important. §§ 1-4 cover the case
where/is increasing. Now we shall see that the case of a decreasing /can be reduced
to the former.
Let f(x) be a strictly decreasing continuous function in a submodulus interval
I. Then besides a unique ?e(?i[/] and the points belonging to (?0[f], there may
occur in I points from G2[/]- In the sequel we shall assume that there is only a finite
number of points belonging to G2[/] in I. Then those points can be matched in pairs:
to every a eS2[/], <*<?, there corresponds a unique /?=/(<*)><!;,/?e (?2[/]> and
conversely. Thus, let
C.31)
О More exactly, the class Ф[Г\ should be replaced here by the class of functions <p(x) fulfilling
C.29) with x0 and d suitably chosen.
B) It is the solution q>(x)=2. In order to obtain the uniqueness we write equation C.30) in the
log (pixjl)
form ?>(*)= H and apply theorem 3.4.
log 2

78 CHAPTER III. Continuous solutions
be the full sequence of points in / which belong to G2[/]- Then (*)
C.32) f(«d=pt, /(&)=«», f(Ad = Bt, f{BD = Au i = l,...,«,
C.33) f{A0)cB0, f(B0)cA0,
where
C.34) A{=(cc{,ai+1), Bt=(fit+1,0d, K+i=?„+! = ?], i=l,...,n,
C.35) A0 = (e,«i), Bo = (/?i,b)
(the possibility ax = px = ?, included), a and Ь being the endpoints of/. For the sake
of simplicity we assume that the interval I=(a, b) is open (finite or not).
Summarizing, we make the following assumptions regarding the function fix).
Hypothesis 3.8./(x) is a strictly decreasing continuous function in a submodulus
interval I=(a, b). In I, besides the unique point ? e GiJ/], there are finitely many
points C.31) (possibly none) belonging to GJ2[/].
We are going to investigate equation @.49) in the form A.1) (a similar treat-
treatment is also possible for equation A.2)). Under hypotheses 3.1, 3.3 and 3.6 we
may form a new function
C.36) G(x,y)^glf(x),g(x,y)-].
Function C.36) is again defined and continuous in the strip C.2). Putting EX=GX(QX)X
where Gx(y) = G(x, y), we have by hypothesis 3.3
C.37) Ех=дПх)(Гх) с дЯх)(пПх)) = ГЯх) с Qf4x) .
The following theorem allows us to reduce the present case to those formerly
dealt with.
Theorem 3.8. (Kordylewski [2].) Let hypotheses 3.8, 3.1, 3.3 and 3.6 be fulfilled.
Then every solution <p e Ф[1] of equation A.1) in I satisfies also the equation
C-38) 9,[/2(x)] = G(x,9>(x)),
where G(x, y) is given by C.36). Conversely, if <p0 e Ф[А] is a continuous solution
of equation C.38) in a set A which is a union of some A/s O'=0, ...,n) and <xt's
(i= 1, ..., ri) (cfi C.31), C.34) and C.35)), then the function
(<po(x) for xeA,
C.39) ^UCT'OUPoLT'OO]) for xef(A),
is a continuous solution belonging to the class Ф[Ли/(Л)] of equation A.1) in A
4
Proof. The first part of the theorem is trivial. We shall prove the second.
For a rigorous proof of these simple facts cf. Kordylewski [2].

5. The function fix) decreasing 79
By C.31), C.34) and C.35) we have A<=(a,?). Hence f(A)<= (f, b) and so A
nf(A) = 0. Since 9оеФ[А], (РоУ^Шей,^ and ^СГЧ*), PoLT4*)]) »
meaningful. Thus the function 9>(х) is by C.39) unambiguously defined in the
set A (jf(A). Its continuity results from the continuity of <p0, f and g.
If xeA, then <p(x) = <po(x) eQx by definition. If xef(A), then #>(x) =
5 (/" 1(x), <Po\J~ 1(x)])e ^f-4x) •= Ц* by hypothesis 3.3. Consequently, <реФ [Akj/(A)].
Now we shall verify that cp(x) satisfies equation A.1) in Akj/(A). If xeA,
then fix) ef(A) and we ha.ve<p(x)=<po(x) and ?>[/(*)]=<?(/-'(/(x)), ^[Г1 (/(*))])
=0(х,9>о(*))=0(*= ?>(*))¦ If xef(A), then f'^eA and by C.32) and C.33)
f(x) e ^4. Thus we have
and, since <p0 is a solution of C.38),
' \x), 9-0 [Г
Remarks. 1. If 9>0(x) is defined in ^40> then the function C.39) is defined in
A0Kjf(A0)<=A0KjB0, not necessarily in the whole of AokjBo. In the set B0—f(A0)
(p (x) should be determined (if it is possible) from the relation
C.40) Po[/(*)] = »(*, ?(*))¦
Let us note that f(B0-f(A0)) cz Ao (cf. C.33)).
2. In theorem 3.8 the case ?, e A may also be admitted. Then one must require
additionally that
C-41) <Po(O = ri,
where r\ is a number fulfilling C.24).
As we see, the problem of solving equation A.1) is reduced to that of solving
equation C.38). The latter is of the form discussed in the previous sections or may
easily be reduced to it.
We shall prove here one theorem, an analogue of theorem 3.1. A number of
further theorems can be proved similarly.
Theorem 3.9. (Kordylewski [2].) Let J be a set of indices i, O^i^n, and put
A = \J At, B= U Bt. If hypotheses 3.8, 3.1, 3.4, 3.6 and 3.7 are fulfilled, then equa-
equals iEJ
tion A.1) has in AkjB a continuous solution <p e Ф[АиВ] depending on an arbitrary
function.
Proof. Since AtKjBi, ieJ, are disjoint open sets, it is enough to prove our
theorem for a single set А^Ви i being fixed.
As we pointed out previously, the function G(x,y) defined by C.36) fulfils
hypothesis 3.6. In virtue of hypothesis 3.4 the inclusions in C.37) may be replaced

80 CHAPTER III. Continuous solutions
by equalities, and so we have Ex=QfZ(x), which is an analogue of hypothesis 3.4
with g and/replaced by G and/2, respectively. Finally, hypothesis 3.7 implies that
for every fixed x e / the function G{x, y) as a function of у alone is invertible. Let
H(x, y) be its inverse.
It follows from hypothesis 3.8 that the function /2(x) is strictly increasing
in At, f\x)=?x in At, PiA^cAi. Thus if iVO, then/2 belongs either to Д°[4?],
or to R°t+i[At], depending on whether f2{x)<x or/2(x)>x in At. By theorem 3.1
equation C.38) has in At a continuous solution <poe Ф [At] depending on an arbi-
arbitrary function. This solution can be uniquely extended onto AtKjBt (cf. C.32)) by
means of theorem 3.8.
If/=0, then f2 e R^JAo] provided f\x)>x in Ao. If/2(x)<x, then f~2 e
e R0ai[A0]. Equation C.38) is equivalent to
C.42) 9>[/-2(х)]=Я(/-2(х),9>(х))
and it can be verified just as in the previous case that theorem 3.1 applies to equa-
equation C.42).
Thus equation C.38) has in Ao a continuous solution <p0 e Ф[-40] depending
on an arbitrary function. This solution may be extended onto A0Kjf(A0) in virtue
of theorem 3.8. In Bo—f{A0) the solution can be determined from relation C.40),
since g is invertible. Denoting by h{x, y) the function inverse to g{x, y) with respect
to the second variable, we have
valid for x e Bo. Thus <p(x) is continuous. ¦
Similarly, one can deal with the case where A contains an a. Without going
into details we give here the result regarding continuous solutions of equxtion
A.1) in the sets <a;, ai + 1)u(/?,-+1, #> and (a;,ai+1>u</?;+1, /?,) under the ad-
additional hypothesis that the derivative дд/ду exists and is continuous.
Let yt e пщ be a root of the equation
and put
Theorem 3.10. (Kordylewski [2].) Let hypotheses 3.8, 3.1, 3.4, 3.6 and 3.7 be
fulfilled and assume that the derivative dgjdy exists and is continuous in the set C.2).
Then the number of continuous solutions (p e Ф[<а;, ai+1)u(/?;+1, /?;>] or
Ф[(аг,аг+1>и</?i + 1,/?i)] О of equation A.1) in <a(, ai+1)u(/?i+1, A> or
(a;, ai+1>u<jSi+1, /?,-) respectively, which fulfil the condition
(!) The theorem is true also for the set (a, a.{yKj(fiub). If i=n, i.e. аг+1 = /?т=?> then condi-
condition C.44) must be replaced by C.41) where tj is a root of C.24).

5. The function fix) decreasing
81
C.43)
resp.
C.44)
is determined by the table below.
<p(eci+1)=yi+1
Continuous solution
in the set
<a4,ai+1)u(ft+i,/?4>
(a(,a4+1>u</?4+1,/?4)
fulfll-
lingthe
con-
condition
C.43)
C.44)
in the case where
dg dg
dy dy
da da
^(«4,»)-^-(/M<)
>1
<1
дд дд
g- («f+l. П+l) • g- (A+i > 8M)
dg dg
J- («1+1 ' fi+l) • g- (fii+l > SUl)
>1
<1
f2(x)>x
in
(«i> «й-i)
depends on
an arbitrary
function
is unique
is unique
depends on
an arbitary
function
/4x)<x
in
(«*.«i+i)
is unique
depends on
an arbitrary
function
depends on
an arbitrary
function
is unique
In the uniqueness case the unique solution may be obtained in <аг,аг+1) or
(аг, ai+1> as the limit
(po(x) = -1im (р„(х)
П->ОО
of the successive approximations
and in (/?i+1, /?;> or </?j+1, A) with the aid of formula C.39).
§6. Examples. 1. Let F(x) be a bounded, continuous function on (—со, со). Consider the equation
C.45) V(x)=b(p(ax)+F(x),
where 0<6<l, 0<аФ\.
Let M be the upper bound of \F(x)\ on (— со, со) and put d'=MI(l—b). Then we may apply
theorem 3.5 with f(x)=ax, h(x, y)—by+F(x), I=(— со, со), d= + oo. Consequently, there is a
unique bounded solution of equation C.45) in (— со, со), which is given by formula C.16). Taking
)=O we obtain from C.11)
n-l
*.(*)=?*№*),
k-U
6 Functional equations

82 CHAPTER III. Continuous solutions
whence
CO
C.46) ?>(*) = ?**F(Ac).
ifc-0
Since the series converges uniformly in (— oo, oo), the solution obtained is continuous.
If we take F(x) equal to the distance from x to the closest integer:
and b=a~1,a being an even integer, then function C.46) is nowhere differentiable in (— oo, oo)(').
Similary, if we take F(x)=cos x, a being an odd integer and ab>l+~n, then C.46) yields an
example of a continuous, nowhere differentiable function, first given by Weierstrass. It is interesting
to note that, although the unique bounded solution is not differentiable at any point, equation C.45)
has in this case a C°° solution in (— oo, oo) depending on an arbitrary function (cf. theorem 4.1).
2. By a combinatorial problem one is led to the equationB)
C.47)
C.48) . tp(x)=—s/-<p(xZ)-2x,
Here f(x)=x2, g(x, y) = -y2~2x, h(x, y)=-y/-y-2x. If we take fl=@,1) x (- oo, 0), we shall
have Qx=(— oo, 0), Гх={— oo, — 2x) so that Гх С пхг. By lemma 3.1 equation C.47) has a contin-
continuous solution depending on an arbitrary function in every interval @, xo>, 0<x0<l. But since
Гхфпхг, we cannot apply theorem 3.1. And, in fact, no solution of C.47) is defined in the whole
@,1). To see this, let us take an arbitrary xo s @,1) and let us put
xn+i = \/xn, Уп=Ч>{хп), и=0,1,2,...,
where <p(x) is an arbitrary solution of equation C.47). From C.48) we obtain the recurrence
Уп+\=— \/ —Уп —
whence —уп+\<\1—Уп- Putting rn= — yn/xn+i we have rn+i<*Jrn. Thus, for n sufficiently large
rn<2, i.e. yn> — 2xn+i, and the values (p{xn) no longer exist (in the real domain, of course).
We shall also investigate the behaviour of continuous solutions of equation C.47) near x=0.
Equation C.13) (equivalent to C.24)) becomes (?=0)
and has two solutions: ^i = 0and г\г= — 1 (to these also г\г= — oo could be added). We have дд/ду=
— 2y, so that
да
— @,0) =0<l and in accordance with theorem 3.7 C) equation C.47) has a con-
tinuous solution fulfilling the condition p@)=»/i=0 depending on an arbitrary function. These
solutions are defined in a neighbourhood of zero. They cannot be continued to the right beyond
a certain critical point (depending on the solution). Cf. fig. 4.
(') De Rham [4]. (For a= 10 this is the example given by van der Waerden.) Cf. also de Rham
[2], [3], [5], Julia [25], Wunderlich [1].
B) Etherington [1]. Cf. also Etherington [2], Weddebura [1].
C) Here tix does not belong to flo but lies on its boundary. However, this fact does not affect
the validity of theorem 3.7.

6. Examples
83
The situation is different for r\2. Here we shall be able to apply theorem 3.4. For this purpose
we assume /=<0,|), fl=(-l, l)x(- oo, -}). Then QX=Q*=(- oo, -}), Ax= -(<x>,-\J±-2x)
I dh 1
and, since for 0<x<J we have -¦sji-2x<-i,AxQ QX forxel. Moreover, — = -
' дУ 2у/-у-2х
Fig. 4
dh
so that — @, — l)=i' and condition C.14) is-fulfilled in a neighbourhood of the point @, —1).
by
By theorem 3.4 there exists exactly one continuous solution (p{x) of equation C.47) in / fulfilling
the condition p@)= — !• This solution may be obtained as limit C.16), where
and as q>o(x) we may take, say, the function poW = — 1. In fact, the solution obtained is defined
in a somewhat larger interval, at least for 0<x<i (cf. Ethenngton [1]); compare fig. 4.
A number of other solutions of equation C.47) tends to — oo as x->0+0.

CHAPTER IV
DIFFERENTIABLE SOLUTIONS
§ 1. Preliminaries. We are going in turn to discuss differentiable solutions of
equations A.1) and A.2). The results of the present chapter are essentially due to
B. Choczewski (*).
We shall make use of the notation introduced in the previous chapter. Hypotheses
3.1-3.4 will be valid, but they will be specified in each instance. Hypotheses 3.5-
3.7 will be replaced by the following ones:
Hypothesis 4.1. The function h(x,y) is defined and of class С in set C.1).
Hypothesis 4.2. The function g{x,y) is defined and of class С in set C.2).
Hypothesis 4.3. The function h(x,y) is defined, of class С and h'y(x,y)^0 in
set C.1).
Hypothesis 4.4. The function g(x,y) is defined, of class C\and д'у(х,у)ф0 in
set C.2).
Also the function/(*) will be assumed to be of class С in a submodulus interval
/. In fact, we shall confine ourselves to the case/e Sr or feRr. The case of a de-
decreasing / may be dealt with as in Chapter III, § 5, and will not be given a separate
treatment here.
We define functions Hk(x, y, yx, ... yk) by the recurrent relations
Hi (x, у, Ух) = К{х, y)+h'y(x, y)f'{x) у у ,
D.1)
„ . .dtdHk ...fdHk dHk
dx 'J v"\dy "¦ dyk
k=l,
Similarly, we define functions Gk(x,y,ylt ...,yk) by the relations
C1) Choczewski [2], [3]. Cf. also Popovici [10], Bodewadt [1], J. M. Whittaker [6], van der
Berg [1], Bidecki, Kisynski [1], Kuczma [15], [24], Kordylewski, Kuczma [5], Pelczar [1], de
Rham [5], Choczewski [4]-[7], Anczyk [1].

1. Preliminaries 85
Gi (x , у, yt) = [/'(*)]"l (9x(x, У) + g'y(x, y) yi) ,
D.2)
rv-v.-i/'56*,50* . , ,8Gk
Gk+1(x, У, У1, ¦¦¦[^д <Г
k=i, ...,r.
The role of the functions Hk and Gk is seen from the following lemma.
Lemma 4.1. Suppose that the function f(x) is of class С in a submodulus interval
I and let hypotheses 3.1 and 4.1 or 4.2 be fulfilled. If(p(x) is а С solution of equation
A.2) or A.1) belonging to ФЩ, then its derivatives <p(k\x) satisfy the equations
D.3) <p«Xx)=Hk(x
or, provided thatf'(x)=?0 in I,
D.4) <p«Xf(x)-] = Gk(x,<p(x),<p'(x),...,<p(kXx)), fc=l,...,r.
Proof. Induction.¦
The next lemmas say something about the structure of the functions Hk
and Gk.
Lemma 4.2. Suppose that /(x) is of class С in a submodulus interval I and let
hypotheses 3.1 and 4.1 be fulfilled. Then the functions Hk(x ,У,У\, ¦ ¦ ¦, yk) (k = 1, ...,r}
are defined and of class С ~kfor (x, y) belonging to set C.1) and for arbitrary yx, ...,yk.
Moreover, we have
D.5) Hk(x,y,yi, ..., Ук) = Рк(х, у, yy) + Qk{x, у, y^+R^x, у, yt,..., yk_t),
where
D-7) Qk{x ,y,yk) = д^Л [f>ix)T yk ,
D.8) Rk(x,y,yy, ...,yk-1) is a polynomial in the variables yt, ...,yk-l, whose
coefficients are functions of the variables (x,y) of class C'~k with respect to
x and of class Cr~k+1 with respect to y.
Proof. At first we prove decomposition D.5). For к = 2 we have by D.1) and
in view of D.6) and D.7)
H2{x,y,yy, y2)=P2(x,y,y1) + Q2(x, y,y2)+f"{x) h'y(x,y) yt .
It is enough to put R2(x ,y,yt) =/"(*) h'y(x,y)y1.
If D.5) holds for k=p, 2^p<r, then by D.1) and according to D.6) and D.7)
we get

86 CHAPTER IV. Differentiable solutions
and it is enough to denote by J?p+1(x, j>, j^, ...,yp) the contents of the big square
brackets on the right-hand side of the above formula. Properties D.8) are easily
verified. The rest of the assertions of the lemma follows directly from D.5)-D.8).и
Similarly one can prove the following
Lemma 4.3. Suppose that f(x) is of class С in a submodulus interval I,f'(x)^0
in I, and let hypotheses 3.1 and 4.2 be fulfilled. Then the functions Gk(x,y,yi, ..,yk)
(k=l, ..., r) are defined and of class C'~k for (x,y) belonging to set C.2) and arbi-
arbitrary уi, ..., yk. Moreover, we have
D.9) Gk(x,y,y1,..., yk) = Sk(x,y,yi,..., yk-i)
We shall prove one more lemma.
Lemma 4.4. Suppose that f(x) belongs to Щ[Г\ (*), f'(x)^0 in I, and let hypo-
hypotheses 3.1, 3.4 and 4.4 be fulfilled. Let h(x,y) be the inverse function to д(х, у) with
respect to у and let the functions Hk and Gk be defined by D.1) and D.2), respectively.
If for a system of values xel, у eQx,y1, ...,yr we have
{4.10) z=g(x,y), zk = Gk(x,y,yu.7!,yk), k=l,...,r,
then z e Qf(x) and
<4.11) y = h(x,z), yk = Hk(x,z,z1, ...,zk), k=l,...,r;}
conversely, D.11) implies D.10).
Proof. Let us suppose that D.10) holds. Then ze Qf(X) by hypothesis 3.4 and
y=h{x, z) in virtue of the definition of the function h. We have further
hy(x,z)=— , hx(x,z)=— (y=h(x,z)),
д,(х>У) в,(х,У)
whence by D.1), D.10) and D.2)
+g'y(x,y)y1~])}=y1.
Now suppose that D.11) holds for к = 1, ...,p<r. Thus we have
Hp(x,g(x,y),G1(x,y,y1),...,Gp(x,y,y1,...,yp))=yp.
(!) ? may belong to / or not.

1. Preliminaries 87
Differentiating both sides of the above relation with respect to x,y,yt, ...,yp,
consecutively, we obtain
|
¦ дх ду дх дух дх "' дур дх
| дНр
ду ду дух ду ' дур ду
дНр dGp_
дУР дуР
We multiply the second of the above equations by yx, the third by y2, etc., the last
by yp+1 and add them. After taking into account D.2) and D.10) we get
f+/(*)^ z,+/(*) ^ za + ...+/W ^ zp+1=yp+1 ,
дх ду дух дур
whence D.11) (with k=p + l) results in view of D.1). The converse implication
may be proved in a similar manner. ¦
§ 2. Solution depending on an arbitrary function. In the present section we shall
prove analogues of theorems 3.1, 3.2 and 3.3 for differentiable solutions of equation
A.2). Also analogues of lemmas 3.1 and 3.2 could be proved in a similar way. We
leave this task to the reader.
Theorem 4.1. (Choczewski [2].) Let fe Щ [I] A ^r^oo), ?,$I{1), f'(x)^0 in I,
and let hypotheses 3.1, 3.4 and AA be fulfilled. Then equation A.1) has in I a C solu-
solution <p e Ф [/] depending on an arbitrary function. More precisely, for any x0 e I and
an arbitrary function <poe Ф[10]пС[10], where I0 = (x0,f(x0)y or </(x0), xo>,
and fulfilling the conditions
D.12)
^[/Оо)] = Gk(x0, <po(xo), <p'0(x0),..., p
there exists exactly one function q> e Ф[1\, satisfying equation A.1) in I and fulfilling
condition C.4). This function <p{x) is of class С in I.
Proof. By theorem 3.1 there is a unique function <p e Ф[Г\ satisfying equa-
equation A.1) in / and fulfilling C.4). It is continuous in /; we need only prove that this
function is in fact of class С in /.
(') In theorems 4.1,4.2 and 4.3, <J may also be infinite. Cf. the footnote on p. 47.

88 CHAPTER IV. Differentiable solutions
We may assume that ?, is the left endpoint of /, so that /0 = <J(x0), xo>. We put
xn=/"(x0)and we shall prove that q>{x) is of class С in every interval (xn, x0). For
л = 1 this is true in view of C.4). If л = 2, then <p(x) evidently is of class С
in (x2, Xx) и (xl, x0), since we have for xe(x2)x1)
Further, for x e (x2, xt) we have (cf. lemma 4.1)
As x^xt — 0, f~l{x)-*x0 — 0 and thus, according to lemma 4.3 (which asserts the
continuity of Gt) and conditions D.12),
D.13) lim <p(kXx)=Gk(xo,<po(xo), ..., p?)(*o))=?'(o)(*i) ¦
x-*xi — 0
On the other hand, in virtue of the hypotheses regarding q>0, we have
D.14) lim <p(k\x)= lim ^)(x)=9lo\xl).
x-»*i + 0 x-»xi + 0
Relations D.13) and D.14) show that the derivatives cp'(x), ..., <p(r\x) exist and
are continuous at the point xt. Consequently, q>{x) belongs to C[(x2, x0)]-
Now we suppose that <p (x) is of class С in (xn, x0) for an n > 2. For xe(xn+1, xn)
<p(x)=g(f~l(x), ^[/"Ч*)]) evidently is of class C. Further, we have for xe
(xn+1, х„)и(х„,х0)
and since all the derivatives (p'(x), ..., g)(r\x) exist and are continuous at the point
*„_!, there exist limits lim <p(k)(x), k=\, ...,r, which proves (the function cp(x)
х-*х„
being continuous in /) that <peC[(xn+1, x0)]. Induction then shows that
q, e CM, x0)].
In order to prove that <p e C'[{xx, b)] (where b is the other endpoint of Г), we
must write equation A.1) in the equivalent form A.2), where h{x,y) is the inverse
function to g{x, y) with respect to y. (This function exists and is of class С in view
of hypothesis 4.4.) Then, by lemma 4.4, we have
<Po(xo) = h(x
D.15)
<p(o\xo)=Hk{xo, <p0 [/(x0)]
k=\, ..., r .
Making use of A.2), D.15) and of relation D.3), valid whenever <p is of class С in
a domain containing x and/(x), we prove as before that q> is of class С in every
interval (xt, x_n). If, for an N,f~N{x0) is defined but/"*" \x0) is not, there is only
a finite number of intervals, and the last should be (xt, b) instead of {xx, x_N_ t). m

2. Solution depending on an arbitrary function . 89
Remark. If r= oo, then the above proof shows that <p(x) has derivatives of all
orders in /, i.e. 9>eC°°[/]. D.12) contains then infinitely many conditions; never-
nevertheless every function <p0 of class C00 in a closed interval contained in (f(x0), x0) or
(xo,f(xo)) may be extended onto Io, so that it will fulfil conditions D.12) (Whit-
(Whitney [I]).
Theorem 4.2. Let feS^rK1), let /0=/n (%—S, ?,+5), where 5 is a positive
number and let hypotheses 3.1, 3.2 and А Л be fulfilled. If q>o e Ф[/о] is а С solution
of equation A.2) in Io, then there exists exactly one function q> e Ф [I] satisfying equation
A.2) in I and fulfilling condition C.4). This function is of class С in I.
Proof. Let ? be the left endpoint of/. The function q>{x) exists and is continuous
in / on account of theorem 3.2; we need only prove that q> e Cr[/]. We adopt the
notation from the proof of theorem 3.2.
Suppose that <p(x) is of class С in (?, xn) for an л>0. This is certainly true
for л=0. We have for x e <xn, xn+ x) <p(x)=h(x, <p [f(x)]). In view of C.8) and C.9)
f(x) e (?, х„). Consequently (p(x) is of class С in (?, xn+1). Induction shows that
<p(x) is of class С in /. ¦
Theorem 4.3. Let the assumptions of theorem 4.2 be fulfilled and let <po(x) be a
function belonging to the class ФЩ. Define the sequence <pn(x) by relation C.11).
If <pn(x) converges in Io to a function belonging to Ф [Io] n C[I0], then <pn(x) converges
in the whole of I and its limit provides а С solution of equation A.2) in I belonging
to the class Ф [I].
Proof. The proof results directly from theorems 3.3 and 4.2.¦
We conclude this section with an example showing the importance of the assumption/'(*) ^0
in / for the validity of theorem 4.1. At first we define an auxiliary function
b—a 2nx—n(b+a) b+a
F(x;a,b) = —~sin—— — + —— , a<b .
4 2F — a) 4
This function has the properties (for F{x; a, b) we write simply F{x))
D.16) F(a)=ia, F(b)=ib, F'(a)=0, F'(b)=0;
moreover, F(x; a, b) is of class C1 and strictly increases in (a, 6).
Let pn be a dense sequence from the interval Q, 1) (e.g. the sequence of the rationals from
(i, 1)) and let the sequence х„ be defined by
Xl = l , Xn+l=iXn-
We define a sequence of points cn and a sequence of functions fn(x) on (xn+i, хпу by the relations
ci=Pi, cB=/B_i(/B_2(...(/i0«))...)), »>2,
, p(x; xn+i, cn) for x e <xB+i, cn} ,
[F(x;cn,xn) for x e (cn,xny.
0) i may belong to / or not.

90 CHAPTER IV. Differentiable solutions
Finally we define a function f(x) on @, 1):
D.17) f{x)=Mx) for xe(xn+i,xn), «=1,2,...
It follows from the properties of F (cf. in particular D.16)) that fe /fj [@,1)] and, moreover,
D.18) J*(Pn)=c«+i, /'(cn+i)=0 for и=1,2,...
Now let /=@, 1), fl=@, 1) x (— oo, oo), and consider the equation
D.19)
where f{x) is defined by D.17). Then all the conditions of theorem 4.1 are fulfilled (r=l) except
that /' (x)=0 for some x e /. We shall show that p(*)=0 is the only C1 solution of equation D.19) in /.
Let (p{x) be a C1 solution of D.19) in /. Then <p'(x) satisfies the equation
D.20) f'WWx)]=W{x).
Hence it follows that if, for an xo el, we have p'[/(xo)]=O, then also <p'(xo)=O. Now, on setting
x=cn+i equation D.20) becomes in view of D.18)
i.e. <p'[f"(pn)]=0, whence it follows that <p'(pn)=0, «=1,2,... Since the sequence pn is dense
in (i, 1), it follows that q>'(x)=0 in ft, 1) and consequently (cf. D.19)) p(*)=0in(i, i). In virtue
of theorem 3.1, p(x)=0 in /, which was to be proved.
Hovewer, let us note that in theorem 4.1 (as well as in theorems 4.8 and 4.9
in §6) it is enough to assume that Г(х)ф0 in a neighbourhood /0 of ?, (or, if ?, el,
that/'@^0)- Theorem 4.1 thus furnishes а С solution depending on an arbitrary
function in /0, and this solution can be extended onto / by the use of theorem 4.2.
§ 3. Existence theorem. In this and in the next section we prove two theorems
concerning С solutions of equation A.2) in / in the case where ? e /. We do not
admit ?, = ± oo, or r= oo.
If <p(x) is а С solution of A.2) in /, then the value r] = <p(g) must fulfil
equation C.13). Let r\ be a fixed root of this equation such that r\ ей,,. On account
of lemma 4.1 the values r\k = (f(K){?), k = l, ...,r, must fulfil the system of equations
D.21) Пк=Нк(^,П,П1,—,Пк), k = l,...,r.
Theorem 4.4. (Choczewski [3].) LetfeS\[I\, ? e/, and let hypotheses 3.1, 3.2
.1 be fulfilled. Suppose, further, that the inequality
D.22) .
holds in a neighbourhood oft; and, moreover,
D-23) \h'y
where r]eQ^ is a root of equation C.13). Then for any system of r]t, ..., r]r fulfilling
equations D.21) there exists at least one С solution (ре Ф\Т\ of equation A.2) in I
fulfilling the conditions
D-24)

3. Existence theorem 91
Proof. Since the proof of this theorem is rather complicated, we shall confine
ourselves to the simplest case r = 1. We assume that ? is the left endpoint of /.
There exist numbers 9, 0<9<l, с' el, с'ф?, and d'>0 such that
D.25)
Let К be an arbitrarily chosen positive number. We put
D.26, K0 = (bff
We may assume that с', d' have been chosen in such a manner that the inequalities
D.27)
hold in <<J, с'} x <fl - d', r\ + d'}. Next we put
D.28) - M = K+\r]1
and choose a c,
D.29) c<min (c',
so that the set
be contained inOnQ*.
To a given e > 0 we assign
D.30)
1 + M
The first derivatives of the function h are uniformly continuous in the closed set D
and/'(x) is uniformly continuous in <?, c>. Consequently there exist positive numbers
St =Si(s) and 32=S2(s) such that the inequalities
\K(X,y)-K&,y)\<e',
D.31)
\К(х,у)/Хх)-к'у(х.,у)Г(Щ<в'
hold in D whenever |3c-x|<<51( \y-y\<S2- We put
The number J depends only on the choice of s.

92 CHAPTER IV. Differentiable solutions
Now we shall define three function spaces: ^", IF and Sf. We define ^ as the
space of those functions a(x) which are defined in <<!;, c> and fulfil the condition
|a(x) — aCc)|<? whenever \x —x\<5,
where 5 = 5(s) is defined above. The set S~ is not empty (it contains at least constant
functions) and the functions meJ are equicontinuous in <<!;, c>.
Next we define IF as the space of functions u(x) which are defined and of class C1
in <?, c>. For м e J^ we define the norm
IImII=max ( sup |и (x)|, sup ]и'(*)|) •
«,c> <«,c>
Thus J*" is a normed vector space over the field of real numbers and the convergence
of a sequence une^ means the uniform convergence of the sequence of the functions
ы„(х) and of u'n(x) in <?, c>. Hence it follows that J*" is complete and therefore it
is a Banach space.
In turn we define a subset S? of IF. It consists of those functions <p&& which
fulfil the following conditions:
D.32)
D.33) (p(x)eQx for
D.34) |p'(*)-4i|<K for
D.35) p'e5
Let us note that if ^, (p2 ? ^> then
D.36) \\(р
In fact, according to D.32) and the mean-value theorem we have
|?>i(*)-<Рг(х)\ = (x-0 \<p'i(x)-<p'2(x)\
with ?<х<х, whence in view of D.29)
SUp \<Pl(.X)-?2(.x)\<(c-05Up \<p'i(x)-tp'2(x)\
Furthermore we have for <p e У
D.37) |p'(*)|<Af for xe«,c),
D.38) \<p(x)-r,\<M\x-Z\ for xe<?,c>.
D.37) results from D.34) and D.28), D.38) results from D.32) and D.37).

3. Existence theorem 93
We shall now prove the following statement.
(*) The set S? is a compact and convex subset of the space IF.
If q>n e?f, then by D.35) and D.37) the functions cp'n{x) are equicontinuous and
equiboundedon<^, c>. By a well-known theorem of Arzela one can choose a subse-
subsequence q>'kn uniformly convergent on <<!;, c>. According to D.36) the sequence (pkn
converges in SF. We shall prove that its limit belongs to ?f. For simplicity, let us
assume that the sequence <pn itself converges to a p0. Evidently <p0 eSF, since the
sequences cpn{x) and <p'n(x) are uniformly convergent in <?, c>. Conditions D.32), D.34)
and D.35) for q>0 follow readily from those for <pn on passing to the limit. Futher,
it follows from D.38) that for x e <?, c> and n=\, 2, ... we have (x, <pn(x)) eD
and, since D is closed, (x, <po(x))eD. But D <= Q, whence (po(x) e?2x. Thus У is
compact. The convexity of ?f is obvious.
We consider transform C.21) for cpe ??. We shall prove that:
(**) C.21) maps У into itself.
(***) C.21) is continuous in ?f.
Let 0> e Sf and let y/ be given by C.21). Differentiating C.21) we obtain
D.39) W'{x)=H,{x, <p [/(x)], ?'[/(*)])
= h'Jx,
y/'(x) is continuous in <<!;, c> by lemma 4.2. Hence у/ е IF. Setting x = % in C.21)
and D.39) we obtain by D.32), C.13) and D.21)
i.e. y/(x) fulfils condition D.32). Further, <p[f(x)] e QfM=?2*, whence by hypothe-
hypothesis 3.2
y/(x) = h(x,<p [/(x)]) e Л х czQx,
i.e. y/(x) fulfils condition D.33).
Now, by D.39) and D.21) we have
<\h'x(x, р[/-(х)])-Л^, 4)| + |4l| \h;(x, <р[Лх)-])Пх)-КД, t,)f'(Z)\
+ \K(x,
In view of D.38) and D.29) we have |{»[/(x)]-/;|<Af |с-?|<йГ and consequently
we can use inequalities D.25) and D.27). Thus, according to D.34) and D.26), we
obtain
i.e. y/(x) fulfils condition D.34).

94 CHAPTER IV. Differentiable solutions
In order to verify that y/ fulfils D.35), let us take an arbitrary ?>0. Let Зс, x
e<?, c> and let |3c-x|<5(e). We write shortly y=<p[f(x)], y=<p[f(x)]. Thus we
have
|«/(х)-«/(;)|<|й;(х, у)-К(х, у)\ + \<р'ЫЩ \h;(x,y)fXx)-h'y(x,
+ |/'Cc)K(x, y)\ [?'[/(*)]-
On account of the first inequality of D.25) we have
D.40)
and hence (x* being a point between/Cc) and/(x))
Thus \y—p|<M5(fi)<52 and since |3c—x|<5(?)<51( we may apply inequalities
D.31). Moreover, it follows from D.40) and from condition D.35) for cp that
D.41) \<p'U(x)-]-<p'lf(x)-]\<a.
Hence by D.31), D.37), D.25), D.41) and D.30)
which proves that y/(x) fulfils condition D.35). Thus statement (**) has been com-
completely proved.
Now let <pn e ?f, (pn-*<p0 (in the sense of the convergence in J^") and write y/n(x)
= h(x,<pn[f(x)]), y/0(x)=h(x,g>0[f(x)]). It follows from what has already been
proved that g>0, у/„, y/0 e ?f. The convergence q>n-+<Po means that <pn{x)-*q>o(x) and
<Pn(x)-*<p'o(x) uniformly in <?, c>. By D.38) the points (x, <pn(xj), (x,y/n(xj), (x, <po(x)),
(x,y/0(x)) belong to D. So the uniform convergence y/n(x)-*y/0(x) and y/'n(x) =
= H1(x,9>n[f(x)],(p'n[f(x)])^V'o(x) = H1(x,g>o[f(x)lg>'o[nx)]) in <?,c> (which is
equivalent to the convergence \\у„—Vo[[-*0) results from the unifonn continuity
of h(x, y), h'x(x,y) and h'y(x,y) in D and from the corresponding convergence
of cpn. This proves statement (***).
On account of the theorem of Schauder there exists in Sf a fixed point of transform
C.21), i.e. there exists a function <pe Sfcz Ф [<?,c>] n Cx[<^, c>] (and thus fulfilling
conditions D.24), r=l) and satisfying equation A.2) in <<!;, c>. This solution can
be extended onto the whole interval / on account of theorem 4.2. ¦
§ 4. Uniqueness theorem. A slight modification of the assumptions of theorem
4.4 allows us to prove the existence of a unique С solution of equation A.2) in /.
We start with a lemma.
Lemma 4.5. Suppose that feS\[I] and let hypotheses 3.1 and 4.1 be fulfilled.
Suppose further that there exist a closed neighbourhood U= <?, c> x (n — d, n+d} of

4. Uniqueness theorem 95
the point {?,, n), where n еЦ* is a root of C.13), and constants 3 and L such that
D.42) \K(x,y)lfXx)J\<9,
drh(x,y) d'h(x,y)
D.43)
ia,f-l
8xl8yr
<L\y-y \, i = 0,l,...,r,
holds for (x, y), (x, y), {x, y) e U. Then to arbitrary constants ak,bk, — со <ak<bk< со,
k=\, ...,r, there exist constants L0,Li, ...,Lr-i independent of x and such that
for (x, y, yu ..., yr), (x, y, yu ...,yr) from the set
D.44) Z=Ux(a1,b1yx...x(a,,bry
we have
_ _ _ r
D.45) \Нг(х,у,У!, ...,yr)-HXx,y,yi., •~,yr)\<L0\y-y\+ ^Lk\yk-yk\,
where
D.46) L,=$.
Proof. Let us fix a set Z of form D.44). By lemma 4.2 the function Hr is con-
continuous in Z and can be expressed in form D.5). Thus it fulfils a Lipschitz condition
with respect to the variables y1, ..., yr in Z, viz. it is a polynomial in these variables
with coefficients which are continuous functions in a closed set U. It remains to
prove that the function Hr fulfils a Lipschitz condition also with respect to the
variable y. But this is a consequence of formulae D.5)-D.8). In fact, on account of
D.6) and D.43) the function Pr(x, y, yi) fulfils a Lipschitz condition with respect
to y, and the functions Qr(x,y,yr) and ЛДх.у.у!, ...,yr_{) are at least of class
C1 with respect to у and at least continuous with respect to the remaining variables
(cf. D.7) and D.8)).
Relation D.46) results from D.5), D.7) and D.42). ¦
Theorem 4.5. О Letfe ^[7], ? e /, and let hypotheses 3.1, 3.2 and 4.1 be fulfilled.
Further, suppose that inequality D.23), where n e Q$ is a root of equation C.13), is
fulfilled and inequalities D.43) hold in a neighbourhood of(?,, rj). Then for any system
'h, •¦¦> *}r fulfilling equations D.21) there exists exactly one function (реФ[Г\ of class
С in I, satisfying equation A.2) in I and fulfilling conditions D.24). This function is
given by formula C.16), where the sequence <pn(x) is defined by C.11) and <po(x) is
an arbitrary function belonging to Ф[Г\п С[Г\ and fulfilling conditions D.24).
Proof. We may assume that that ?, is the left endpoint of/. We choose numbers
eel, d>0 and 3, 0<3< 1, such that in U=<?,c> x (n-d,n+d} inequalities D.42)
and D.43) are fulfilled and U<=?2n?2*. Then, for a suitable positive constant K, we
Choczewski [3]. Cf. also Kuczma [15].

96 CHAPTER IV. Differentiable solutions
have
D.47) \HAx,ti,tilf...,rir)-Htf,ri,rii,...,rir)\< — K for xe<(,c>,
Let us put
D.48) Mk= ? У + К, k=l,...,r-l, Mr=K,
i = *+l
and
ak=r]k-Mk, bk=rjk+Mk, k=l,...,r.
With the aid of the above constants and of the set U we define the set Z by formula
D.44). By lemma 4.5 inequalities D.45) are fulfilled in Z.
Now we choose positive numbers a and 9< 1 in such a manner that the following
inequalities are fulfilled:
D.49) ff<min(c-{,l),
D.50) i\nk\^+K^<d,
r-l r-k ffi r ._* 1—9
D.51) Z ZLk\r,k+i\- + KZLr-k-< — K,
*=o i=i i\ *=i K\ 2
D.52) rYL
YLk
u=0 (r-fc)!
We put /=<<!;, ^+ff>, and we define & as the space of the functions <p(x) on J
which belong to Ф [J]nC[J], and fulfil conditions D.24) and the condition
D.53) \<p(r\x)-rir\<K for xeJ.
In IF we introduce the metric
P L<P l. 4>i\ = SUP |?>ir)(*) - 9>2r)W| •
One can easily verify that the metric postulates are fulfilled. In particular conditions
D.24) guarantee that if p[<p1} (p2\=0,then(pl(x) = (p2(x)m J. Furthermore, on account
of Taylor's theorem, for (pl,q>2e^r and 0<A:<r-1 we have
where x is a point between ?, and ?,+<j, whence
r-k
|»>()»>()|
D.54)

4. Uniqueness theorem 97
The above relation shows that the convergence of a sequence <pn in IF is equivalent
to the uniform convergence of functions <pn(x) together with all derivatives up to
order r (inclusively) in the interval J. Consequently, & is a complete metric space.
We consider transform C.21) for pe^. At first we shall show that C.21)
maps & into itself. In fact, y/(x) is of class C[J] and by hypothesis 3.2 у/ е Ф[Т\.
It follows from C.13) and D.21) that y/ fulfils conditions D.24). We must show
that y/ fulfils condition D.53).
Since peF.we have for x eJ(we write ?/0=/;)
k=0,1, ...,r-
Since by D.53) |/r)Cc)|< |//r|+K, we get hence
^K/
D.55) \<p(kXx)-r,k\<Zh+i\^-.+K/---, xeJ, fc=0, 1, .... r-l
Thus, in view of D.49), D.48) and D.53), we have
D.56) \(p(k\x)-nk\<Mk for xeJ,k=l,...,r.
On the other hand, we obtain from D.55) for A:=0
and hence according to D.50)
D.57) \<p(x)-rj\<d for xeJ.
Now, we have
According to D.49), D.56) and D.57), for xeJ the points (x,<p[f(x)], <p'[f(x)],
..., (р(г)[/(х)]) and (x,//,//!, ...,rir) belong to the set Z. Thus we can use D.45)
and D.47). We obtain
Taking into account D.55) and D.53) we obtain hence
7 Functional equations

98 CHAPTER IV. Differentiable solutions
r~k ст! '-1 <f~k
!] *fi (r-fc)!
r-1 r-fc J r Jt
This together with D.51) yields
1-9
i.e. function (v(x) fulfils condition D.53).
Lastly, by D.45) and D.46) we have for arbitrary plt (ргв^ and
VW~ VB'X4 = \Hr(x,
*=1
since by D.49), D.56) and D.57) the points (x,g>j[f(x)l Vj[f(x)],...,q>lp[f(?c)]),
7 = 1,2, belong to Z. Further, in virtue of D.54), we obtain from the above inequality
r-l
a
P Oi» V2] < Z Lfc 7 tt;P [^i > ^2] + ^P [?>i, 9»2],
k=o (r—k)\
i.e. by D.52)
Thus C.21) is a contraction map. On account of the Banach theorem there exists a
unique fixed point of C.21) in IF, given as the limit of successive approximations.
In other words, there exists a unique function <p e SF с Ф [J]nC[J], satisfying equation
A.2) in /and fulfilling conditions D.24); it is given by C.16). It follows from theorem
4.3 that formula C.16) defines a Cr solution q>{x) of A.2) in the whole of /. The
uniqueness of (p{x) follows from the fact that every С solution <p e Ф [I\ of equation
A.2) in /, fulfilling conditions D.24), restricted to the interval J belongs to J5" provided
that a had been chosen sufficiently small. ¦
§ 5. Regularity of solutions. Now we shall prove two theorems to the effect
that the unique solution of equation A.2) furnished by theorem 3.4 or 4.5 is as
regular as the functions/(x) and h(x,y). In theorems 4.6 and 4.7 r may also be
infinite (but not ?).

5. Regularity of solutions 99
Theorem 4.6. (Choczewski [3].) Let feS\[I], ?e/, and let hypotheses 3.1, 3.2
and 4.1 be fulfilled. Further, suppose that inequality D.22) holds in a neighbourhood
oft; and, moreover,
D.58) \h'^\
where n e Q$ is a root of equation C.13) and
D.59) q<r.
Then for any system qlt ...,nq fulfilling the equations
there exists a unique function <pe Ф[Г\ of class С in I, satisfying equation A.2) in I
and fulfilling the conditions
D.60)
This function is in fact of class С in I.
Proof. Since the function h(x,y) is of class С and q<r, the #th derivatives
of h fulfil a Lipschitz condition in a neighbourhood of (?, n). Thus the existence
of a unique <p follows from theorem 4.5 (with r replaced by q). We need only prove
that <p e C'[I].
At first, let us assume that r< oo. In view of D.58) and D.22) we have
D.61) №,4)[/'@]'|<l for p=q,...,r.
We shall show that there exist unique nq+1> ...,nr such that the system п1г ...,nq,
nq+i> ¦¦¦> nr fulfils equations D.21). Suppose that we already know p—l^q of the
n's: rii,..., //p-i- The pth equation of D.21) is
i.e. by lemma 4.2 (cf. formulae D.5) and D.7))
Hence we can find np, since in view of D.61) 1 -h'y(?, n) [/'(?)PVO.
Relation D.61) is, in particular, valid for p=r. Consequently we may apply
theorem 4.4. Thus there exists а С solution <p* e Ф[Т\ of equation A.2), fulfilling
conditions D.24). According to D.59) it is also a Cq solution of A.2) fulfilling con-
conditions D.60). Since such a solution is unique, it must be the solution found formerly,
i.e. (p* = <p and <p e C[I].
Now, if r= oo, then the hypotheses of our theorem will be fulfilled for each r',
q<r'<oo. Consequently, <p e С for every r'<oo, i.e. <p e С°.ш
Similarly one can prove the following
Theorem 4.7. (Choczewski [3].) LetfeS^l], ?el, and let hypotheses 3.1, 3.2
and 4.1 be fulfilled. Suppose further that inequality D.22) holds in a neighbourhood

100 CHAPTER IV. Differentiable solutions
of'? and, moreover, \h'y(i, n)\< 1, where n e Q^ is a root of equation C.13). Then there
exists a unique continuous function <pe Ф[1] satisfying equation A.2) in I and fulfilling
condition C.15). This function is in fact of class С in I.
§ 6. Lack of uniqueness. In the present section we shall prove that if we replace
D.23) by the opposite inequality, then, in general, equation A.2) has а С solution
depending on an arbitrary function. It will be necessary to deal with the equation
written in the form A.1) and therefore we shall have to make somewhat stronger
assumptions. In order to be able to introduce the functions Gk we must assume
thatГ(х)фО in/, which together with/e ?,<[/] implies fe Щ[1]. That the assump-
assumption f'(x)Ф0 is essential may be seen from the example in § 2.
Here again ? and r must be finite.
Lemma 4.6. Suppose that fe Щ[Г\, ?el, f'(x)^0 in I, and let hypotheses 3.1
and 4.2 be fulfilled. Let n eQ^be a root of equation C.24) and let
D.62) |0;«,ч)[/шг|<1.
Then there exists a unique system of numbers r/i, —,nr fulfilling the equations
D.63) *lk=G&,ri,Tii,...,TiJ, /c=l,...,r.
Proof. Since feR\, we have |/'(?)|<1. Hence, according to D.62), we get
D.64)
For rit we have the equation (cf. D.2))
which has a unique solution, since, by D.64), 1 -[/'(?)]'WyiZ* ^^O. Having deter-
determined tii,...,rip-i, we may determine np uniquely from the equation (cf. D.9))
since by D.64) 1 -д'Д, п)
Theorem 4.8. (Choczewski [3].) LstfeЩ[Г\, ?el, f'(x)^0 in I, and let hypo-
hypotheses 3.1, 3.3 and 4.4 be fulfilled. Further, let n be a root of equation C.24) such that
neQ;. and suppose that condition D.62) is fulfilled. Then there exist positive numbers c,
d such that for every xoel, 0<|xo-?|<c, and for every solution <p{x) of equation
A.1) in I which is of class С in /-{?} and fulfils condition C.26) we have
D.65) limp(x) = f/, lim?>'(x) = j|1, ..., Vimq>(r\x) = r]r,
x->i x->i x->S
where nt, ..., nr are the unique solution of equations D.63).
Proof. Let the numbsrs c, d be those occurring in theorem 3.6. Then lim g>(x)=tj
x->{
in virtue of that theorem (cf. D.64) for p=0).
In view of lemma 4.1 we have

6. Lack of uniqueness 101
i.e. according to D.2)
D.66)
If we put <p(?)=r\, <p'@="i, then the function cp'{x) so extended satisfies the linear
equation D.66) in /. (Here (p{x) is regarded as a given function, continuous in /.)
On account of theorem 2.9 and in view of relation D.64) forp= 1, <p'{x) is continuous
in/, i.e. \im(p'(x) = ri1.
x->{
Now suppose that
= n, litn<p'(x) = ri1, ..., Iim(»(p~1)(x)=?;p_1,
and define the functions <p, <p', ..., #>(p) at x = ? by putting
Thus the functions (p(x), (p'(x), ..., (p(p~1)(x) are continuous in /, and the function
cp(p\x) satisfies in / the equation
9>(p)[/(x)] = GP(x, p(x), <p'(x), ..., <plp\x)),
i.e. by lemma 4.3
<P(PXf(x)l-g'y(x, 9>(x))[/'(x)]->(p)(x)=Sp(x, p(x), ?>'(x), ..., P0^)).
On account of theorem 2.9 and in view of relation D.64), #>(p)(x) is continuous in /,
p
Thus relations D.65) are fulfilled. ¦
Together with theorem 4.1 the above theorem gives
Theorem 4.9. (Choczewski [3].) LetfeR\[I\, ?e/,/'(x)^0 in I, and let hypo-
hypotheses 3.1, 3.4 and 4.4 be fulfilled. Further, let neQt-be a root of equation C.24)
and suppose that condition D.62) is fulfilled. Then equation A.1) has in I a Cr solution
<p e Ф[/](х) depending on an arbitrary function.
§ 7. An application: the Goursat problem for a hyperbolic equation. Let G(x,y)
be a continuous function in a rectangle
D={(x,y): 0<x<a, 0<y<b}
and let #(x) and h(y) be functions of class C1 in the intervals <0, a> and <0,Z>>,
respectively, with the only exception that the derivative h'@) may be infinite. Further,
let 0(О) = Л(О)=О and 0^g(x)<b, 0<A(j)<a. Moreover, the curves
D.67) y=9(x) and x=/i(j)
should have no point in common except the origin (cf. fig. 5). Lastly, let P(x) and
Q(y) be functions of class C1 in <0, a} and <0, by, respectively, fulfilling the con-
condition J°@) = Q @).
0) Cf. the footnote on p. 77.

102
CHAPTER IV. Differentiable solutions
D.68)
We are going to consider the Goursat problem for the equation (*)
дхду
Find a function u(x,y), of class C1 in D, possessing a continuous derivative u"xy in D
and satisfying equation D.68) and the conditions
D.69)
u(x,g(x)) = P(x) for x6<0,a>,
u{h(y),y)=Q(y) for
a x
Fig. 5
The integration of equation D.68) yields
D.70) u(x,y)
where
U(x,y)=j \G{S,t)dtds,
x у
и
О О
and g>(x) and у/ (у) are functions of class C1 to be determined. Put
D.71) V(x)=P(x)-U(x,g(x)), W(y) = Q(y)-U(h(y), y).
Inserting D.70) into D.69) and making use of D.71) we obtain
D.72) fW+!C[9(x)] = ^W, 9 [h (y)~\ + y/ (y) = W (y).
The elimination of ц/ from the above equations yields the equation
q>(x)-q> (h [g (x)]) = V{x)-W[_g (x)] ,
C1) Bielecki, Kisynski [1]. The Goursat problem for partial differential equations often leads
to functional or integro-functional equations. Cf. Goursat [1], Picard [7], d'Adhemar [1], [2],
Popovici [1], [9], [13], Myllsr, Valcovici [1], Sjostrand [1], Cim.-nino [1], Colombo [2], Cioranescu [1],
Kisynski [1], Ghermanescu [22], Majchsr [2], [4].

7. Goursat problem for a hyperbolic equation 103
which, on setting f(x)=h[g(x)], F(x)=V(x)-W[g(x)], can be written as
D.73) <p{f(x)\-<p(x)=-F(x).
Now, if we find a C1 solution cp{x) of equation D.73) in <0, a) and then determine
the function y/(j) from the second equation of D.72), then function D.70) actually
yields a solution of the original problem, which is thus reduced to solving equation
D.73) in the class CxK0, a}].
The function/(x) is continuous in <0,a> and of class C1 with the possible
exception of x=0. Moreover, f(x) < x in @, a}. To see this, suppose that f(x0) = x0,
xoe@,a>, and put Уо = д(хо). Thus h(yo) = xo, which means that curves D.67)
meet at the point (x0, y0) ф @, 0), contrary to the assumption. Consequently,/(x) ф х
in @, a), and, since evidently/(a)<a, we have/(x)<x in @, a}. Hence it follows
that f(x) belongs to Sg[<0, a}] and to 5q[@, a}].
It follows from theorem 2.11 that <p(x) must have the form
D-74) ?>(*) = '/+?f [/"(*)]•
n = 0
By D.69) we have и@, О) = Р(О) = 0(О), and so in view of D.70) <p@) + y/@) = P@)
= <2@). Hence the constant tj in D.74) disappears in D.70) and may therefore be
assumed to be equal to zero. This proves the uniqueness of the solution of the
problem considered.
However, the conditions formulated above are not sufficient to prove the
existence of a solution. If we assume, in addition, that
D.75) 0<д'@)й'@)<1,
then the existence of a unique solution results directly from theorem 4.5 (r=l). (Let
us note that D.75) implies that/e 5q[<0, a}]). Geometrically condition D.75) means
that curves D.67) do not have a common tangent at the origin. If D.75) is not fulfilled,
then stronger assumptions must be made in order to ensure the existence of a solution.
It may be proved (Bielecki, Kisynski [1]) that in the case where g'@) h'@) = l a
solution exists provided the functions — f\x) are equibounded in <0, a) and
dx
there exists a function /(x,y) of class C1 in D whose derivatives %'x and Z, fulfil
in D a Lipschitz condition with respect to у and x, respectively, and we have
P(x)=x(x,g(x)), Q(y)=x(h(y),y).
§ 8. Other examples. 1. Consider the equation (l)
D.76)
in /=(— oo, oo). Here Q is the whole plane and the function h(x, у)=2уг—\ fulfils hypotheses 3.1,
Van der Corput [1], Forder [1], Cooper [1], Mikusinski [1].

104 CHAPTER IV. Differentiable solutions
3.2 and 4.1 with an arbitrary r. The function f(x)=x/2 belongs to Щ[1]. Equation C.13) becomes
D.77) ti=2tj2-l
and has two solutions: q= 1 and r/= — -J-. Further, h'y@,1)=4, h'y@,—?)= —2. In the case of ti= 1
theorem 4.5 is applicable for r=3. We have Щ(х, у, yi) = 2yyi, H2(.x, y, yu y2)=y2+yy2,
Нз(х,У, У\>У2,Уз)=Ъ\У2+%ууъ- System D.21) (r=3) has the solutions
»7i = 0, ^arbitrary, »7з = О.
By theorem 4.5 for every t\z there exists exactly one C3 solution (fix) of equation D.76) fulfilling
the conditions
D.78) p@)=l, (p"@)=r,2.
As can easily be guessed, q>(x)=coshyjt}2X if r/2>0, c/>(x)=cosyj — tjzx if ^2<0, c/>(x) = l if ^2=0.
In the case of n=—\ theorem 4.5 is applicable already for r=2. The function q>(x)= —i is
the only C2 solution of equation D.76) fulfilling the condition чЩ= — J .
Unfortunately we cannot apply theorem 4.9 to show that equation D.76) has in /a C1 solution
depending on an arbitrary function since the function h(x, y) is not invertible. It will be invertible
if we restrict п to the set (— со, со) х A, со). In this set C1 solutions of equation D.76) may be con-
constructed depending on an arbitrary function (theorem 4.1), and they will all fulfil the conditions
lim <p (x) = 1 , lim p'(x) = 0
г-*0 г-*0
(theorem 4.8). Uniqueness will not be achieved even by the additional requirement that <p(x) be convex
(Cooper [1]).
Our theorems do not allow us to decide the number of C2 solutions of equation D.76) fulfilling
conditions D.78). For r=2 we have h'v@, l)[/'@)]2=l, and so this is a typical indeterminate case.
However, following H. G. Forder [1] we shall show that in this case C2 solutions are unique.
Let <p(x) be a C2 solutions of D.76) fulfilling D.78). Thus <p(x)>0 in a neighbourhood U of
x=0, and we have by de 1'Hospital's rule
D.79) ш^-Ш^-^Щ-и.
2 2x 2 2
Let us fix an xo e (/and suppose that p(xo)<l. There exists a smallest positive к such that (p(x0)
= cos kx0. By D.76) we get pB"nxo)=cos 2~nkx0 and
<p(x)-\ ipB-»xo)-l cos2-»/«o-l 1 ,,
D.80) lim = lim = lim = к2 .
,-e x2 •-. B-"*oJ n^ro B-»xoJ 2
D.79) and D.80) imply ri2——k2. If ^2>0, this shows that the assumption p(xo)<l was false and
necessarily p(xo)>l. This case is handled similarly by putting 0>(xo)=coshfcto. И ^2<0, then
k=^]—ri2 and c/>(xo)—cosy/ — Ц2 *o- Since xo has been chosen arbitrarily in U, this proves that
D.81) p(x)=cos ^J'- tj2x
for x e U. In virtue of theorem 3.2 relation D.81) holds for all x e (— со, со).
2. In view of theorem 4.1 the equation
D.82)
has in @, со) a C°° solution depending on an arbitrary function. An application of theorem 4.5 is

8. Other examples 105
impossible: f(x)=x+\ does not have a finite fixed point ?. The theorems of Chapter III do not
allow us to choose a unique solution, either. Evidently D.82) has no bounded solution in @, oo)
(otherwise <p(x+V)—<p(x) would have to be bounded) and, since .F(oo)=lim F(x)=lim log x#0,
the theorems of Chapter II, §7, are not applicable either. So, although the theorems proved so far
have a very vast field of applications 0), there still occur cases where they fail. Therefore we
must find conditions of another kind B), applicable to problems like D.82). Now we turn to this
question: in the next chapter we shall establish several theorems stating the uniqueness of
solutions of some equations of form @.49) under such conditions as monotonicity, convexity, asymp-
asymptotic behaviour for x->(, etc. In § 10 we shall also show that these theorems are well suited for proving
the uniqueness of solutions of equation D.82).
(!) In particular, theorem 4.5 always allows us to choose a solution (or a finite-parameter
family of solutions) characterized by the highest regularity at ? if the functions/and h are of class C00,
€ is a finite fixed point and 0 </'(?) <1. Condition D.23) is then certainly fulfilled for r sufficiently
large.
B) Even the requirement of the analyticity does not furnish a unique solution of equation
D.82). The general analytic solution of D.82) in @, oo) is given by p(x)=log Г(х)+п(х), where
Г(х) is Euler"s Gamma function and n(x) is an arbitrary periodic analytic function of period 1.

CHAPTER V
MONOTONIC AND CONVEX SOLUTIONS
§ 1. Fundamental theorem. Here we shall confine ourselves to discussing some
particular linear equations C1). The equation
E.1) <p{f(x)-]-<p(x)=F(x)
is of particular importance. The following theorem B) seems to be fundamental for
the theory of equation E.1).
Theorem 5.1. Let f(x) be defined in a submodulus interval I and let F(x) be a
function on I. If a function q>(x) satisfies equation E.1) in I and fulfils the conditon
E.2) limp[/"(x)] = 0 for xel,
n-* со
then
E.3) ?>(*)=-?ПГ(*)].
n = 0
Proof. We obtain from E.1)
?(x)=g> [/m(x)] -"X F [/"(*)] for m=0,1, 2, ...
n = 0
whence E.3) results in view of E.2) on letting т-юо.ш
Similarly one can prove C)
Theorem 5.2. Let f(x) be defined in a submodulus interval I and let g{x) be a
function on I. If a function q>(x) satisfies the equation
E.4) ?>[/(*)] = <
in I and fulfils the condition
(') Monotonic and convex solutions of some non-linear equations are dealt with in Baj-
raktarevic [7], Cooper [1], Kuczma [22], [33], [38], Vajzovic [1], Rozmus-Chmura [1], Pelczar [3].
B) Kuczma [14], Burek, Kuczma [1]; cf. also Steinberg [1]. Let us note that the part of theo-
theorem 2.11 concerning equation E.1) results from theorem 5.1.
C) Theorem 5.2 can also be derived from theorem 5.1. Cf. Burek, Kuczma [1].

1. Fundamental theorem 107
x)] = l for xel,
then
§ 2. Monotonic solutions. Now we shall prove theorems (*) concerning the
uniqueness and the existence of monotonic solutions of equations E.1) and E.4).
For convenience we shall assume that I=(a,b) is an open interval (so ?=a or
?=b). One or both of its endpoints may be infinite.
Theorem 5.3. Suppose that fe R° [Г\, ? ф1, and let F(x) be a function defined on
I and fulfilling the condition
E.5)
If<p(x) is a monotonic solution of equation E.1) in I, then
E.6) P(x) = 4o ?
n=0
where x0 is an arbitrarily fixed point from I, and t]0 = (p(x0). If, moreover, F(x) is mono-
monotonic in I, then equation E.1) actually has a unique one-parameter family of monotonic
solutions in I (furthermore, if FeM+, then q> eM2, and conversely, if Fe M.°,
then cp e M+). These solutions are given by formula E.6), where tj0 is an arbitrary
constant.
Proof. We assume that ? is the left endpoint of /. We fix an arbitrary xoe 1
and we put xn=f\x0), n=0, 1,2, ... By theorem 0.4 the sequence х„ is strictly
decreasing and tends to ?, as n-> oo.
Let (p(x) be a monotonic solution of equation E.1) in /. For an arbitrary
xe(?, xo)we denote by x = x(x) the greatest term of the sequence х„ not exceed-
exceeding x, and we put
4/(x) = <p(x)-<p(x), H(x)=F(x)-F{x), xe(?,x0).
Since f(x)=f(x), we obtain hence
vUW]-V(x)
Further, since the function q>(x) is monotonic in (?, x0), we have
whence we get by E.5)
ton у [/"(*)]=0.
(i) Kuczma [14]; cf. also Kuczma [7], [20], [27], [25], [13], [24], Burek, Kuczma [1], John
[1], Hamilton [1], A. Smajdor [5].

108 CHAPTER V. Monotonic and convex solutions
On account of theorem 5.1
n=0
i.e.
E.7) <р(х) = <р(х)-? {F [/"(*)]-F [/"(*)]} for xe(?,x0).
n = 0
Let us fix an xe(?, x0). There exists an index k>0 such that 5c=xk. Thus E.7)
may be written as
cW=?W-f{f[/"W]-ffeJ} for xe(?,x0).
n = 0
Now,
n = 0
=9(xJ- Z {f[/W]-fW}+ Z f (xm+n)- Z f (*»-i)-
n=0 n=l n=l
But
Z F (х„_)) = ? {<P(*n) ~<p(xn- 0} = 9(xk) - <p (x0),
whence in view of E.5) we obtain formula E.6) (with rjo = (p(xo)) valid for xe(?, x0).
But one verifies immediately that if a solution q> of equation E.1) is of form E.6)
for x=f(t), then it is of form E.6) also for x=t. Thus E.6) is valid in the whole
of/.
Now we assume that F(x) is monotonic in / and let x0 be an arbitrarily fixed
point in /. For an x e /we can find aye /and an index Л?> 1 such that x e (fN(y), уУ
and x0 e (fN(y), y} (cf. theorem 0.4). Since F(x) is monotonic, we have
The series
Z \FUn(y)l-\
n=0 n=0
t = 0 n=0
(where со = +1 or — 1 according to whether F(x) is increasing or decreasing) con-
converges in view of E.5), and thus E.6) actually defines a one-parameter family of
solutions of equation E.1) in /. The monotonicity of <p(x) results from that of F(x)
and/(x).B

2. Monotonic solutions 109
Remarks. 1. In theorem 5.3 the monotonicity of cp and F in / may be replaced
by monotonicity in a neighbourhood of ?. In such a case the above proof applies
to an interval Itd in which cp{x) (or cp{x) and F(x)) is monotonic, and then it is
shown that formula E.6) is valid in the whole of /. The same remark applies also
to further theorems in this chapter.
2. Equation E.1) may have a unique one-parameter family of monotonic solutions
though the function F(x) is not monotonic. E.g. the equation
x+l
cp{x+l) — cp{x) = I e~"(sinuJdu
x
has a unique family of monotonic solutions in (—00,00) (the uniqueness results
from theorem 5.3, though not the existence)
g>(x)= I e~u(sinuJdu,
а
but the function
x+l
F(x)= I e"u(sin ufdu
x
is not monotonic (J).
Theorem 5.4. Suppose that feR°[I], ?$I, and let g{x) be a positive function
on I fulfilling the condition
E.8)
Ifcp(x) is a monotonic solution of equation E.4) in I, then
E-9) »М-*Д шш •
where x0 is an arbitrarily fixed point from I, and y0 = q>(x0). If, moreover, д (х) is mono-
monotonic in I, then equation E.4) actually has a unique one-parameter family of monotonic
solutions in I. These solutions are given by formula E.9), where y0 is an arbitrary
constant.
Proof.B) Let q>(x) be a monotonic solution of equation E.4) in /. If д>(х)=0,
then E.9) holds for yo=0. So let us assume that <p(x)^?0. If we had p(xo)=0 for
an x0 eI, then by E.4) g> [f\x0)]=0,n=0, 1,2, ..., and in view of the monotonicity
of cp and of theorems 1.1 and 0.5 we would have q> (x)=0, contrary to our assumption.
Function g>(x) cannot change the sign, since in that case there would have to exists
О In this connection cf. also A. Smajdor [5].
B) We derive here theorem 5.4 from theorem 5.3, but a quite independent proof (similar
to that of theorem 5.3) can be given based on theorem 5.2. Such a proof would make no use of the
logarithmic function.

110 CHAPTER V. Monotonic and convex solutions
a te/such that <p(t) <p[f(t)]<0, which implies g(t)<0, contrary to the hypotheses
of the theorem. Thus we may put co = sgn cp(x), and a> is constant in /.
Put
y/(x)=loga><p(x), F(x)=logg(x), xel.
Then y/(x) is a monotonic solution of the equation
E.10)
in /, where by E.8) the function F(x) fulfils condition E.5). Consequently, in virtue
of theorem 5.3,
n=0
whence we immediately obtain E.9) with yo = co exp //0. On the other hand, if g{x)
is monotonic in /, then so is F(x)=logg(x), and by theorem 5.3 equation E.10)
has a monotonic solution y/{x) in /. The functions <p(x) = y0 exp y/(x) then provide a
one-parameter family of monotonic solutions of equation E.4) in /. It follows from
the first part of the proof that these solutions must be of form E.9).¦
§ 3. Lack of uniqueness. Condition E.5) is essential for the uniqueness of mono-
monotonic solutions of equation E.1). If E.5) is not fulfilled, then equation E.1) may
have a monotonic solution depending on an arbitrary function. For simplicity^
we shall assume throughout this section that ? is the left endpoint of I=(a, b) (and
so ? = a) and that the function F(x) is positive (it follows from E.1) that if q> is mono-
monotonic in /, then F(x) must have a constant sign in I). If F(x) is negative, we need
only consider the equation
instead of E.1).
Lemma 5.1. Suppose that fe /?° [/], I=(a, b), and let F(x) be a positive, mono-
monotonic function on I. Then every monotonic function <po(x) in {/(xo)> xoX xoe^ fulfil-
fulfilling the condition
E.11) Po[/(*o)] = ?>o(*o) + f(*o)
can be uniquely extended to a solution <p(x) of equation E.1) in I. This solution is
decreasing in (a, x0) if F(x) is decreasing, and it is decreasing in (f(x0), b) if F(x)<
is increasing.
Proof. The possibility of such an extension follows from theorems 1.1 and.
0.5. Putting xn=f(x0) for those n for which f(x0) is defined, we have
E.12) p(x)=Po[/""W]+i;1f[/l""W] for xe(xn+1,xn},
t = 0

E.13) ?>(x) = ?>o[/"W]-Zf[/iW] for
t=0
3. Lack of uniqueness 111
/2 = 1,2,
(If x_jv is defined but x_N_1 is not, then the interval <х-#> х-я-^ should be re-
replaced by <х_л,,й).)
Since F(x) is positive, it follows from E.11) that <po(x) is decreasing in <xx, xo>.
Thus it is easily seen from E.12) and E.13) that q>(x) is decreasing in each <х„+1, х„>,
/2=0, 1,2,..., whenever F(x) is decreasing in /, and <p(x) is decreasing in each
<x_n+!, х_„>, и=0, 1,2, ... (as far as х_„ are defined), whenever F(x) is increasing
in /. Hence the conclusion of the lemma follows directly. ¦
Theorem 5.5. (Burek, Kuczma [1].) Suppose that feR°a[T\, I=(a,b), and let
F(x) be a positive, monotonic i1) function on I. Further, let the following conditions
be fulfilled C)
E.14) limF(x) = c, 0<c<oo,
x-*a
E.15) lim/(x)</3 or lim F(x) = c'>0.
Then equation E.1) has in la monotonic solution depending on an arbitrary function.
The theorem remains true if feR°b[T\ and if instead o/E.14) and E.15) we have
lim F(x)>0 and lim f(x)>a or lim F(x)>0.
x-*b x--*a x-*a
Proof. I. Suppose that F(x) is increasing. Then necessarily c< oo. The equation
E.16) <plf(x)-]-9(x) = F(x)-c
has, according to theorem 5.3, a decreasing solution cp^x).
Let us fix an arbitrary xoe I and a decreasing function <po(x) on </(x0), xo>
such that
Po [/(*(>)] = Po(*o)+C-
Since the constant function is both increasing and decreasing, cpo(x) can be extended
by lemma 5.1 to a decreasing solution q>2(x) of the equation
E.17) ?>[/(*)]-?>(*)=с
The function д>(х) = д>1(х) + д>2(х) is evidently a decreasing solution of E.1) in /.
Let Q [J], 7c=/, be the class of the functions cp(x) of the form g>(x) = fi(x) + y/(x)
in J, where y/(x) is a decreasing function on J. What has just been proved implies
(!) If the condition of the monotonicity of F is dismissed, then the conclusions will no longer
be valid. Equation E.1) may have a unique monotonic solution (up to an additive constant) though
condition E.5) is not fulfilled (John [1])!
B) If \imf(x)=b and limF(x)=0, then/-ie/fg[fl and it follows from theorem 5.3 applied
я->& х-*ь
to equation E.19) that E.1) Has in / a unique one-parameter family of monotonic solutions.

112 CHAPTER V. Monotonic and convex solutions
that every function <p0 e$2[</(x0), xo>] fulfilling condition E.11) can be extended
to a solution g> еA[Г\ of equation E.1) in /. Evidently !2[Г\сМ2[Г\. But, on the
other hand, every monotonic solution g>(x) of equation E.1) in / belongs to S2[T\.
To prove this, put y/(x) = q>(x) — q>1(x) and suppose that y/(x) is not decreasing
in /. Consequently, there exist xlt x2el, x1<x2, such that w{x1)<y/{x1). Since
y/(x) (as the difference of a solution of E.1) and a solution of E.16)) satisfies equation
E.17) in /, we have
Consequently,
E.18)
Choosing а. у el and an N>0 such that fN(?)<x1<x2<y, we have, since g>t is a
decreasing solution of equation E.16),
-9i [/"GO]
1
i=0
whence in view of E.14) and E.18)
Urn
But this is impossible because <p(x) is decreasing and/"(x1)</"(x2) for all n.
II. Now suppose that F(x) is decreasing and limf(x)=bo<b. We may extend
fix) and F{x) onto (a, b} by putting fib) = b0, FF)=limF(x). The function/(x)
thus extended belongs to /?°[(a,6)]. Choosing xo=b we may apply lemma 5.1.
Thus every decreasing function g>oix) on (Jb0, b), fulfilling the condition
lim po(x)^g>oibo)-Mm Fix)
x->b x->b
can be extended to a monotonic solution of equation E.1) in /.
III. Lastly, suppose that Fix) is decreasing, lim fix)=b and limF(x) = c'>0.
X-.6 x-,b
Then/ e Rb[I\- Replacing equation E.1) by the equivalent equation
E.19) \\
we proceed as in case 1.И
By an argument similar to that employed in the proof of theorem 5.4 one can
obtain hence the following
Theorem 5.6. Suppose that /e /?„ [/], /=(a, b), and let gix) be a monotonic func-
function, gix)>\ on I. Further, let limg(x)>l and either lim/(x)<6 or li
x-ta x->b

3. Lack of uniqueness 113
Then equation E.4) has in la monotonic solution depending on an arbitrary function.
The theorem remains true iffeR°[f\, limg(x)>l and either lim/(x)>a or
x->b x-ta
lim g(x)>l.
x-*a
For the special case g(x) =const we have the following, slightly stronger result.
Theorem 5.7. Suppose that fe R® [/], ?$I, and let s^l be a positive constant.
Then for every x0 el and every monotonic function g>0(x) on <xo,/(xo)> or </(x0), xo>,
fulfilling the condition
E.20) <PoU(xo)l =
the unique function <p(x) satisfying the equation
E.21) 4>U(x»*9(x)
and such that <p(x) = q>0(x) in (хо,/(хо)У resp. </(x0), x0), is monotonic in I.
Proof. We may assume that ?, is the left endpoint of / and that s> 1. It follows
from E.20) that <po(x) has a constant sign in </(x0), xo>; so we may put ca = sgn <po(x),
?o(x)=l°g coq>0(x), x 6 (J"(x0), xoy,c=log s. By lemma 5.1 there is a unique extension
y/(x) of y/0(x) onto / satisfying equation E.17), and y/(x) is monotonic in /. The
function ср{х) = оз exp y/(x) is monotonic in / and is the desired extension of <po(x).
(The uniqueness of this extension results from theorems 0.5 and 1.1.) ¦
Corollary. Under the conditions of the above theorem, every solution of equation
E.21) which is monotonic in a neighbourhood of ? is also monotonic in the whole of I.
It is easily seen from formulae E.12) and E.13) applied to equation E.17)
that both the theorem and the corollary remain true when "monotonic" is replaced
by "strictly monotonic".
§ 4. Further uniqueness theorems. Now we shall obtain some consequences of
theorems 5.2 and 5.4 (Anastassiadis [6], [7], Burek, Kuczma [1]).
Theorem 5.8. Let fe R°[I], ?ф1, and let g{x), Q(x) be positive functions on I.
If<p(x) is a solution of equation E.4) in I such that lim Q(x) q>(x)= 1, then
q>(x)= lim (<2[/"(*)] EU UUT1 •
Theorem 5.9. LetfeR\[I\, €ф1, and let g(x), Q(x) be positive functions on I
fulfilling the condition
lim — = 1 .
*-« aw
If (fix) is a solution of equation E.4) in I such that the function Q(x) <p(x) is monotonic
in I, then
8 Functional equations

114 CHAPTER V. Monotonic and convex solutions
where x0 is an arbitrarily fixed point from I and yo = 9(xo). Moreover, if the function
б [/001 000/600 " monotonic in I, then equation E.4) actually has a unique one-
parameter family of solutions q>(x) such that Q (x) q>(x) is monotonic in I. These solutions
are given by formula E.22), where y0 is an arbitrary constant.
Theorems 5.8 and 5.9 follow from theorems 5.2 and 5.4, respectively, on the
grounds of the observation that the function y/(x) = Q(x) <p(x) satisfies the equation
in/.
As an application of theorem 5.9 we shall prove the following
Theorem 5.10. (Burek, Kuczma [1].) Suppose that feR°[I\, where ?,el is a
finite endpoint of I, and let g(x) be a positive function on I such that д(Е)ф\ and
n-i.
If cp ? Ml[I\ is a solution of equation E.4) in I, then
= y0 hm jsr-г—, П
/ [X) Q i 0
where x0 is an arbitrarily fixed point from I and yo =
Proof. Let q>(x) be a convex solution of equation E.4). It follows from the
condition g(C)?= 1 that p(?)=0. Consequently, the function
is monotonic in Io=I-{?,}. Assuming Q(x)=co/(x—?), where <a = sgn(x-?), we
obtain from theorem 5.9 formula E.23) valid for x e /0. For x=? E.23) is trivial.¦
§ 5. A finite difference equation. In the present section we shall be concerned
with convex solutions of the finite difference equation (x)
E.24)
Theorem 5.11 below is due to W. Krull [1] (and so is theorem 5.12). It can simply
be derived from theorem 5.3, but the argument involves the Lebesgue measure
and integral (Hamilton [1], Kuczma [27]). The proof given below is entirely elemen-
elementary and does not differ essentially from that given originally by Krull.
Theorem 5.11. Suppose that Fe MX[I\ I=(a, со), а> - сю. Further, let
E.25)
(i) Krull [1], Kuczma [1J, Dinghas [1]. Cf. also John [1], Hamilton [1], Kuczma [3], [27], [13],
[24], Rozmus-Chmura [2].

5. A finite difference equation 115
and let xoe(a, oo) be arbitrarily fixed. Then for every t]0 e (— oo, сю) there exists
exactly one function cp e M1 [/] (moreover, if Fe Ml, then g> e M+, and conversely)
satisfying equation E.24) in I and fulfilling the condition
E.26)
This function is given by the formula
E.27) ^(x)
n = 0
Proof. Let q>(x) be the required solution. Then we have for 0<f < 1 (*)
1 f
(fig. 6), whence
)-<p(x0
i.e.
E.28)
Now
xo + n + t)-<p(xo + n)
n-l B-l
= X F(xo + t + k) + <p(xo + t)-r,o- X F(xo
k=0 k=0
Writing x=xo + t and inserting the formula obtained into E.28) we get
n-l
i.e. in short notation
0() "?
x-xo)F(xo)-
k=0
we have
Фп(х)-(х-х0)
@ We assume here that <peM\[I]. If <peMl[I], the argument is the same, but the inequal-
inequalities must be turned into the opposite ones.

CHAPTER V. Monotonic and convex solutions
The above inequality shows in view of E.25) that cp{x) must be of form E.27) for
x?<xo,xo + I), which together with theorems 1.1 and 0.5 proves that there can
be at most one function <p(x) fulfilling the conditions of the theorem. To prove the
xo+n-l
Хо+П+1 X
Fig. 6
existence, it is enough to show that the sequence Ф„(х) converges in / to a function
q> б Ml[T\ satisfying equation E.24) in /and fulfilling condition E.26).
Put un=F(x0+n + l)-F(x0+n). The series occurring in E.27) may be
written as
(x-x0)
But for xe(xo,xo
n = 0\ X—Xo
we have (for FeMl)
—F(xo + n)
<"„-!>
whence it follows that the series in E.29) is majorized by ? (m«_!— м„). The
latter series converges, since by E.25) Пти„=0. Consequently series E.29), and
n-»oo
thus also the sequence Ф„(х), converges for xe(xo,xo + l).
Now we have
E.30) Ф„(х +1) - Ф„(х) = F(x) + F(xo+n)-F(x + n).
The function F(x), being convex, is monotonic for x sufficiently large. Consequently
E.25) implies that F{t+h)-F{t) tends to zero as f->oo, for every fixed h, and thus
lim {F(xo+n) — F(x+n)} = 0. Thus E.30) implies the convergence of the sequence
Л-.00
Ф„(х) in the whole of / to a solution of equation E.24). Relation E.26) results
from the fact that Ф„(хо)=^о. Since all the functions Ф„(х)аге convex (moreover,
if FeMl, then Ф„ eM+, and conversely), the same is true about the limit.
Theorem 5.12. Suppose thatFeM1 [7]nC[/], I=(a, оэ),а>- oo, r> 1. Further,
suppose that Fir\x) is monotonic for x sufficiently large and let condition E.25) be
fulfilled. Then the unique functions q> e Л/х[/] satisfying equation E.24) are of class

5. A finite difference equation 117
С in I and we have
q>'(x) = F(x0)- t {F'(x + n)-[F(xo + n + l)-F(x0 + и)]},
n=0
<p(r\x)=~t Р(гЬ + п) if
n = 0
Proof. At first let us note that all the derivatives F(l\x), i=\ r, are mono-
tonic for x sufficiently large. For i=r this is so by hypothesis. If F(i+1\x) is mono-
tonic for x>X~^a, l</<r— 1, then F(\x) is convex in (X, oo) and thus monotonic
for x sufficiently large. The monotonicity of the derivatives F@(x) and condition
E.25) imply that
E.31) limF("(x)=0, i=l,...,r.
x-* oo
For the proof of our theorem it is enough to show that the sequence Ф^\х) is
uniformly convergent in every compact interval contained in /. In view of the relation
Ф<г)(х +1)=Ф<г)(х) + F(r\x) - F(r\x+n)
(resulting from E.30)), it is enough to show that Ф^\х) uniformly converges in
<xo,xo + l>.
I. r = 1. In virtue of the mean-value theorem we have for x e <x0, x0 +1)
ад = ^(хо)-П?
= F(xo)-"l
where x e (x0, x0 +1). Since F' is monotonic (even in the whole of /, as the derivative
of a convex function), we have
(where co= + l or —1 according to whether F'eM+ or Ml), and the uniform
convergence of Ф'„(х) follows from the convergence of the series ? [F'(xo + k+l)
-F'(xo+k)].
II. r>2. We have
In view of E.31) and Ihe monotonicity of F(r) the function [F(r)(x)[ is decreasing
for x sufficiently large. Thus we have for x e <x0, xo+ 1) and к sufficiently large

118 CHAPTER V. Monotonic and convex solutions
and the uniform convergence of Ф['\х) follows from the convergence of the series
Corollary. IfF(x) is-completely monotonic in I, then —q>(x) is also completely
monotonic.
§ 6. Generalization of the previous result. As a generalization of the considera-
considerations of the preceding section, we are going to investigate Mp solutions of equation
E.24). We start with some results concerning polynomial solutions (cf. e.g. Norlund
[1], Ghermanescu [22]).
Since every polynomial is monotonic for x sufficiently large, it follows from
theorem 5.3 that the only polynomials satisfying equation E.24) with F(x)=0 are
constant functions. Consequently equation E.24) may have at most one polynomial
solution q> (x) fulfilling q> @) = 0. Put
B0(x) = 0, BJix) = -x"--"^
n n i=2
It can be proved by induction that В„(х) is a polynomial of degree n satisfying the
equation
and that Д,@) = О. Hence we have the following
p-i
Lemma 5.2. IfF(x) = ? a,(x—x0)'is a polynominal of'degree 4,p—l,then there
( = 0
exists exactly one polynomial q>(x) satisfying equation E.24) and such that <p(xo) = O.
This polynomial is of degree </> and is given by
To an arbitrary integrable function g(x) we assign a new function g*(x) defined
by the formula
df x xo+1
g\x)=\g(t)dt-(x-x0) I g(t)dt.
XO XO
The reader will easily verify the following lemmas (here I=(a, со), a^-co, xoel):
Lemma 5.3. If g e Mp[l], then g* e Mp+1[/].
00
Lemma 5.4. If the series ? gn(x)=g(x) of integrable functions converges uniformly
n=0 oo
in every compact interval contained in I, then so does the series ? fif*(x), and its sum
n=0
equals g*(x).
Lemma 5.5. If the functions <p(x) and F{x) are integrable and q>(x) satisfies equa-
equation E.24) in I, then cp*(x) satisfies
in /, and <p*(xo) = O.

6. Generalization of the previous result 119
Let polynomials Р„(х) be defined by the relations
E.32) P1(x) = x — x0, Рп+1(х) = Р„(х), n = l,2,...
We shall prove the following
Theorem 5.13. (*) Suppose that FeMp[I], I=(a, oo), a> -oo, />>0. Further,
let
E.33) limApF(x)=0,
x-*oo
and let xoe(a, oo) be arbitrarily fixed. Then for every ri0e(—co, oo) there exists
exactly one function <peMp[I] (moreover, ifFeME, then q>eMp+, and conversely)
satisfying equation E.24) in I and fulfilling condition E.26). This function is given by
the formula
E.34) q> (x) = r,0 + Qp(x) - f {Я (x + n) -H (x0 + n) -
n=0
V «
i=0
where Qp(x) is the polynomial satisfying the equation
E.35) (
i = 0 I !
and fulfilling the condition
E.36) <2p(*o) = O
(c/. lemma 5.2), ?Ae polynomials Pt(x) are defined by relation E.32)
E-37) H(x)=F (x) - ? —±-2. (x-хо)\
i=o i!
df p~1
(If p = 0, then H(x) = F(x) and in formula E.34) the polynomial Qp and the sum ?
are omitted.) i=0
Proof. The proof is by induction. We shall also prove that the series occurring
in formula E.34) converges uniformly in every compact interval contained in /.
For/> = 0 our theorem results from theorem 5.3, for p= 1 it reduces to theorem
5.11. The verification of the uniform convergence in every compact interval contained
in / of the series occurring in E.6) and E.27) is left to the reader.
Now we suppose that our theorem is true for p — 1 ^ 1.
I. To begin with, we shall prove the existence of the desired solution. Let us
put D(x)=Ap1~1F(x). Thus (cf. lemma 0.12) De M^I] aadAp-1F'(x) = D'(x). (The
@ Kuczma [25]. Cf. also Krull [1], Dufresnoy, Pisot [1].

120 CHAPTER V. Monotonic and convex solutions
derivative exists in virtue of lemma 0.9, since p^2.) It follows from E.33) that
liraA\D(x)=0. Hence UraD'(x)=0, since D'(x) is a monotonic function. Con-
x->0 x-.cc
sequently,
E.38) liraAp1-1F'(x)=0.
JC-.00
Let H(x) be the function denned by E.37). In view of lemma 0.11, He МР[Г\.
Furthermore, A\~~ xF(x) and A\~ lH(x) differ only by a constant (namely by F(p~ 1}(х0))
and consequently Ap1~1F'(x)=Ap1~1H'(x) and lira Ap1H'(x)=0. By lemma 0.10,
JC-.00
H' e МР~1[Г]. On account of the induction hypothesis there exists a function у/ е
6 Mp~x [/] satisfying the equation
E.39) yf(x+l)-yf(x)=H'(x).
As foHows from lemmas 0.11, 5.3 and 5.5 and from formulas E.35), E.36) and E.37),
the function
E.40) <p (x)=t]0 + Qp(x) + y/*(x)
is the desired solution of equation E.24).
II. To prove the uniqueness, let us suppose that there are two functions of
class MP[T\, q>i(x) and <p2(x), satisfying equation E.24) in / and such that
E.41)
Their derivatives belong to class MP~1[I] and satisfy the equation
Since F'ejlf!/] and fulfils condition E.38), it follows from the induction hypo-
hypothesis that the functions <p[ and q>'2 must be identical up to an additive constant:
Hence, according to E.41)
q>i(x)=<p2(x) + c(x-
But, since both these functions satisfy E.24), <Pi(x+l) — <p1(x) = <p2(x+l)—<p2(x),
whence it follows that c=0. Consequently, q>i(x)=(p2{x).
III. Lastly we prove formula E.34). On account of E.40) it is enough to show
that
E.42) y,*(x) = - J {Я (x + и)-H(x0 + n)-
n = 0
P?Pt+ t(x) [Я<°(х0 + n +1) -Я(!)(х0 + и)]}.
i = 0

6. Generalization of the previous result 121
Now the function y/(x) satisfies equation E.39) and belongs to Mp-1[7]. Moreover,
by E.37) we have
E.43) H'(x) = F'(x) - ? -i^ (x - x0)' •
I!
i=0
Thus formula E.43) may be obtained from E.37) by replacing H{x), F{x) and p
by H'(x), F'(x) andp— 1, respectively. On account of the induction hypothesis
E.44) y/(x)=- ? {H\x + n)-H'(xo + n)-
0
г=о
and the series on the right-hand side of E.44) uniformly converges in every compact
interval contained in /. Hence by lemma 5.4
E.45) yf*(x)=- ? {H(x + n)-H(xo + n)-
n = 0
-(x-
E [
i=0
and the series on the right-hand side uniformly converges in every compact interval
contained in /. Relation E.42) results from E.45) in view of E.32).
Finally let us note that if FeMl[I], then, as can easily be seen from E.34),
<p ? M" [/], and conversely. ¦
§ 7. Recurrent sequences. In the considerations concerning the equation
E.46) P [/(*)]+ p(x)=F(x)
the best suited function class is that of functions {/}-convex of order n (cf. e.g.
theorem 2.14). Before dealing with this equation we shall prove some theorems on
sequences defined by recurrence relations (Kuczma [16], Brydak, Kordylewski [1],
A. Smajdor [3], Czerwik [I]).
The difference operator A" is defined, as usual, by
A°an=an, JX=^-4+i-^-4, v=l,2,...
Lemma 5.6. For an arbitrary sequence а„ we have
p (- U+1
(iy (
E ^{лЧ+1+^Л=дп-^—^т-лр+4, P=o,i,2,...
v=0 •? •?
Proof. Inductions
Lemma 5.7. If a sequence х„ satisfies the recurrence
E.47) xn+1+xn=bn

122 CHAPTER V. Monotonic and convex solutions
and for some
E.48)
then
E.49)
integer p^
Xn = '.
V
I fulfils
уЧ-К
h 2v+i
the condition
lim А х„ ^ 0 ,
И-*00
^ Avb |(~1)P У
2 v=o
Proof. Putting у„=Арх„, с„=АрЬп, we have according to E.47)
У«+1+У« = са-
Hence (induction)
Jn= I (-lYcn+v + (-l)k+1yn+k+1.
v = 0
In view of E.48) lim у„=0. Thus, letting k-^co in the above relation, we obtain
00 П-Ю0
Jn=E(-1)Vcn+v,i-e.
v = 0
E.50) A'xH = ? (-l)M"bn+v.
v=0
Now, on account of E.47) and of the definition of the operators A1, we have
А'хп+1 + А'хп = А\, А'хп+1-А'хп = А1+1хп,
whence
E.51) А1хп = $1А'Ьп-А1+1х„].
Using successively formula E.51) for /=0, ...,p—\ and taking into account E.50)
we obtain E.49).¦
Theorem 5.14. (Kuczma [16].) If for ap^ 1
E^52) limApbn=0,
П-*0О
then there may exist at most one sequence xn satisfying E.47) and such that the terms
Apxn have a constant sign. If such a sequence exists, then it must be of form E.49).
If, moreover, the terms Ap+ 1bn have a constant sign, then formula E.49) actually defines
a unique sequence fulfilling E.47) and having the differences Apxn of a constant sign.
Proof. Suppose that a sequence xn has the required properties. Applying to
both sides of E.47) the operator A" we obtain
A'xH+1+A'xH=A'bH,
whence by E.52)

7. Recurrent sequences 123
Since the terms Арх„ have a constant sign, this implies E.48) and relation E.49)
results from lemma 5.7.
Now suppose that Ap+1bn have a constant sign. Thus the sequence Apbn is
00
monotonic and consequently the series ? ( — l)vApbn converges. Thus formula E.49)
v=0
actually defines a sequence xn. We have
P-if-iy (_iy » _.
E.53) xn+1= ? ——.Aybn+1+—^ ?t(~1)V J"b«+v>
whence
p-i(_iy (-1)" °° _! (-1;
v=0 ^ ^ v=l
v=0 ^ Z V=0
( — 1) Л О_ + „ ^ D_
f-2 C—1
= z
v=0 •? Z' v=0
Repeating this procedure, applied successively to A1xn, A2xn, etc., /> times, we
finally obtain
v = 0
which may be written as
00
Apxn=- X [Apbn+2v + 1 _
v=0 v=0
Since the terms Ap+1bn+2v have a constant sign, the same is true about Apxn. More-
Moreover, we have by E.49) and E.53)
"-!(-!)* (-1)"
xn+i+xn= ? -^+т№Ьп+1+А b-]+ ~^р-ЛрЬ„,
whence, according to lemma 5.6, xn fulfils E.47).¦
§ 8. General linear recurrence. Now we turn to the study of a more general
linear recurrence
E.54) х„+1-Хх„=Ъп.
E.47) is the case X= — 1 of E.54). However, we shall confine ourselves to the study
of monotonic sequences fulfilling E.54) (i.e. to the case/? = 1 of the preceding section).

124 CHAPTER V. Monotonic and convex solutions
Theorem 5.15. (*) Ifl<0 and
E.55) Hm-f=0,
then there may exist at most one monotonic sequence xn satisfying E.54). If it actually
exists, then it is given by the formula
E
E
.56)
Proof.
.57) x
By
induction
,=X"xn +
one
И-1
i=0
xn =
proves
= —
the
i-i
» bn+v
v=0 л
formula
Let х„ be an increasing sequence satisfying E.54). Thus, for arbitrary n>0, k^l,
we have
whence, since A<0,
Setting m=2kin E.57) we obtain
2/t-l
Хп+ L A°n+2k-i-l^ ] T-,
Я—1
U
1lU *? un+2k
i=o
whence
^K+jk-i-l K+2k
Xn^ 2j :2k-i /i f\ -i2k'
i=0 / (Л—1)/.
and after a change of the index of summation
2k— 1 U U
E58) ^?
Similarly, starting from the relation
we get the inequality
" bn+v bn+2k+1
n> if1 (Al)
Relations E.58) and E.59) together give
"n+2k bn+2k+l
bn2kl
0) Brydak, Kordylewski [1]. Cf. also A. Smajdor [3].

8, General linear recurrence 125
Hence, as A:->oo, according to E.55) we obtain formula E.56). If we assume that
xn is decreasing, the proof is analogous. ¦
Corollary. If X ^ -1 and
E.60) lim [bn+1 — Ь„] = 0 ,
П-* 00
then there may exist at most one monotonic sequence fulfilling E.54).
For X< — 1 this results from theorem 5.15, since then condition E.60) implies
E.55); for X= -1 from theorem 5.14 mthp = l.
Theorem 5.16.0 If
E.61) |A|> lir
П-*00
and either
(a) X>0 and the sequence bn is increasing [decreasing],
or
(b) X<0 and the sequence и„=Ь„+1 + ХЬ„ is increasing [decreasing],
then formula E.56) actually defines a decreasing [increasing] sequence satisfying
E.54). In case (b) it is the unique monotonic sequence fulfilling E.54).
Proof. The convergence of the series occurring in E.56) follows from E.61)
in view of the Cauchy-Hadamard theorem on power series. Inserting E.56) into
E.54) we verify that the sequence obtained actually fulfils E.54). If X>0 and bn is
monotonic, then xn is monotonic as a sum of monotonic sequences. If X<0, then
we write E.56) in the form
00 U 00 U 00
32v+l 2j -2V+2 ~ 2-1
v=0 л v=0 Л v=0
oo ,,
v^ "n+2
/;2\v+l '
and the monotonicity of xn results from that of un. The uniqueness then follows
from theorem 5.15.И
A sequence fulfilling E.54) is completely determined by the initial term x0.
Let us write
sk=- i
i=oXi+1 (X-l)Xk
and
ae= sup S2k, ae= inf
o0 = sup S2jt_ i , Сто= inf
@ Brydak, Kordylewski [1]. Cf. also A. Smajdor [3].

126 CHAPTER V. Monotonic and convex solutions
Let X be negative. Then (cf. Brydak, Kordylewski [1], A. Smajdor [3]) a sequence
xn fulfilling E.45) is increasing if and only if ffe<x0<<70; and is decreasing if and
only if d0 < x0 <<re.
§ 9. Consequences for functional equations. Results of §§ 7-8 allow us to draw
some conclusion concerning the solutions of certain linear functional equations
(Kuczma [6], [8], Brydak, Kordylewski [1], A. Smajdor [3]).
Theorem 5.17. Suppose that f'e R°[I], S, $ I, and that the function F(x) is defined
in I and for a p^O fulfils the condition
If a function <peMp{f}[I] satisfies equation E.46) in I, then
p (_ry f_iy+1 со
E.62) ф)= I У—± Al
v=0 *¦
Moreover, if Fe Mp+1 {/}[/], then formula E.62) actually defines a unique so-
solution (p e Mp{f) [7] of equation E.46) in I.
Proof. Let us choose an xel and put xn=(p[f\x)], bn=F[f(x)]. Then the
sequence х„ fulfils E.47) whenever <p(x) is a solution of equation E.46). Further-
Furthermore, Л{х„=Л\п(ру\х)}, Aibn=A\I)F\f{x)]. Thus the present theorem results
from theorem 5.14.И
Theorem 5.18. Let I=(a, oo), 0^a> — oo, and suppose that FeMp+1[I] and
limAp+1F(x) = 0. Then there exists exactly one function (peMp[I\ satisfying the
X -ЮО
equation
E.63)
in I. This function is given by the formula
E.64) P(*)=V(±*)
where y/(x) is a solution of the equation
E.65) y(x+l)-y/(x)=-FBx)
belonging to the class МР+1[Г\. Function E.64) is also the unique solution of equation
E.63) belonging to the class Mp{x+l} [/].
Proof. On account of theorem 5.13 equation E.65) has a Mp+1 solution uni-
unique up to an additive constant. Consequently the function (p(x) is by formula E.64)
unambiguously defined in /. Moreover, <p e MP[I]. In fact, by lemma 0.9 and for-
formula E.64) <p e CP[I] and

9. Consequences for functional equations 127
Since y/ip\x) is convex, <pip\x) is monotonic as a difference quotient of a convex
function, and <p e Mp in virtue of lemma 0.10. Further, we have by E.64) and E.65)
i.e. (p(x) actually satisfies equation E.63). The uniqueness follows from theorem
5.17 and relations @.33).¦
Theorems 5.15 and 5.16 also have analogues for the functional equation
E.66) v[f(*)]-fy(*) = F(x) .
Theorem 5Л9-С1) Suppose that f e В%[1), ?$I, and F(x) is a function on I. If
and lim F[/"(x)]/A" = 0 for x el, then there may exist at most one solution
(p(x) of equation E.66) {f}-monotonic in I. If it does exist, then it is given by the for-
formula
E-67) V(x)=- ^ Р-Щт^-
v=0 /
Corollary. If A<-1 and limJ{1/}F(x)=0, then equation E.66) may have at
most one solution {f}-monotonic in I.
Theorem 5.20. (*) Suppose thatfe R° [I], ?, $ I, and F(x) is a function on I. If for
every xel, \1\ > lim sup y/\F[f"(x)]\ and either
(a) A>0 and F(x) is {f}-increasing [{f}-decreasing] in I,
or
(b) l<0 and the function F[f(x)] + lF(x) is {f}-increasing [{f}-decreasing] in I,
then formula E.67) actually defines an {f}-decreasing [{f}-increasing] solution
of equation E.66) in I. In case (b) it is the unique {f}-monotonic solution of E.66)
in I.
Remark. In theorems 5.19 and 5.20 the {/} in the expressions {f}-monotonic,
{f}-increasing, {f}-decreasing may be omitted (both in the hypotheses and in the
statements, of course). In case of theorem 5.19 this results from relation @.32).
In case of theorem 5.20 an independent proof is necessary, but it does not differ
from that of theorem 5.16 and is therefore omitted.
§ 10. Eider's Gamma function. The function Г(п)=(п—1)! is defined for posi-
positive integers n by the recurrence ГA)=1,Г(и+1)=лГ(л). If one attempts to
extend this function onto arbitrary positive values of the argument B), it is nat-
(!) Brydak, Kordylewski [1]. Cf. also A. Smajdor [3].
B) The problem of defining Г(х) for other values of the argument will be dealt with in Chap-
Chapter VIII.

128 CHAPTER V, Monotonic and convex solutions
ural to seek such an extension among the solutions of the equation
E.68) <p(x + l)=x<p(x), xe@,oo),
fulfilling the condition
E.69)
By theorem 4.1. equation E.68) has in @, oo) a C00 solution depending on an arbi-
arbitrary function. The requirement of analyticity would not yield a unique solution
either (cf. Chapter IV, § 8). Therefore we adopt the following definition (*).
Definition. The logarithmically convex function B) satisfying equation E.68)
and fulfilling condition E.69) is called Euler's Gamma function and is denoted
by Г(х).
It follows from theorem 5.11 that there exists a unique function fulfilling the
conditions of the above definition. Since log x is completely monotonic, the func-
function - logT(x) is also completely monotonic (cf. theorem 5.12, corollary) and
thus Г(х) is analytic in @, oo). Further, according to formula E.27), we obtain
- (n+iHn+2)*-1 1 1
= П
whence, since lim = 1 ,
( + l)( + )
n\n
E.70) Г(х)= lim
Further properties of the function Г(х) can also easily be deduced from the above
definition (Artin [1], [2]; cf. also Chapter XI, § 3).
The function Г(х) may also be characterized by other properties.
Theorem 5.21. If a function (p(x) satisfies equation E.68) and, moreover,
lim {-] J
then
(!) Bohr, Mollerup [1]. The idea was then developed by E. Artin [1], [2]. Cf. also van der
Waerden [1], Courant [1], Mayer [1], Caratheodory [1], Bourbaki [1], Anastassiadis [2], [7], Ding-
has [1], Schafke [1] and also Kuczma [13], [24]. Let us note that N. E. Norlund [1] denned Euler's
function using the notion, which he introduced himself, of a principal solution of equation E.24).
Since in this book we do not develop a theory of finite difference equations, we cannot go into
details here.
B) A function F{x) is called logarithmically convex in an interval I if F(x)>0 in / and
log F(x) is convex in /.
C) A similar characterization of Г(х) was given by A. Eagle [1].

10. Euler's Gamma function 129
x
Proof. Taking in theorem 5.8 Q(x)=[ — I /—, we obtain
\x \2
E.71)
,. J2He\x + n) n\nx
= lim
lim .
n-oo Itln y/X + n Х(Х+1)...(Х
Further,
^e\x + n)x
+n+1
\ n
In view of Stirling's formula (cf. A1.27))
E.73) n!=nV2
expression E.72) tends to 1 as n-+oo. Thus <р(х)=Г(х) results by E.71) and E.70).¦
It follows from Stirling's formula E.73) that we do not have too much liberty
in choosing a function with which Г(х) might be compared. As G. Szekeres [3]
has remarked, Г(х) is the only function fulfilling E.68) and E.69) which is asympto-
asymptotically equal to an L-function. In fact, we have (р(п)=Г(п), n=l,2,3,..., for
every solution <p(x) of E.68) with E.69). Thus theorem 5.21 shows that the limit
lim (<p(x)jr(x)), finite or infinite, does not exist for any such (рфГ. On the other
Х-юо
hand, any two members of Hardy's scale H are asymptotically comparable with
each other.
Theorem 5.22. (Anastassiadis [3], [7].) If a function <p(x) satisfies E.68) with
E.69) and, moreover, (e/x)x<p(x) is monotonici1), then <р(х) = Г(х).
Proof. Put Q(x) = exx~x. This function is decreasing in @, oo) and
xQ(x + i)
By theorem 5.9 there is exactly one function <p(x) fulfilling the conditions of the
theorem. Formula E.22) with xo = 1 gives in view of E.69)
,<™ , , ,. e(l + n) Z}
E.74) <p(x)=hm —т -,—г П
' * ' n^o ех+\х + пУ(х^ ,Ц
= lim
n-юо
,Ц x + i
(x + n)x+n+1 n\nx
(J) It is enough to assume monotonicity for large x.
9 Functional equation»

130 CHAPTER V. Monotonic and convex solutions
Further,
lim t j-^r = lim I 1 • I I • I ——. I • e1 ~x= 1,
n->ao e n(l + n)nn->a>\nj\nj \l + nj
whence <р(х) = Г(х) follows according to E.70) and E.74).¦
Generally, one can obtain various functions related to Г(х) as solutions of
equations of the form
where co= +1 or -1 and R(x) is a rational function, under additional monoton-
icity or convexity conditions (*). Here we shall discuss only two examples.
Definition. The function B(x, y) which is defined in the first quadrant, mono-
tonic B) with respect to x for each fixed y, and fulfils the conditions
^(x,y), x,ye@,oo),
y)=-
У
is called Euler's Beta function.
Theorem 5.23. C) We have
В(х,у).
V '" Г(х + у)
Proof. Keeping у fixed, we have in virtue of theorem 5.4
у n=o l + n+y x + n
lim
,. x + n n\nx n\ny
hm • • x
and the theorem follows in view of E.70).
(!) Mayer [1], Thielman [1], Anastassiadis [l]-[7], Bajraktarevic [2], Kuczma [6], [7], [17].
Cf. also Vajzovic [1].
B) It is enough to assume the monotonicity for large x.
C) Anastassiadis [4]. Cf. also Anastassiadis [7], [2], Kuczma [7].

10. Euler's Gamma function 131
Theorem 524.(*) Let (p(x) be an {x+\}-decreasing function on @, oo) satisfying
the equation
E.75) <p(x + \)<p(x)=—, xe@,oo).
x
Then
E.76)
\J2 ПЦХ + 1У]'
Proof. Since <p(x) is {x+ l}-decreasing, it follows from E.75) that <p(x)>0
in @, oo). Thus A(x) = log (p(x) is also {x+ l}-decreasing and satisfies
E.77) A(x + l) + A(x)=-logx.
By theorem 5.18 (p=0) equation E.77) has a unique {x+l}-decreasing solution
where y/(x) is the convex solution of
y/(x + l)-y/(x) = log
By theorem 5.11
i//(x) = r]0+xlog2+logr(x),
whence
E.78) X (x) = - \ log 2 + log T(ix) - log ГЩх +1)]
and E.76) follows.B
Since the function l/x belongs to M+[@, oo)], its primitive logx belongs to
M+[@, oo)]. Obviously limJi logx=0. Thus, in virtue of theorem 5.18 with
Х-Ю0
p= 1, function E.78) is the only M+ solution of equation E.77). Since the function
exp x belongs to M+nMf, function E.76) is also in M + B). Generally, it follows
from theorem 5.18 that the function — l(x) is completely monotonic in @, oo).
§ 11. Application to branching processes C). Let A'be a discrete random variable which takes
the value j with the probability pt, y=0,1,2,... Thus
GO
E-79) Ел = 1.
(We say that {pt} is an honest probability distribution if 0</>3<l and E.79) is fulfilled.) The function
OO
E.80) /(*)=?/>,*>, 0<*<l,
is called the generating function of the probability distribution {p}}.
@ Anastassiadis [1]. Cf. also Mayer [1], Bajraktarevic [2], Anastassiadis [7].
B) Moreover, E.76) is the only solution peM4@, oo)] of equation E.75). (In this form
theorem 5.24 was originally stated by A. Mayer [1].) This follows from the fact that every convex
function is monotonic for x sufficiently large, and, since <peMj, it follows from E.75) that <p must
be decreasing, and thus also {x+l}-decreasing in @, oo).
C) Heathcote, Seneta, Vere-Jones [1]. Cf. also Jaglom [1], Bellman [2], Zolotarev,
Koroljuk [1], Th. E. Harris [1], Seneta [1], [2].
9*

132 CHAPTER V. Monotonic and convex solutions
If Xi and Xj are two independent random variables with the same probability distribution
{Pi}, we may form a new variable X=Xi + X2. Let {/>#} be the probability distribution of X. Since
1
X\ and Хг are independent, we have pjt= LjPiPj-u and hence the g;nerating function of {рг;} is
i-0
GO 00 j
? P2tx'= ? ? AAV= LA*)]2 •
By induction, the generating function of ths probability distribution {pnj} of the variable X=X\ +
+ ... + Xn, where Xi are independent random variables with the same probability distribution
{Pi), is
CO
E.81) ?pw*'= [/(*)]".
i-o
Now consider a sequence of mutually independent random variables X\,Xz,..., each
with the same generating function E.8 J). We form a new variable
E.82) V=Xl + ... + Xn,
where the number л of summands in E.82) is again a random variable with ths probability distri-
distribution [qj] and the generating function
00
E.83) </(*)= ?«*>•
i-o
CO
Let {vj} be the probability distribution of E.82). Then fj= 2j Wnj» and the generating function
of {vj} is given by n=0
CO CO CO CO CO
? »***= ? ? qnpnjx}= ? ?n ? /W
y-o y-o n-o n-o y-o
n-0
according to E.81) and E.83). (We may change the order of sum-nation, sines all ths terms of the
occurring sequences are non-negativj.)
We are going to consider a branching process, where a single object has the probability p]
of giving / offsprings of the same typj. If W3 start from a single ancestor, which constitutes the
zeroth generation, we have the probability pi that first generation consists of у msmbres. The po-
population at the second generation is the random variable E.82), where ths probability that the
number of summands is я equals pn and each of the summands has the probability distribution
{Pi}. Consequently the generating function of V is /[/(*)] =f4x) and we infer by induction that
the population at the nth generation (dsscsnied from a single ancestor) has the generating func-
function/n(x).
We shall prove the following theorem on the conditional distributions of ths population size.
Theorem 5.25. (!) If {pi} has the generat'ng function E.80) and the mean
E.84) ™
(i) Heathcote, Sensta, Vere-Jonss [1]. Cf. also Jaglom [1].

11. Application to branching processes
133
then the sequence
E.85)
converges to a limit function y/(x) with the following properties:
(i) y/(x) generates an honest probability distribution, ^@)=0.
(ii) y/(x) satisfies the functional equation
E.86) 1 - xji [fix)] = т[1-ч/ (*)].
(iii) y/(x) is the only function fulfilling (i) and (ii).
Proof. Condition E.79) means that /A)= 1, condition E.84) may be written as m=f'(l)<\.
Since all p] are non-negative, all the derivatives of f{x) are non-negative on {0,1). Hence/(x)
Fig. 7
is strictly increasing and convex and consequently feR^[@, 1)]. (Cf. fig. 7, which represents the
graph of f(x).)
We put
E.87)
and
E.88)
1-х
(*)=
- y/n(x)
1-х
The convergence of sequence E.85) to a function y/(x) satisfying E.86) is equivalent to the con-
convergence of sequence E.88) to a function <p(x) satisfying the equation
E.89)
<?[/"(*)]=
1 \x)
As x tends to 1 (from the left), T(x) tends to f'(\)=m and consequently the coefficient of <p(x) in
E.89) tends to 1. Since /"(x)>0, we have/eAf?[<0, l>] and hence T(x) is increasing as the dif-
difference quotient of a convex function. By theorem 5.4 equation E.89) has in @, 1) a unique mo-
notonic solution <p(x) fulfilling the condition p@)=l, given by
E.90)
4>(x)=Y[
я-0
Г [/"(x)]
= lim (р„(х).

134 CHAPTER V. Monotonic and convex solutions
Consequently sequence E.85) converges in <^0, 1^> (the convergence for x=\ is obvious) to a
function
V(x)=\-(l-x)q>(x),
which fulfils ^@)=0, y/{\)=\, and satisfies equation E.86).
Since, for every n, fn(x) is a generating function of a probability distribution, also y/(x) is a
generating function of a probability distribution. As the limit of sequence E.85) of monotonic
functions, y/(x) is itself monotonic. Hence it follows that lim y/(x) exists and by E.86) and E.84)
z->l-0
equals 1. Consequently <//(x), being a limit of probability generating functions, itself generates
an (honest) probability distribution.
Now suppose that a function y/(x) fulfils conditions (i) and (ii) of the theorem. Then y/(x)
is convex in<p,iy,4/(l)=l, and consequently the function
<P(x) =
1-х
is monotonic in <0,1) and satisfies equation E.89). Condition yr(O)=O implies that p(O)=l.
Thus, by theorem 5.4, <p(x) must be given by E.90), which proves the uniqueness statement. ¦
Remarks. 1. Function E.87) is the generating function of the "tails" of the offspring distri-
distribution; i.e. if
1-0
then tj is the probability that X>j, where X is the variable with the distribution {pj}.
2. If m>\ and /(х)фх, then either f(x)>x in @, 1), or there is a (unique) f e @, 1) such
that /еЩ[ф, l)]. In the former case we must have m=l ani function E.90) still exists. Equa-
Equation E.86) now becomes
E.91) V [/(*)] = V(*).
By theorem 5.4 the only monotonic solutions of equation E.91) are constant functions. Since y/ @)=0,
we must have y/(x)=0.
In the latter case we have lim f"(x)=? for every x e (p, 1) (theorem 0.4) and consequently
П-*гх>
sequence E.85) tends to zero on <0, 1), provided it is defined (i.e. />o=/(O)#O or, what amounts
to the same, ?#0). Thus in both cases we arrive at the same conlusion:
If /n>l and рофО, then sequence E.85) tends to zero on @, 1).

CHAPTER VI
SCHRODER'S EQUATION
§ 1. Preliminaries. The Schroder equation
F.1) 4>lf(x)l=s9(x),
where s^O, 1 is a constant, belongs to the most important particular cases of equa"
tion @.49). It appeared for the first time about 1870 (Schroder [2]) in connection
with the problem of continuous iteration (cf. Chapter IX). A fundamental theorem
about the existence and uniqueness of analytic solutions of F.1) was then proved
by G. Koenigs [4], and therefore equation F.1) is often called also the Koenigs
equation or the Schroder-Koenigs equation. It has been extensively studied since A).
The authors have mainly paid attention to analytic solutions on the complex plane.
The Schroder equation for functions of a real variable has been dealt with only
in the present century B).
Here we start with studying equation F.1) in the real case, postponing the
treatment of the complex-variable case till §§7 — 10. §§3 and 11 contain results
for both real and complex functions.
We start with the following result of a general character.
Lemma 6.1.C) Letf(x) be defined in a submodulus set E and let g(x) be a one-to-
one mapping of E onto a set Et and put
F.2) h(x)=g(f[g-\x)]).
A) Schroder [1], [2], Tanner [1], [2], Ellis [1], Rawson [1], Rausenberger [1], Koenigs [3]-[6],
Farkas [1], Appell [6], [7], Leau [1], [2], Bourlet [4], Fatou [1], [5], [13], Lattes [3], [12], [15], [16],
[17], van Uven [1], Kasner [1], Pfeiffer [2], [3], Julia [2]-[5], [8], [12], [13], [14], [18], Pincherle
[10], Cremer [l]-[8], Wolff [1], [2], [3], [6], [8], Tambs Lyche [4], [7], Valiron [5], [6], Tschebo-
tarev [1], Barba [1], Siegel [3], [7], Wright [1], Bradley [2], Topfer [2], [4], Pastides [1], [4], Kneser
[1], Jabotinsky [3], Szekeres [1], Klingen [1], McKiernan [3], Th. E. Harris [1], Gumowski,
Mira [1], [2], Targonski [4]-[6]. Cf. also Pincherle [3], [4], Picard [10], Montel [4], [7], [13],
Ghermanescu [22], Kuczma [24], Gotz [1].
Also various generalizations of F.1), especially to the case of several, real or complex, vari-
variables, were studied by Leau [1], [2], Julia [12], Fatou [9], [14], Cartan [1], Tambs Lyche [9],
Hukuhara [1], Urabe [1], [4], Pastides [2], [3], Myrberg [7], J. Moser [1], Siegel [8], Montel
[13], Sternberg [5], Kuczma [20], Mizumoto [1], Schubert [1].
B) Oeconomou [1], Crum [1], Kneser [1], Sternberg [1], Szekeres [1], Kuczma [19], [20], [21],
[38], Lundberg [1], Coifman [1], [3], [4]. Cf. also Kuczma [24].
C) Schroder [2], Montel [13], Szekeres [1], Kuczma [20].

136 CHAPTER VI. Schroder's equation
If y/{x) is a number-valued A) solution of the equation
F.3)
in Ex, then the function
F.4)
satisfies equation F.1) in E.
This fact is directly checked by inserting F.4) into F.1).
The above lemma is often used when we wish to transform E or f(x) into a
more convenient form. Let us note that if/is analytic and g is analytic and schlicht
in a domain E on the complex plane, then relation F.4) establishes a one-to-one
correspondence between the analytic solutions of equations F.1) and F.3). Simi-
Similarly, if /e C[E] and g is a C-morphism B) from E onto E^, then we have a cor-
correspondence between the С solutions. Lastly, if g is linear on an interval /, then
correspondence F.4) preserves convexity.
We may use lemma 6.1 to transfer a fixed point ? of/to the origin. If ? is finite,
it is enough to put д(х)=х—%. If ?= oo, we take g(x)= l/x. Therefore in the sequel
we shall assume that the fixed point of/is at 0.
We shall assume that/e R% [I], and in most cases even that/e Rq [I]. If |s\ ^ 1,
then it follows from theorems 2.7 and 2.11 that q>(x)=0 is the only continuous
solution of equation F.1). On the other hand, if |s|<l, then equation F.1) has a
continuous solution depending on an arbitrary function (theorem 2.10). Therefore
in order to obtain non-trivial particular solutions distinguished by a certain prop-
property, we must turn our attention to the C1 solutions of F.1). Now, if <p{x) is a C1
solution of F.1), then nl = q>'@) must fulfil the equation
Thus, either r\x = 0 or we must have
F.5) /'@)=5,
in which case n^ may be arbitrary. If F.5) does not hold, then in accordance with
Chapter II, §2, C) only two cases are possible: equation F.1) may have a C1 solu-
solution depending on an arbitrary function, or tp(x)=0 is the only C1 solution of F.1).
Since /e Rq, we have 0</'@)< 1. The case where
F.6) 0</'@)<l
will be called regular. In the regular case we shall always assume that F.5) holds.
The case where /'@) = 0 or 1 will be referred to as singular.
(!) Real or complex. Here the set E may be quite arbitrary.
B) I.e. д is invertible and both д and gf are of class C.
C) Cf. the proof of theorem 6.2.

2. Regular case 137
§ 2. Regular case. At first let us note the following simple result.
Theorem 6.1. (Kuczma [20].) Suppose that feRr0[I], Oe/, r^2, and that
relations F.5) and F.6) hold. Then to every ne(— oo, oo) there exists exactly one
С solution <p of equation F.1) in I fulfilling the condition
F.7) ?'Ф)=п.
This solution is given by the formula
F.8) <p(x)=r,lims-nf\x)
n-*co
and for n ф 0 is strictly monotonic in I (increasing for n>0 and decreasing for r\ < 0).
Proof. The existence and uniqueness of cp and formula F.8) follow from the-
theorem 4.5. Since <peC[I], for n^O the derivative <p'(x) is of a constant sign in a
neighbourhood of 0. The strict monotonicity of <p in / then follows from the corol-
corollary to theorem 5.7. ¦
In order to study C1 solutions let us put
We may distinguish cases (i), (ii) and (iii) as in Chapter II, § 2.
Theorem 6.2. (Kuczma [20].) Suppose that /e«j[i], Oe/, f'(x)^0 in /,
and F.5) and F.6) hold.]
In case (i) to every n e (— oo , oo) there exists a unique C1 solution q> of equation
F.1) in I fulfilling condition F.7). This solution is given by formula F.8) and for пфО
is strictly monotonic in I (increasing for n>0, decreasing for n<0).
In case (ii) equation F.1) has in I a C1 solution depending on an arbitrary func-
function. In this case every C1 solution <p(x) of equation F.1) in I fulfils the condition
p'@)=0.
In case (iii) the function q>(x)=0 is the only C1 solution of equation F.1) in I.
Proof. Let <p(x) be a C1 solution of equation F.1) in / fulfilling condition
F.7) and put a(x) = q>'(x). Then the function a(x) is a continuous solution of the
equation
F.10) a[/(i
a[/(x)]a(x)
J \x)
in /, and <x(x) fulfils the condtition
F.11) a@)=i/.
Conversely, let a(x) be a continuous solution of equation F.10) in / fulfilling con-
condition F.11). Then the function
X
q>(x)=\oi(t)dt
о

138 CHAPTER VI. Schroder's equation
is a C1 solution of equation F.1) in / fulfflling condition F.7). In fact, we have
by F.10)
<P [/M] = J a @ dt = J a \_f(uj\f'(u) du = s J a (и) du = s<p (x).
oo о
Thus there is a one-to-one correspondence between the C1 solutions of equa-
equation F.1) and the continuous solutions of equation F.10). Consequently our the-
theorem, except for the monotonicity of <p when n^0 and for relation F.8), follows
from theorem 2.2 applied to equation F.10).
The strict monotonicity of <p(x) in the case where пфО follows from the corol-
corollary to theorem 5.7. In order to prove formula F.8) we assume that <p(x) is a C1
solution of F.1) fulfflling F.7) with an n^0 and we form the function fi(x) = q>(x)/x
for x?=0,i@)=n. It follows from F.1), F.5) and F.6) that <p@)=0. Hence /?(x)
is continuous in /. Furthermore, /?(x) satisfies
F.12) PU(x)']=g(x)p(x),
where g(x)=sx/f(x) for x^0, g@)=l. We may apply theorem 2.2 to equation
F.12). Since /?@)^0, it is necessarily case (i) that occurs. Consequently
P(x)=— \ims~"fn(x),
whence we immediately obtain F.8). For ^=0 formula F.8) is evident. ¦
Theorem 6.3. (Szekeres [1], Cram [1].) Suppose that /e/?J[/],0 el andthat there
exist positive constants К and ц such that
holds for \x\ sufficiently small, xe I. Then case (i) occurs.
Proof. Condition F.13) implies F.5) and F.6). By lemma 0.3, x=0 is a strongly
attractive fixed point of/(x). Moreover, we have on account of F.13)
-1
fix)
for small |x|, where M is a positive constant. Thus our theorem results from the-
theorem 2.5. If the condition/'(x)^0 is not fulfilled in the whole of/, then we obtain
a unique one-parameter family of C1 solutions of F.1) in a neighbourhood of zero
(where/'(x)^0); these solutions are then extended onto the whole of/according
to theorem 4.2.И
A weaker condition than F.13),
F.14)
, is already sufficient for the existence of solutions q>{x) of equation F.1)
in /, differentiable at zero and fulfflling F.7) (Kneser [1]; cf. theorem 6.4). However,
those solutions need not be of class C1 in / or strictly monotonic (cf. Szekeres [1]).

2. Regular case 139
All the three cases (i), (ii) and (iii) can actually occur for functions/e/?J. Case
(i) occurs whenever /eRr0, r^2, or feR\, and fulfils condition F.13) (theorems
6Л and 6.3). For the function
F.15) fix)=s f A - -^-\dt for x e @, a), /@) = 0,
о
where a is sufficiently small and 0<$<l, we have f(x)>sx in @,a) and, since
both, f(x) and sx, are strictly increasing,
for хе@,я), п = 1,2, ...
Hence, for expressions F.9) we get
G()<
„
П( - l/log/vW) П A - l/(vlogs+logx))
v=0 v=0
and consequently Gn(x) tends to zero as n-*co- Since each of the functions Gn(x)
is decreasing in <0, a), the convergence is uniform in <<5, a) for every <5>0. Thus
function F.15) provides an example of case (ii).
Similar estimations show that for the function
F.16) f(x) = sUl + ~)dt for xe@,a),/@)=0,
о
we have lim Gn(x) = oo for every xe@,a) and consequently case (iii) occurs.
Л->00
§ 3. Koenigs existence theorem. In the present section x may be either complex
or a real variable. We shall assume that f(x) is analytic and is representable by a
power series
F-17) №=sx+ftanxn
n=2
convergent in a neighbourhood of the origin A). The coefficients of series F.17)
may also be complex.
In the regular case
F.18) 0<|s|<l
we have the following theorem, due to G. Koenigs [4] (cf. also Picard [10], Cremer
|4], Kneser [1], Siegel [7]).
(*) The case where fix) has an asymptotic expansion of form F.17) in an angular domain
xm the complex plane was investigated by G. Szekeres [1].

140 CHAPTER VI. Schr6der's equation
Theorem 6.4. Let F.17) with F.18) hold in a neighbourhood of the origin.
Then formula F.8) yields a unique one-parameter family of analytic solutions of equa-
equation F.1) in a neighbourhood of the origin (n is here the value of the derivative of q>
at 0, i.e. F.7) holds).
Proof. It follows from F.17) that there exist positive constants К and ц such
that F.14) holds in a neighbourhood of the origin A). By F.18) there exists a q
such that
F.19) ||
and so
F.20) |/(*)|<eN
in a neighbourhood U={x: \x\<r} of the origin. By F.20)
F.21)
for x e U and in view of F.14)
holds for n sufficiently large. According to F.19) q*+ll\s\<l and therefore the
sequence s~"f(x) uniformly converges in U. Its limit is thus analytic in U. Since
all the derivatives (s~"f(x)y at x=0 equal 1, the same is true about the limit.
Now let q>{x) be an arbitrary analytic solution of equation F.1) in a neighbour-
neighbourhood of the origin and let q>0(x) be the function F.8) with rj=\. Consequently
F.22) 1 = <po(x)=km = lim .
x->0 X л-юо J (X)
By F.22) сро{х)фО in a neighbourhood of the origin. Hence
m=шп ii
since both <p(x) and (po(x) are solutions of F.1). Consequently <p(x) must be giver*
by F.8) with F.7).¦
This theorem has a local character. A global theory of equation F.1) and of
related equations, based on Montel's theory of normal families, was developed
in the nineteen twenties by P. Fatou, G. Julia and others B). Unfortunately, lack
(•) In fact the argument below shows that F.14) suffices to prove the existence of limit F.8).
Then <p(x) need not be analytic, but F.7) still holds.
B) Julia [2]-[5], [7], [9], [11], [12]-[14], [17], [18], Lattes [15]-[17], Ritt [2], Fatou [4], [5],.
[11], [15],Brolin[l].

3. Koenigs existence theorem 141
of space prevents us from reproducing their investigations here. Let us note, how-
however, the following
Theorem 6.5. Letf(x) be analytic in a submodulus region G, OeG, and let F.17)
with F.18) hold in a neighbourhood of the origin. Then equation F.1) has a unique
one-parameter family of solutions analytic in the domain of attraction Af@), given
by F.8).
Proof. With the notation of the preceding proof, let x be an arbitrary point
of Af@) and let m be such that/m(x) e U. We define <p at x by
<p being already known in U according to theorem 6.4. Then q> is analytic at x, just
as fm and q> are analytic in G and U, respectively. The details of the proof are left
to the reader. ¦
In view of F.7) functions F.8) with цфО have inverses if/(x) = tp~i(x), analytic
in a neighbourhood of the origin. These yield analytic solutions of the Poincare
equation A)
F.23) V(sx)=/I>(x)],
and, except for the constant one у/(х)=0B), they are the only analytic solutions
fulfilling (y@) = 0. In fact, if (/@)^0> then (p(x) = y/~1{x) exists and is analytic
in a neighbourhood of the origin and satisfies F.1). If if/'@) = 0, then differentiating
F.23) twice and setting x=0, we obtain
i.e. in view of F.17) s2y/"@)=sy/"@), whence (/'@) = 0 follows in virtue of F.18).
Proceeding thus further, we show that all the derivatives of y/ at x=0 vanish, i.e.
W(x) = 0.
Thus the Schroder equation F.1) and the Poincare equation F.23) may be
reduced to each other and may be regarded as equivalent, at least as long as we
are interested in local solutions.
§ 4. Convex solutions. We return to the case of a real variable. Theorem 5.9
applied to equation F.1) yields the following result.
Theorem 6.6 (Kuczma [21]. Cf. also Lundberg [1].) Suppose that fe R°0[I]C),
/'@) exists and fulfils F.5) and F.6). If<p{x) is a solution of equation F.1) in I such
A) Poincare [3], [4], Kaba [1], Picard [6], [10], F. de Brun [1], Levi [1], Giraud [l]-[3], Vali-
ton [1], [6], Lattes [15]-[17], Fatou [7], [9], Julia [8], [12], [13], [14], [18], Ritt [6], [14], Touchard
[2], Wolff [11], Siegel [3], [7], Gerst [1], Wittich [1], Pastides [3], Myrberg [11], [14], Ghermanescu
[22], Klingen [1], Gotz [1].
B) More generally, for every fixed point ? of f(x) the function y/(x) = g satisfies equation
F.23).
C) 0 may belong to / or not.

142 CHAPTER VI. Schroder's equation
that the function <p(x)/x is monotonic in I, then
F.24) ^ (*) = ,, lim Z_AZ
> л->»; (x0)
where n is a constant and x0 ф 0 is an arbitrarily fixed point from I. Moreover, if
f(x)/x is monotonic in I, then F.24) (where л is an arbitrary constant) actually de-
defines a unique one-parameter family of solutions <p(x) of F.1) such that <p(x)/x is
monotonic in I.
Theorem 6.7. (Lundberg [1].) Suppose that fe R%[I], 0 being the left endpoint
of 1(*), /'@) exists and fulfils F.5) and F.6), and, moreover, f{x)jx is increasing
in I. Then formula F.24) defines a one-parameter family of solutions of equation
F.1) which are continuous and for пфО strictly monotonic in I {increasing for n>0
and decreasing for n < 0).
Proof. It follows from theorem 6.6 that F.24) defines a family of solutions
of equation F.1) in /. We must prove that for пфО (р(х) is continuous and strictly
monotonic in /. We may confine ourselves to the case //>0.
Since family F.24) is independent of the choice of x0, a function obtained
from F.24) with another x0 may differ from <p{x) only by a constant multiple. In
particular, if
/0 /"(*)
q>1(x)= lim —— and <p2(x) = lim —— , x1,x2el, хгф0, х2ф0,
n->oo J (Xi) л-»со J (X2)
then <p2(x) = c<p1(x) and, since q>1(x1)= 1, c=<p2(x1).
Since the functions/"(x) are strictly increasing in /, functions F.24) with n>0
are increasing in /. Let us take xt<x2, xlf x2 el, хгф0, х2ф0, and let us choose
an x0 e I, xt <x0 <x2. We have
Further,
f\x2)
whence it follows that the sequence /"(*i)//"(*0) is decreasing, whereas f(x2)/f(x0)
is increasing. Writing
F.26) 1М
we obtain hence by F.25) (pix^^cp^Xi). The same is true for all functions F.24)
with n>0, which proves that they are strictly increasing.
To prove the continuity, let us put xx =f(x2). Then the function q>i(x) is lower
semi-continuous in (xlfx2y as a limit of an increasing sequence of continuous
functions, and the function <p2(x) is upper semi-continuous in (x1} x2y as a limit
0 may belong to / or not.

4. Convex solutions 143
of a decreasing sequence of continuous functions. But <p2(x) = c<p1(x), i.e. both
functions are upper and lower semi-continuous, and hence continuous in <x1; x2}.
It follows from theorem 2.10 that functions F.24) are continuous in /.¦
It is interesting to note that the condition that f(x)/x should be increasing
cannot be replaced by the condition that f{x)jx should be decreasing. In the latter
case functions F.24) need not be strictly monotonic.
Theorem 6.8-О Suppose that /e J^/JnM1!/] (г), /'@) exists and fulfils F.5)
and F.6). Then F.24) defines a unique one-parameter family of convex solutions of
equation F.1) in I. These solutions are continuous and for rj фО) strictly monotonic in I.
Proof. All the functions f\x) belong to the same class M+[I] or Ml[I],
and so does the limit function F.26). Consequently <p is continuous in /. It is also
increasing in / and, being convex, is either constant or strictly increasing. But
q>(xo) = l, while lim^(j:)=0 by theorem 2.8; consequently q> cannot be constant.
x->0
The uniqueness follows from theorem 5.10.Ш
Let us note that the theorems of this section often allow us to choose a unique
one-parameter family of solutions of equation F.1) when theorems 6.1-6.3 fail.
E.g. theorem 6.8 is applicable in the case of the function f{x) given by F.15)
or F.16).
§ 5. Principal solution. Suppose that /eJ?JJ[/], /'@) exists and fulfils F.5)
and F.6). If the sequence f\x)lf\x0), x0 e/, xo#0, converges in /, its limit satisfies
equation F.1) in /. Function F.26) is then called the principal solution of equation
F.1) in / (Szekeres [1], Kuczma [20]). As \vas pointed out in the proof of theorem
6.7, the principal solution of the Schroder equation is determined up to a multiplic-
multiplicative constant.
If the limit lim s~"fn(x) exists, it is identical with F.26), but F.26) is slightly
more general, as may be seen from the example of functions F.15) and F.16). Con-
Consequently, theorems 6.1-6.8 say that if equation F.1) has a unique one-parameter
family of solutions characterized by a particular property, it is the family of prin-
principal solutions of F.1).
The significance of the principal solution of the Schroder equation will be
seen also in Chapter IX. Here we would like to stress that the principal solution
behaves better near the origin, than the other solutions of F.1). It need not, how-
however, be the most regular solution in /. We shall illustrate this by an example.
We put
g(x)=i(x+x2)
and we choose an xo e @,1). We define the double infinite sequence xn by the formula
F.27) xn = gn(xt)), n=±X,+2, ...
A) Kuczma [19], [20], [21], Lundberg [1], Coifman [4]. Cf. also Kuczma [38].
B) 0 may belong to / or not. If 0 is an endpoint of /, then/'@) necessarily exists, since f(x)
is convex.

144 CHAPTER VI. Schroder's equation
Next we define the function f(x) in <0,1) by the conditions:
(a) f(xn)=xn+i, n=0, ±1, ±2,...,
(b) f(x) is linear and continuous in every interval <xn+i, xny,
(c) Д0)=0.
The function f(x) belongs to Д?[<0, l)]rWj[<0,1)] and /'@) exists and fulfils F.5) and
F.6) with s=i- f(x) is not of class Ci [<0,1)], since it is not differentiable at the points xn, but is
of class C° in the set
+ 00
<S= U (xn+i,xn)-
— 00
Since f(S)=5, all the functions/"(x), и=0,±1,±2,...,аге of class
We choose a, be (x\, x0), a<b, and we put
exp[-(x-a)-Kx-b)-2] for xe(a,b),
0 for xe(i
The function q>o(x) is of class C° [(x\, xo"}]. By theorems 1.1 and 0.5 it may be uniquely extended
to a solution <p*(x) of the equation
F.28) <plf(x)]=i<p(x)
in <0,1). This extension is given by
F.29) ,w (
10 for
Thus q>* (x) is of class C° in 5. It is also of class C° at every point xn, since it vanishes in a neigh-
neighbourhood of every xn. Consequently tp* e C° [@,1)].
By theorem 6.8 there exists a principal solution F.26) of equation F.28) in <0,1) and it is a
convex function. But, just like f(x), <p(x) is not even of class C1 in @,1). In fact, for arbitrary
x, у e (xk+l, xk) (k fixed) we have
(x+y\ J"(.x)+f"(y)
I 2 / 2
(this can be shown by induction), whence
(x+y\ (p(x)+q>(y)
\ 2 / 2
Thus in each interval (xk+l,xk) <p(x) satisfies Jensen's functional equation, which means (cf. Aczel
[5], p. 49) that (p(x) is linear in each (xk+l,xk). On the other hand, it is not linear in the whole of
@,1), since linear functions do not satisfy F.28). (To see this, note that for лг=лгп f(x)—g(x).)
Consequently q>(x) is not differentiable at the points xn-
This shows that there can exist solutions of the Schroder equation which have higher regular-
regularity for хфО than the principal solution. The situation changes when we approach the origin. We
shall show that in the above example <p*(x) is not differentiable at x=0, whereas <p(x) is.
By theorem 6.1 there exists a unique regular function
F.30) v(x)= ton 2"g"(x)
П->0О
which satisfies the equation

5. Principal solution 145
and the condition (y'@)=i. Thus, in view of theorem 5.7 (corollary), v(x) is strictly increasing in
@, 1) and hence also strictly positive in @,1).
Now p*(xn)=0 for every n, whence
F.31) lim =0.
On the other hand, we have by F.29) and F.27) for у е (a, b)
V* If" (У)} > V* [/" (y)] _ 2~" у oCv) _
/ЯО) "" xn xn 1\
whence by F.30)
F.32) hminf > >0.
Relations F.31) and F.32) show that lim (y*(x)jx) does not exist.
l->0+0
Since the function <p(x) is convex, the limit lim {<p(x)\x) exists. We have
z->0+0
.. fW .. (P(xn) 2-"(p(x0) <p(x0) <p(x0)
lim = lim = lim = lim =
х-ю x n_>a х„ п^ю xn п^ю2"дп{х0) i//(x0)
and thus this limit is finite and positive. However, it is not always so. The principal -solutions of
the Schroder equation with the function F.15) or F.16) have the derivative at the origin equal to
0 or +oo, respectively.
§ 6. Singular case. Multiplier zero. Here we shall confine ourselves to the case
where
F.33) /'@) = 0.
The case where /'@) = 1 and л: is a real variable is more conveniently handled by
reducing F.1) to the Abel equation. If л: is a complex variable, the case where
|/'@)[ = l will be dealt with in next sections.
If F.33) is fulfilled, we cannot expect equation F.1) to have a non-trivial C1
solution in a neighbourhood of the origin. The derivative <p'{x) of a C1 solution
of F.1) satisfies the equation
which by theorem 3.4 has <p'{x) = Q as the only continuous solution in a neighbour-
neighbourhood of the origin. Nevertheless, we shall construct a one-parameter family of
solutions of equation F.1), which is of importance in the iteration theory (cf. Chap-
Chapter IX).
Theorem 6.9. (Szekeres [1].) Suppose that fe Rl[<0, c)], c>0, and that there
exist positive numbers a, S, M and ц> 1 such that
F.34) ||
holds for x sufficiently small. Then the formula
F.35) Ш "\
\ое-
/ (X)
10 Functional equations

146
CHAPTER VI. Schroder's equation
defines a one-parameter family of continuous and strictly decreasing solutions of the
equation
F.36) 9> [/(*)] =W(*)
in @, c).
Proof. It follows from F.34) that there exist positive numbers Klf K2 such
that the estimates
F.37)
F.38)
hold for x sufficiently small. Set
*W=[-log/(e-]
Writing R(x)=logf(x) — log а—ц log x we have
|log/(x)-
xf'(x)
h(x) =
1
-\oga+nlx-R{e
\oga+R{e~llx)
ц2 — x log a—jjlxR (e~
-x2.
By F.37) the coefficient of x2 in the above relation tends to Qoga)/n2 as x->0+0
and is therefore bounded for small x. Consequently
F.39)
for small x.
Further, we have
h(x)--
h\x)=-2\h{x)\
e-1/xfl{e-i/X)
Making use of F.38) and F.39), we show by estimating in the same way as above
that
h\x)--
for small x, where A" is a positive constant. By theorem 6.3 equation F.3) with
j= \JH has a unique family of C1 solutions, strictly monotonic in <0, c):
Now F.2) holds with g{x) = [-log x] and we have h"(x)=g(f"[g~1(x)]). The func-
functions (p*(x) = i//[g(x)]=nlimnng[fn(x)] are continuous and strictly monotonic in
(О, с) and by lemma 6.1 satisfy equation F.1) with s=\jn- <p(x) = [<p*(x)]~1 are thus
continuous and strictly monotonic solutions of equation F.36) in @, c). ¦

7. Case |s| = l. A review of the results 147
§ 7. Case |s|=l. A review of the results. The case where
F.40) |*| = 1
is much more complicated than F.18). In the present section we are going to review
the most important results concerning the analytic solutions of the Schroder equa-
equation in case F.40). Some proofs will be supplied in the next sections. We assume
that fix) is of form F.17), the series being convergent in a neighbourhood of the
origin, and F.40) holds. In the present section, and also in §§ 8-10, x is a complex
variable.
Theorem 6.10. If s is a p-th root of unity, then F.1) has an analytic solution
in a neighbourhood of the origin if and only if
F.41) f(x) = x.
If s is not a root of unity, then, by assuming <p(x) of the form
F.42) ?(x) = x+? cnx"
л=2
and inserting F.42) into F.1), it is possible to calculate recurrently the coefficients
cn, and thus to obtain a. formal solution of the Schroder equation, i.e. a formal power
series, possibly divergent, which, inserted into F.1), satisfies this equation formally.
The convergence of series F.42), however, is a very delicate matter. E. Kasner [1]
conjectured that F.42) converges whenever s is not a root of unity. This was dis-
disproved by G. A. Pfeiffer [2], [3], who first demonstrated the existence of f(x) for
which the corresponding series F.42) diverges. His result was strengthened by
H. Cremer [7] (cf. also Cremer [2], [3], [4] and Siegel [7]), whose theorem may be
formulated as follows:
Theorem 6.11. There exists a set M such that for every function F.17) with se M
complex numbers е„ with |е„| = 1, л = 2, 3, ..., can be constructed such that the formal
00
series F.42) corresponding to the function f*(x) = sx+ ? Епа„х" diverges for every
n=2
хфО. The set M has the power of the continuum and is dense on the circle |x| = l.
A certain condition for the convergence of F.42) was already found by G. Julia
[2] (cf. also Pastides [4]):
Theorem 6.12. Series F.42) corresponding to function F.17), with s fulfilling F.40)
and not being a root of unity, is convergent in a neighbourhood of x = 0 if and only
if the iterates f(x)form a normal family at x = 0.
This theorem, however, gives no practical means of verifying whether series
F.42) converges or not. Julia himself (Julia [6]; cf. also Fatou [5]) expressed the
erroneous conjecture that F.42) diverges whenever f(x) is a rational function not
identically sx. Finally С L. Siegel [3] settled this question by proving the following
Theorem 6.13. If there exists a constant K>0 such that
F.43)
10*

148 CHAPTER VI. Schroder's equation
then series F.42) corresponding to function F.17) converges in a neighbourhood of the
origin.
Condition F.43) is fulfilled for all s with F.40) except for a set of linear Lebesgue
measure zero.
In the next sections we shall present proofs of theorems 6.10, 6.11 and 6.13
following the lines of Siegel's book [7]. Instead of using equation F.1) we shall deal
with equation F.23), which, as was pointed out at the end of § 3, is equivalent to
equation F.1).
§ 8. Case of a root of unity^1) Suppose that sp=\ and that equation F.23) has
a solution
F.44) V(x) = x+f, bnxn
in a neighbourhood of the origin. Setting x(x)=(p(f[i//(x)]), where cp(x) = i//~1(x)
is analytic in a neighbourhood of the origin, we have
F.45) /(*)
But it follows from F.23) that z(x)=sx, whence
F.46) f(x)=spx = x.
Thus F.45) and F.46) show that q>(fp[y/(x)]) is the identity function, i.e. if we re-
replace x by <p{x), <p(fp{xj)=(p{x) follows, which implies F.41).
On the other hand, assume that F.41) holds. Then the general solution of equa-
equation F.1) in a neighbourhood of the origin may be written as
<6.47) p(x) = Vs-'0[/'(x)],
i = 0
where д is an arbitrary function defined in a neighbourhood of the origin. In fact,
F.47) evidently satisfies F.1) and, on the other hand, if <p(x) is a solution of F.1),
then F.47) holds with g(x)=p~1 q>(x).
Taking all functions g (x) which are analytic in a neighbourhood of the origin,
we obtain from F.47) all analytic solutions of the Schroder equation (in a neighbour-
neighbourhood of the origin); moreover, if g'@)=p~1, then y/(x)=<p~1(x) exists and has de-
development F.44), yielding an analytic solution of the Poincare equation F.23) in a
neighbourhood of the origin.
We may note that in the present case there is no uniqueness of analytic solu-
solutions of equation F.23), contrary to the so called irrational case (where s is not a
root of unity, i.e. the argument of s is an irrational multiple of 2л), where the ana-
analytic solutions, if they exist at all, form a family containing only one parameter,
viz. y/@)-
Rausenberger [1], Leau [2], Tschebotarev [1], Pastides [1], Siegel [7], Muckenhoupt [1].

9. Divergence case 149
§ 9. Divergence case. We shall prove a little less than was stated in theorem 6.11,
namely that for every s (not a root of unity), such that
F.48) |s"-l|<(«!)-2
holds for infinitely many n, there exists a series F.17) for which the corresponding
series F.44) diverges. Also, we shall omit the proof that the set of s fulfilling F.28)
for infinitely many n has the power of continuum and is dense on the unit circle
(cf. e.g. Siegel [7]).
If we write F.23) in the form
Ц/ (sx) - si// (x) =f О (л:)] - si// 00,
then we obtain from F.17) and F.44)
F.49) ? (sn-s)M"= ? «„[УМ]",
л = 2 n-2
whence it follows that every coefficient bn can be calculated as
Now we assume а„= ±(л!)~1, where the sign is so chosen that
F.50) |&j>i|s»_s|-i = l|s»-i_i|-*, „ = 2,3,...
Series F.17) with an chosen in such a manner obviously converges on the whole
complex plane, since \an\ = l/nl for л=2,3, ... But series F.44) diverges, since in
view of F.50) and F.48) bn>(n-l)\/n for infinitely many n.
§ 10. Convergence case. In this section we shall present a proof of theorem 6.13.
If we put s=e2n '", a € <0, 1), then condition F.43) is equivalent to the existence of
positive numbers X, \i such that
F.51) |ла-иг|>1л"д
holds for all positive integers m, n. At first we shall show that the set of such a has
measure 1.
Let B{X, ft) be the set of those а € <0, 1) for which the inequality
F.52)
has at least one integral solution m, n. So B{X, /i) is a union of at most denumer-
00
ably many intervals. Thus the set B= f~]B(k~1,2) is Lebesgue measurable and
evidently
6.53) BczB(X,2) for every A>0.

150 CHAPTER VI. Schroder's equation
For any m, n fulfilling F.51) we have
F.54) -1<яг<л + 1,
and the interval of a's fulfilling F.51) has the length 21л~д~\ If we fix an л, then
the number of m fulfilling F.54) does not exceed л+ 21+1, whence the measure of
2?(A, 2) is estimated from above by
E 2А(л+2А + 1)л~3<4А(А + 1) f л~2=— А(А+1).
л=1 л=1 3
From F.53) it follows that В has measure zero.
The set A of those a for which F.51) holds, with suitable A>0, /x>0, for all
positive integers m, n, contains the complementary of В and therefore has measure 1.
Let us put
F.55) sn=|s"-ir, « = 1,2,...
Since s is not a root of unity, we have %<sn< + oo. To every positive integer л we
choose an integer m such that — \<m — m<\. Then we obtain
Is" — ll = \e2mix — 11 = \emia — e~mia\
= 2 |sin лла| = 2 sin (n |ла—wj|)^4 |ла —wj| ,
whence by F.52)
s<li _ .-i !„„
sn-"~ V m\ <4ХП '
Choosing a constant ^^^ such that l/41<2<i we arrive at the estimation
F.56) sn<BnY, /1 = 1,2,...
The proof of theorem 6.13 will be based on some lemmas.
Lemma 6.2. Let xt,yj be positive integers such that
F.57) E
>=i
F-58) lyj>~, yj^
l=i 2 2
wherep^O (*). Then we have with t=p + q
0 0
Here and in the sequel E =0. П=1-

10. Convergence case 151
Proof. If n^2t—2, F.59) is trivial; thus we assume that n>2t—2. Note that
F.58) implies q^2, i.e. t^p+2^2, and if л is odd, even t^l and л>2г-2>4,
i.e. л>5.
We write k= [л/2] (where [a] denotes the greatest integer not exceeding a) and
w=p+(y1 + ...+yq). Then
F.60)
Furthermore, we may write
F.61) Х! + ...+хр=л
F.62) 1^хг, l^yj^k for i =
P
We keep w fixed and estimate the products x=]^[x;, y=Y\ У] from below.
x as a function of the variables xt attains its minimum in the set described by F.61),
F.62) for all xt equal to 1 except for one equal to (л — w+p)—p+1= n — w+1.
Thus, if p^ 1, we have
F.63) x^n-w + l,
which is obviously valid also for p=0 (for then w=n and x= 1).
If w — t+l<;k, then у (as a function of y}) similarly has its minimum in the set
described by F.61), F.62) for all yj equal to 1 except one equal to (w—p)—q+l
F.64) y^w-t+1 if w-t+l^k.
In the other case the minimum of у is attained with all yj=\ except two of them
equal tow — t—к+2 and k, respectively. Thus
F.65) y&k(w-t-k+2) if w-t+l>k.
F.63), F.64) and F.65) yield the inequalities
2 ((л-w+lHw-f + lJ if w^k+t-l,
F.66) xy >\(n_w+lKw_t_k+2yk2 tf w>k+t_u
In order to obtain an estimation of the right-hand side of F.66) we consider the
function P{z) = (z-a)p(b-zf', p>0, a>0, a<b. Since
d2
--^logP(z)>0 ta[a,b),
dz
we have for arbitrary a<z1<z2<b
),P(z,)) for ze<z,,z2y.
Taking novtP(z) = (n-z+l)(z-t+1J and z1=k+l,z2=k + t-l, we have by F.60)

152 CHAPTER VI. Schroder's equation
in the first case of F.66) t-l<k+l<;W^k + t—l<n + l, whence
(n-w + lXw-t+lJ7zmin[(n-k)(k-t+2J,(n-k-t+2)k2].
But, since 0^t—2^k—l, we have
(n-k-t+2)k2-(n-k)(k-t+2J = (t-2)[Bn-3k)k-(n-k)(t-2)~]
>(/-2)[B«-3fc)fc-0i-fc)(fc-l)]
whence
(n-w + l)(w-t+lJ^(n-k)(k-t+2J.
Similarly, taking P(z)={n—z + l) (z—t—k + 2J and z1=k+t-\, z2=n, we have in
the second case of F.66) t+k-2<t+k-\4:w4:n<n + l. Here P(z1)=n-k-t+2,
P(z2)=(n-k-t+2J>P{z^), whence
by what has already been shown. Thus in both cases we get
F.67) xy2>{n2
where we have put t—1 =u. Further, we have by F.60), l^u^k, whenceu{k+1 — u)
~$>k and (k + l — i^^ku'1. Consequently, if л is even, n=2k, and we get
If n is odd, я = 2&+1 and then f>3, i.e. м^2, and n ^5, as has been remarked at
the beginning of the proof. In this case
2 22 u~2>{k+\)k2 u~2
-i^Vi-±Y>r^Vi-lV "v '"
\2u) \ n) \2u) \ 5/ \2u) \2t-2
Thus in both cases we have obtained the inequality
which together with F.67) gives the desired relation F.59). i
Lemma 6.3. Let 0<mr< ... <m0 be integers. Then
F-68) П *mi<Lr+1 К П И-i-
П
are defined by F.55), L=225+1, дли? <5 w йе constant occurring in F.56).

10. Convergence case 153
Proof. For r=0 F.68) holds by F.56). We assume that it is true for an r — 1 ^ 0.
For arbitrary integersp>q>0 we have
min(sp,s4)
whence by F.56)
F.69) min(sp, sq)^2sp_q<2d+1(p-q)s.
Let sm, /=0, ..., r, attain its minimum for i=h. Defining m_1 = + 00, mr+1:
— oo, we have by F.69)
F.70) smh < 2e+ * min [(/ил_ t - игА)г, (/ил - mA+ J5] .
Now we must distinguish the cases 0<h<r, h~0 and h = r.
I. 0<h<r. In virtue of the induction hypothesis
and F.68) follows from F.70) and the inequality min(a, b)^2ab/(a+b) applied to
a=mh-1 — mh, b = mh-mh+1.
II. A = 0 resp. A=r. Then by the induction hypothesis we obtain
resp.
F.68) now follows directly from F.70) and, in the former case, from the inequality
mjmo<l.m
Lemma 6.4. Let the sequence dn be defined by the recurrence
F.71) di = l, dn = sn_! max (dkl,dk2,...,dkl), n=2,3,...,
max being taken over all decompositions n = kt + ... + kt, where 0<&;<...<&! are
integers and 1^2. Then
F.72)
Лг=8г1,=25г+1.
Proof. Let us write con=Nn~1n~2S. Then, for arbitrary m, n, we have
i.e.
F.73)

154 CHAPTER VI. SchrSder's equation
The proof of F.72) will be by induction. For л = 1 we have d1 = l = co1, and
so F.72) holds true. Let us assume that it is true for 1, ... , n — 1, and we shall prove
it for n.
By definition F.71) there exist lt integers kllt ..., klh, O<A:1;1^...<A:11, such
that
F.74) dn = sn^dkll...dkl4,
F.75) fc11 + ...+fc1,I = ;i.
If it happens that k11>n/2^1, we continue the process, applying again F.71) to
dkii. (Note that in virtue of F.75) we must have klx^n/2 for a=2, ..., lt.) Then
there exist /2 integers k21, ..., k2h, 0<к212^...^к21, such that
If k21>n/2, we continue the process again. Since the numbers klltk21,... form a
decreasing sequence of positive integers, the process must end after a finite number
of steps, and we arrive at a decomposition
in which 0<A:p+li; +i^.-.^kp+lil^n/2, whereas kpl>n/2. Introducing the nota-
notation п-,=кц, /=1, ...,p, no=n, we may write the resulting representation of dn in
the form
F.76)
f
i=0
where Лг=<4„ ... d^ for j = l, ...,p, Ap+1=dkp+ll ... ^p+1|Ip+1- Here none of the
occurring indices kap exceeds л/2, and we have
In virtue of the induction hypothesis and relation F.73) we obtain the estimations
for i=l, ...,p, and, with the notation lp+1 = q, kp+1J =ys,
F.78) AD+1^ T\coy=Nn>-qT\ yJ2S.
Here yt + ...+yq=np>n/2, and yj^n/2 for y= 1, ..., g.
Inserting F.77) and F.78) into F.76) and making use of lemma 6.3 with /и,
=п:— 1 and r=/>, we obtain

10. Convergence case 155
dn<lF+1(n ft [n^-n^yfl JV"-'-'-1(«i-i-«,r2V'-' П
i=l 1=1 j=l
where we have set xl=nt^1—ni. The numbers xt, ys fulfil the conditions of lemma
6.2. Since, moreover, L"^' and г^-г^г', we obtain hence
Proof of theorem 6.13. Since series F.17) has a positive radius of con-
convergence, there must exist a positive constant a such that {a^^a"'1 holds for
л = 2,3,... Introducing new functions \j/(x)=ay/(a~1x) and f(x)=af(a~1x), we
obtain from F.23) y/(sx)=f[y/(x)], and so again an equation of form F.23). Here
f(x) has a development of form F.17), in which, however,
F.79) [<ф1, и = 2,3,...
Thus we may assume that the function f(x) fulfils condition F.79).
We consider the formal power series
F.80) У(х) =?"„*",
л=1
where the coefficients un are defined by
u1 = \, u^s^^Uk^-'Uk,, n=2,3,...,
and the summation is extended over all decompositions n=k1 + ...+kl, where
...^kt are integers and /^2. We show by induction that
F.81) \Ьп\фп\, и = 1,2,...
For л = 1 both terms in F.81) are equal to 1. Assuming that F.81) holds for 1, ..., л -1
and equating the coefficients of x" in F.49), we obtain
b^is-s)-1 ? a,bkl...bkl,
n = kt+...+ki
whence F.81) follows in virtue of F.79), of the relation \sn—s|~1 = sn_1 and of the
induction hypothesis.
Thus it suffices to prove the convergence of series F.80). For this purpose we
consider the function Ф(х) defined by the equation
6.82)
Writing
F.83)

156 CHAPTER VI. Schroder's equation
we obtain for the coefficients vn the recurrence formula
^ = 1, г;„= Yj Vkt-'-Vk,, и =
n=kt+ — + ki
By induction we now easily obtain the inequality
where the sequence dn is defined by F.71).
Relation F.82) implies that Ф(х)=^[1+х — A — 6х + х2I12], and consequenty
series F.83) converges for |x|< 3 — 2^/2. Thus there exists a positive constant A
such that |г;п|<Л" for л=1,2, ... By lemma 6.4 we get hence \un\^AnN"~1n~15,
which shows that series F.80) converges at least for \x\<(AN)~1.m
§ 11. Conjugacy problem. Let Ф be a function class whose members are all
invertible and such that each function belonging to Ф has also its inverse in Ф.
Definition. We say that functions f(x), g(x) defined in a neighbourhood of
x=0(x) are Ф-conjugate if there exists a function у/ е Ф such that
F.84) <7(х) = У~Ч/1>(*)])
holds in a neighbourhood of zero.
At first we shall deal with the conjugacy problem in the simplest case, where
/'@) and g'@) (exist and) are not equal to zero or one and x is a real variable. Let
Tr be the class of functions which are of class C, are strictly increasing in a neigh-
neighbourhood of the origin and satisfy/@)=0, /'@)^0.
Theorem 6.14. (Sternberg [1], Kuczma [20].) Suppose that /e Г, r>2,/'@)^l.
Then a function g e T is T-conjugate with f if and only if
F.85) /'(O)=0'(O).
Proof. We may confine ourselves to the case/'@)=s, 0<s<l, for otherwise
we replace / by its inverse. The necessity of condition F.85) is obvious. To prove
its sufficiency it is enough to show that f(x) is conjugate with sx. F.84) then be-
becomes
F.86) (ГЧ/йК*)])-».
which, after putting <p(x)=y/~1(x) may be written in form F.1). The theorem then
follows from theorem 6.1.И
Let us note that а у/ е Т fulfilling F.86) is unique up to a constant multiple in
the argument. For r=\ the theorem is false, since the Schroder equation need not
have a C1 solution if /e Iх.
If x is a complex variable, then instead of T we consider the class A of func-
functions/(x) which are analytic in a neighbourhood of the origin,/@)=0,/'@)^0.
@ The choice of zero is not essential. One could also consider global conjugacy. F.84) is
then postulated in a given set.

11. Conjugacy problem 157
Theorem 6.15. Suppose that f в A, f'@) = s and s fulfils condition F.18) or F.43).
Then a function g e A is A-conJugate with f if and only г/F.85) holds.
The proof of the above theorem differs from that of theorem 6.14 only in the
circumstance that this time we apply theorem 6.4 or 6.13.
Further, we are going to discuss Л-conjugacy in the case/'@)= 1С)- Instead of
00
functions, we shall deal with formal power series f(x)=x+ X fln*" which need not
have a positive radius of convergence. (Thus we consider formal A-conjugacy. Cf.
also Fine, Kostant [1].) We may operate with series as if they were functions, e.g.
substitute them into one another, differentiate them, etc. With every series
f(x) = x+amxm+..., m>2, атФ0,
we may associate a unique series of the form Lf(x)=amxm+... satisfying formally
the equation
F.87) L[/(x)] =/'(*) L(x).
(cf. also Chapter IX, § 6). First let us note the following lemmas.
Lemma 6.5. If f(x) = x + xm+bx2m~1 +..., then Lf(x) = xm+(b-%m)x2m~1 +...
This follows by comparing the terms in the series in F.87).
Lemma 6.6. The associated series of g(x) = y/~x (f[if/(x)]) is
F.88) L^ = Lj^-
/'OOO] v'00
Proof. We have g'(x)= L ,r -_v . Hence
? [g 00]
Lg [<? 00] = —,r .-, =/ Vw (x)] -^7jr-—-n- = g (x) Lg (x),
i.e. function F.88) satisfies the suitable equation. Also the first non-zero term of
Lg(x) is identical with the second non-zero term of g(x) and the statement follows
from the uniqueness of the series fulfilling these two conditions. ¦
Lemma 6.7. Given a formal power series f(x) = x + amxm+ ..., иг>2, атф0,
there exist a polynomial y/(x) = cxx + ...+cm-1xm~1, c^O, and a number b such
thaty/-1(f[y(x)])=x + xm+bx2m~1 + ...
Proof. The coefficients ct may be determined by comparing the coefficients of
x?,k = m, ...,2m-2, in f[y/(x)] = y/(x + xm + bx2'1 + ...).¦
Let us note that the number b occurring in the above lemma is unique, though
the polynomial y/(x) is not. This number assigned to a power series/(x) will be
denoted by b(f).
Concerning the case /'@)=0, cf. Kuczma [35].

158 CHAPTER VI. Schroder's equation
Theorem 6.16. (Muckenhoupt [1].) Two power series f(x) and g(x) with linear
coefficients equal to one are formally A-conjugate if and only if their second non-zero
terms are of the same order m and b(f) = b(g).
Proof. To prove the sufficiency it is enough to show that
00
f(x) = x + xm + bx2m~1+ ? CnXn and
F.89) B=2m
g(x) = x+xm+bx2m~1+ ? dnx"
n = 2m
oo
are formally Л-conjugate. Let <p(x)=x+ ? е„У, where en are to be deter-
determined. Comparing the coefficients in
F.90) Р[/ОО] = 0|>ОО]
allows us to calculate successively en's.
Conversely, let us suppose that f(x) = x + xm+bx2m~1 +... and д(х)=х+х* +
00
+ cx2k~1 + ... are formally Л-conjugate and let <p(x)= ? dn x" be a formal power
л=1
series with dt ^0 such that F.90) holds. If k<m, then comparing the coefficients of
xk in F.90) we obtain dk=dk+d™, which is impossible. A similar contradiction is
obtained by assuming m<k. Therefore m=k. The comparison of the coefficients of
xl for m<i<2m — l in F.90) then gives d,-=0 for 2</<иг—1. Finally, comparing the
coefficients of x21"'1, we obtain d1b+mdm+d2m-1 = d2m-1+mdm+d1c, whence
b=c.u
Since two functions which are ^-conjugate are also formally ^-conjugate, the
conditions contained in theorem 6.16 are necessary for the Л-conjugacy of func-
functions. However, they are not sufficient. Let f(x)\ be the function for which Lf(x}
= |x2-^x3. By lemmas 6.5 and 6.6/(x) = x+|x2 + |x3 + ... But, though in virtue
of theorem 6.16 the series/(x) and e(x)=ex— I = x + |x2+|x3 + ... are formally A-
conjugate, the two functions are not conjugate. Supposing the contrary, let у/{х}
be a function from the class A such that y/~1(f[v(x)])=e(x). It will be proved in
Chapter IX, § 6, that there exists a function q> e A such that q>2(x) =f(x). The func-
function oc(x) = y/~1(<p[y/(x)]) then satisfies oc2(x)=e(x) and is in class A, which con-
contradicts theorem 15.13.
However, we shall prove the following
Theorem 6.17. (Muckenhoupt [1].) ///, geA withf'@) = g'@)=l are formally
A-conjugate and their associated series *Lf(x) and Lg(x) have positive radii of con-
convergence, then f(x) and д (х) are actually A-conjugate.
Proof. In view of lemma 6.7 and theorem 6.16 we may assume that/(x) and
g(x) are of form F.89). By theorem 6.16 there exists a formal power series y/{x)-
such that F.84) holds and by lemma 6.6 y/(x) satisfies formally equation F.88)..
Thus it is enough to prove that F.88) has an analytic solution y/(x).

11. Conjugacy problem 159
Let us write y/(x) = x+yxm, y=y(x). Equation F.88) then reduces to
( j dx xmLg(x)
Since by lemma 6.5 both Lf(x) and Lg(x) have the form хт+ф-±т)х2т~1 +..., the
expression on the right-hand side of F.91) is analytic in a neighbourhood of x = 0,
y=0, and the existence of the desired function y(x) (and hence also y/(x)) results
from Cauchy's existence theorem for differential equations. ¦
However, in the light of the recent investigations of G. Szekeres [4] and I. N.
Baker [6] (cf. also Chapter IX, § 6) the family of analytic functions of the form
f(x) = x +... for which the associated series Lf(x) has a positive radius of convergence
is not very large.
Finally, we quote a theorem which allows one to reduce the problem of ^-con-
^-conjugacy for functions/(x) with/'@) being a root of unity to the same problem for
functions f(x) with /'@) = 1. For the proof the reader is referred to Mucken-
houpt [1].
Theorem 6.18. Letf(x) and g(x) be in class A and letf'@) = g'@) = s, where s is
a primitive p-th root of unity. Then f and g are A-conjugate if and only iffp(x) and
gp(x) are A-conjugate.
§ 12. Exponential and logarithmic functions. (x) The logarithmic functions on
<1, сю) may be defined as the only non-trivial functions X eM'[(l, сю)] satisfying
for x e < 1, сю) the functional equation
F.92) l(x2)=2X(x),
or equivalently
F.93) A(Vx)=iA(x).
On account of theorem 6.8 and lemma 6.1 with g(x) = x — 1 there exists a unique
family of such functions
x2'" — l
F.94) l{x)=n lim -i=n
a -1
(a e A, сю) fixed). These functions are continuous and strictly monotonic in <1, сю),
satisfy A(l) = 0 and by F.92) tend to infinity as x does so. Actually, it is readily
seen from F.92), since functions F.94) are strictly monotonic, that
lim JL'=0
х-юо Л
for every <x>0. Further, we have for u>v> 1
u2--v2-" 2.„
= lim ^ = lim v = 1 ,
(J) 1
lim^lim
A(UJV) „-.со (UJV) -1 „-.oo
(i) Kuczma [19], [34]. Cf. also Oeconomou [1], Kuczma [37].

160 CHAPTER VI. Schroder's equation
whence for u/v=x, v=y (x, у e A, сю))
F.95) X{xy) = X{x)+X(y),
which is the fundamental property of logarithms.
Since family F.94) is independent of the choice of a e A, сю), replacing a by
another number b e A, сю) we obtain the same family. This means that for every
b e A, сю) there exists an ц ф 0 such that
F.96) hm = ц hm 2_„ .
п->аоЬ —1 л->оэЯ 1
On the other hand, to every ^>0 there exists a b e A, сю) such that F.96) holds.
This b can be calculated from
F.97) ti= Hm ar^r~. ,
л-юо" —I
since the function lim (x2~" — l)/(a2" — 1) is continuous and ranges from 0 to + сю on
n-»oo
<1, сю). If rj<0, we may find a b<\ fulfilling F.97) in virtue of
- = — lim
л-*оо
i—,i J.J.J.J.J. i — л
а2 -1 „^ооа2 -1
Consequently, in F.94) we may keep ц fixed (e.g. r] = l) and take a as the parameter
ranging over @, l)u(l, сю). The function
x2'"-\
F.98) la(x)= lim -^г-т
n-»oo a 1
is then called the logarithm with the base a. It evidently fulfils the condition Xa(a) = l.
Combining F.96) and F.97) we get according to F.98)
F.99) kb{x)=kb(d)ka{x),
i.e. the formula for a change of the base.
Formula F.98) defines the function la{x) for хе@, сю) and relation F.95) holds
for x, у е @, сю), but we cannot claim that F.98) is the general convex solution of
equation F.92) in @, сю). Though we may apply again theorem 6.8 and lemma 6.1
with g(x)=x — 1 in order to obtain F.98) as the general convex solution of equa-
equation F.92) in @, 1>, but the two families can be combined in various ways. In fact,
a X e M'[@, сю)] satisfying F.92) must be of the form
Па(х) for xe@,l),
[lb(x) for xe<l,oo),
where either b>a>\ or l>a>A>0.

12. Exponential and logarithmic functions 161
However, applying theorems 6.2 and 6.3 with /=(—1, сю) and lemma 6.1 with
g (x) = x — 1 we obtain the following result:
Lemma 6.8. For every пфО there exists a unique function X(x) of class Сх[@, сю)],
satisfying equation F.92) in @, сю) and fulfilling the condition l'(l) = r], namely
F.100) X(x)=rj\im.2\x2'"-\),
Since formulae F.94) and F.100) both define the principal solution of the Schro-
Schroder equation F.93), the two families must be identical. Consequently we have the
following
Theorem 6.19. The logarithmic functions F.98) (x) are the only non-trivial C1
solutions of equation F.92) in @, сю).
In fact by theorem 6.1 functions F.98) are of class C00 in @, сю).
Exactly one of functions F.100) fulfils A'(l) = l> namely
F.101) A(x)= lim 2"(x2""-l) .
n->oo
F.101) is called the natural logarithm. Its base e can be calculated from the relation
A(e)=l. Put dn=2n(e2~"-l)-l. Then 4,->0 as л->-сю. Further, we have
i 2" / 1 \2" / ^r \2"+l
F.102) -,-.„„, , - . 2,
From the estimation A +e/m)m^3e for e>0 it follows that the last two factors on
the right-hand side of F.102) tend to 1 as л->оо. Consequently, e is the limit of
A +2~"J", which is a subsequence of the increasing sequence A + l/и)". Hence
e= lim I 1-|
Differentiating F.95) for l(x)= A (x) with respect to у we obtain
xA'(xy) = A'(y),
whence, on setting y = \,
A'(x)=-.
x
In view of F.99), where we replace b by e, we obtain generally
1
aV ' A(a)x'
whence, since Aa(l)=0,
F.103) Xa(x) =
A(a)J t
l
0) This time considered for x e @, oo).
11 Functional equations

162 CHAPTER VI. Schroder's equation
From F.103) all the properties of the logarithmic functions can be derived. In
particular, it follows from F.103) that Xa(x) are analytic in @, oo). The same conclu-
conclusion could be derived directly from theorem 6.4 (which yields analyticity only in a
neighbourhood of 1) and formula F.95).
The exponential functions satisfy the equation
F.104) coBx) = 0(x)]2.
By theorem 3.4, a>(x) = Q is the only continuous solution of F.104) fulfilling co@) = 0.
1 is the other possible value of co@), and it is easily seen from F.104) that any con-
continuous solution (o(x) such that co@)= 1 must be strictly positive. For, if there were
an x0 such that co(xo) = 0, then by F.104) coB~"xo) = 0, which on passing to the
limit asn->oo yields co@) = 0. Consequently, <p(x) = log co(x) is a solution of
F.105)
having the same regularity as co(x). Hence in virtue of theorems 6.2 and 6.3 we
obtain the following result.
Theorem 6.20. The exponential functions are the only non-constant C1 solutions
of equation F.104) in (— oo,oo). Specifically, co(x) = ex is the only C1 solution of
F.104) fulfilling the condition co@) = co'@) = 1.
Similarly, exponential functions on <0, oo) or ( — oo,0> may be described as
the only logarithmically convex solutions of F.104) on the corresponding interval.
The logarithmically convex solutions of F.104) on (— oo,oo) are composed of two
branches of exponential functions.
However, these results do not allow us to obtain all the properties of the ex-
exponential functions from F.104), as it was possible in the case of logarithms. The
passage from <p{x), a solution of F.105), to a>(x), a solution of F.104), requires a
knowledge of the exponential function and its properties (*).
(*) Equation F.104) together with co(—x)=[co(x)]~1 and the condition co(x)>l+ax for
xe(—со, oo) completely characterize the exponential function co(x) = eax. This set of postulate
allows one to obtain all the properties of the exponential functions (Kuczma [37]).

CHAPTER VII
ABEL'S EQUATION
§ 1. General. Abel's functional equation (x)
G.1) a[/(x)] = a
сап be reduced to equation F.1) by putting (p(x) = sa(x)lc. It follows from theorem
1.8 that equation G.1) cannot have a solution in a domain containing a fixed point ?
of the function/(x). If/(x) is regular in a neighbourhood of i; and \/'{&\ф 1, then
Abel's equation has a unique solution (up to an additive constant) which is regular
in the vicinity of i; and has a logarithmic singularity at ?. This is the function a(x)
= (c log <p(xj)/logs, where <p(x) is the regular solution of equation F.1) (cf. Chapter
VI, § 3) (Koenigs [4].).
In the present chapter we shall consider equation G.1) in the real domain B).
In most cases we shall assume that fe R\[I\ (? ф Г) and lim/'(x)= 1, which is just
the case not covered in our considerations of the Schroder equation in the real
domain. The results of this chapter are mostly due to G. Szekeres [1], [2], [3].
At first let us note the following
Lemma 7.1. Letf(x) be defined in a submodulus set E and let g{x) be a one-to-one
mapping of E onto a set Et and define the function h(x) by F.2). If fi(x) is a number-
valued solution of the equation
G.2) P[h(x)-] = P(x) + c
in Elt then the function a(x) = fl[g(x)] satisfies equation G.1) in E.
The above lemma allows us to transfer a finite fixed point ? to the origin; one
chooses g(x)=x — ?,. Therefore in the sequel we shall often assume that fe /?J[/]>
since this does not diminish the generality of the considerations.
A) Abel [1], Formenti [1], Korkine [1], Koenigs [3], [4], Bottcher [4], [10], Pincherle [3],
[4], Reichenbacher [1], [2], van Uven [1], Bennet [3], [5], Fatou [5], [15], Julia [8], [12], [18], Tambs
Lyche [1], [2], [3], Valiron [2], P. Levy [3], [4], Picard [10], Myrberg [4], Ward, Fuller [1], Montel
[4], [7], Lundmark [1], Wolff [11], Bodewadt [1], Topfer [2], Bajraktarevic [4], [5], [6], de Bruijn [31,
Szekeres [1], [2], [3], Ghermunescu [22], Barvinek [1], Neuman [1], McKiernan [3], Kuczma [22],
[24], [30], [31], Coifman [1], [3], [4], A. Smajdor [2], Seneta [1].
B) For a treatment of similar problems in the complex case cf. Szerekes [1].
n*

164 CHAPTER VII. Abel's equation
If f'(x) exists and is different from zero in a submodulus interval / and a{x)
is a differentiable solution of equation G.1) in /, then (p{x) = a'{x) satisfies in /the
equation
G-3) P [/(*)]=^ ?>(*)•
Conversely, a solution of G.3) usually gives rise to a family (with an additive para-
parameter) of solutions of equation G.1J. Namely, we have the following
Lemma 7.2. Suppose that f'{x) exists and is different from zero in a submodulus
interval I {with respect to f{x)) and let (p(x) be an integrable solution of equation G.3)
fix)
in I such that J q>(t) dt^O in I. Then there exists a solution a(x) of equation G.1)
X
in I, unique up to an additive constant and such that a'(x) is proportional to <p(x).
f(xi)
Proof. Let *! e/be chosen so that y= J <p(t) dt^O, and put
xi
X
G.4) a(x) = - I <p(t)dt, xel,
У J
xo
where xo=/(xi)- Then
f(x) x x
«[/(*)]=- I v(t)dt=- [ ?[/(«)]/'(«) du=- [<p{u
1 ] У J У J
x xo
= — <p{u)du+ (p{u)du =a(x) + c,
i.e. function G.4) satisfies equation G.1). Moreover, a'(x)= — <p{x) and it is readily
с У
seen that the coefficient of the proportionality must be —. Together with a(x),
У
all the functions a{x)+r\ have this property, but no others.¦
The above lemma shows that there is a one-to-one correspondence between
solutions of equation G.3) (determined up to a multiplicative constant) and solutions
of equation G.1) (determined up to an additive constant). Therefore, instead of
dealing directly with equation G.1), we shall often prove the uniqueness or the
existence of solutions (fulfilling some additional assumptions) of equation G.3).
Let f(x) be defined in a submodulus interval / and suppose that a sequence dn
fulfils the condition
G.5) lim = c for xel.
n-*<x> а„

1. General 165
If the limit
G.6) . a 00 = hm ,
where x0 is an arbitrarily fixed point from the interval /, exists for all x e I, then
the function a(x) satisfies equation G.1) in /. In fact, we have by G.6) and G.5)
r,, ,, ,. /п+1О0
« [/(*)] = lim
Г(х)-Г(х0) f+\x)-f\x)
= hm h hm = a(x)+c .
Function G.6) is then called the principal solution of equation G.1) in /(Szekeres [1]).
If another sequence a"n also fulfils G.5), then necessarily lim Di/^n) = 1, and
,. f\x)-f\x0) dn f\x)-f\xo)_ f"(x)-f"(x0)
hm = hm — = lim
»-»» «n «->«> an an в-»» «„
This shows that the principal solution is independent of the choice of a sequence dn
fulfilling G.5). If instead of x0 we take another x* e /, we shall have
,. J KX)~J Kx ) ,. J Kx) J lxoJ ,. J Kx )—J V^oJ
hm = hm hm
Thus, changing x0 into x* we obtain a solution which differs from G.6) by a constant.
Consequently, the principal solution of equation G.1) is defined up to an additive
constant, as it should be.
Lemma 7.3. (*) Let f(x) belong to Щ[Ц and fulfil
G.7) lim/'00 = 1.
x->(
Then for arbitrary x, у elwe have
Proof. Put
G.9) wn(x)=f"(x)-x,
The functions wn(x) are of class C1 in / and by G.7)
G.10) Iimwi00 = 0.
Szekeres [2]. Cf. also Kuczma, A. Smajdor [1].

166 CHAPTER VII. Abel's equation
First we show that for every n we have
G.11) lim — = n.
For n = 1, G.11) is trivial. Supposing that it is true for an n ^ 1, we have
G.12) Wn+1(x)=f"+\x)-x=f(x + Wn(x))-x = Wn(x) + Wi
= wn(x) + wt(x) + wn(x) w[(x + 9(x) wn(x)) ,
Now, if ?, is finite, then it follows directly from G.9) that lim [x+6(x)wn(x)] = ?.
If ?, is infinite, then in view of G.10) and G.11) (assumed for n as true) we have
lim (wn(x)jx) = 0, whence
l+0(x)-^- =?
x J
all the same. Hence we get by G.12), G.10) and G.11)
,. wn+1(x)
hm
i.e. G.11) is valid for all positive integers n.
Now we take arbitrary x0, yoe I and put xn=f(x0), yn=fn(y0)- Formula G.8)
is symmetric with respect to x, y, and so we may assume that x0 is closer to ?, than
y0. Then there exists an integer k^O such that x0 lies between yk and yk+1. Then xn
lies between yk+n and yk+n+i and
l(yk+n) w[(zn),
where zn lies between yk+n and yk+n+1 and the absolute value of
„ _хп~Ук + п _ хп~Ук + п
Щ(Ук+п) Ук+п+1-Ук+п
does not exceed 1. Hence lim —-—-— = 1 in view of G.10). But we have
,. Щ(Ук+Л ,. Ук+п+1-Ук+п ,. Уя + Пк+АУн)-У«-пк(уя)
lim = hm = lim — 1
OO „^oo Wl(yn) "-oo VV^yJ
х„+1-х„ Y/i(xn)
hm = hm =1 .
by G.11). Hence
Since x0, y0 have been arbitrary, this proves relation G.8) for all x, у е I.i
In virtue of the above lemma, if the limit

1. General 167
GЛЗ) a(x)=lim
exists for x e /, it is the principal solution of the equation
G.14)
(provided that/(x) fulfils the assumptions of lemma 7.3). Algorithm G.13) is due
to P. Levy [3] (cf. also Szekeres [2]).
§ 2. Asymptotic conditions at a finite fixed point. Now we are going to investigate
solutions of equation G.1) fulfilling certain asymptotic conditions at a finite fixed
point of f(x). According to the remarks in the preceding section (cf. in particular
lemma 7.1) we may assume the fixed point to be at zero.
Theorem 7.1. Suppose that fe Rq[I], f (x)^0 in I, and lim f'(x) = l. For a given
constant/u>0 there may exist at most one solution (up to an additive constant) a e C1 [I]
of equation G.1) in I for which there exists a limit
G.15) limx"+V(x)^0.
x->0
Proof. If a(x) is a solution of G.1) fulfilling G.15), then
satisfies the equation
Г f(x)Y+
G-16) Ц^«
and may be regarded as continuous at zero. By theorem 2.2 equation G.16) may
possess at most one solution (up to a multiplicative constant) y/(x) continuous at
zero and such that ^@)^0. Hence also a'(x) is unique up to a multiplicative constant
and the theorem results from lemma 7.2. ¦
Theorem 7.2. (Szekeres [1].) Suppose that feR\\I\, /=<0,c),/'(x)^0 inland
there exist positive constants а, ц, М and 8 such that for sufficiently small x el
G.17) \f'(x)-l + a(^ + l)xfl\^Mxfl+d .
Then there exists a solution a(x), unique up to an additive constant, of equation G.14)
in I fulfilling condition G.15). This solution is of class C1 in I, is strictly decreasing
in I, is given by the Levy algorithm G.13) and there exists a constant К such that the
inequality
G.18) |а'(х)-а-1х-"-1|^Юс-'1-1+*
holds for sufficiently small x e I provided that 8<ц.
Proof. Let us fix an arbitrary xoel and put xn=fn(x0), а„ = (х„-х„+1)/х?+1,
^n=x~"-x~_ for и>1, 6о=Хо "• Relation G.17) implies that

168 CHAPTER VII. Abel's equation
for sufficiently small x e I, where Mt is a constant. Hence lim (xn+jx^) = l and
n~*ao
lim an=a. Moreover,
П-* CO
Xn~Xn+l, Xn+1 xn~Xn+l , (Xn+l\ L~(.Xn+llXn)
0O l I
I ГГТ
n n+1 \ Xn ) L \xn
which shows that lim Ь„=/ш. Thus
n-> со
1 1 "
G.20) Urn — x~"= lim — ? Ьг=/ш .
n-»oo И л->оо И ; = о
Write в(х)=№)Г+1/(^+1/'D By G.17) and G.19) (Д*)/х)"+1 = 1-
" + r1(x) and /'(х) = 1-(аЛ-1) c^+r2(x), where |г1(х)|<С1хд+а and
x^+* for sufficiently small xe/. Consequently
where
holds for sufficiently small x el. Hence we get for x e <0, xo>, where x0 e I is arbi-
arbitrarily fixed, and for n sufficiently large
By G.20) there is a constant B>0 such that xa^Bn~lltl, whence
This proves that the infinite product
G-21) щ(х)=
л =
absolutely and uniformly converges in <0, xo> for every x0 e /and therefore defines a
solution of equation G.16) which is continuous in /and fulfils the condition lim y/(x)
= 1. Hence it follows that ^(x)>0 for sufficiently small x el. Then it follows from
G.16) that y/{x) is positive in the whole of /.
The function <p(x) = x~fl~1\f/(x) satisfies equation G.3) in / and is positive in /.
Therefore J q>(t) dt^O for x e I and by lemma 7.2 the function
X
G.22) a(x) = y~
(where x0 e I is arbitrarily fixed and у is a constant suitably chosen) is a C1 solution
of equation G.14) in / fulfilling condition G.15). It follows from theorem 7.1 that
G.22) is the only solution (up to an additive constant) of equation G.14) fulfilling
the above conditions.

2. Asymptotic conditions at a finite fixed point 169
Formula G.21) gives
"пя/'м] ~nx)
G-23)
and the convergence is uniform in every compact interval contained in /. Let us
fix an integer fc>0. For x e<xt, xo> we have х„+к^/"(х)<х„, whence by G.20)
uniformly in-<X/t, xo>. Since x0 may be arbitrary and xk lies arbitrarily close to 0
(provided к has been chosen large enough), the convergence in G.24) is uniform
in every compact interval contained in /. Consequently, we may replace G.23) by
ax
G.25) p(x)=lim—
where dn = (a/m)~1~111', and the convergence is uniform in every compact interval
contained in /. Thus we obtain from G.22) and G.25)
G.26) <*(*)=у hm
dn
i.e. a(x) is the principal solution of equation G.14). In order to determine у note
that G.26) must be identical with G.13), i.e. we must have Hm -^ - = y . But
n-»co dn
in view of G.20)
,. Xn+l~Xn ,. anxn
lim = — hm -l-i/ = ~a »
whence y = —a. This shows in view of G.22) that a(x) is strictly decreasing since
<p(x) is positive in /.
It remains to prove relation G.18). We start from the inequality Kt \u\ s% [log A + и) \
^K2\u\ valid for |u| sufficiently small, where K^ and K2 are positive constants.
Hence
00
n=0 n=0
for sufficiently small x el. Here we have put K3 = CK^ 1K2¦ On the other hand,
we have for x=xm,

170 CHAPTER VII. Abel's equation
? * = K3 ? (xn+my+*=K3 ?
n = 0 n = 0 n=m
with a suitable constant 7sf4 > 0. In view of G.20) there is a positive constant В such
that xl^Bn'6'11 holds for all n provided that x0 has been chosen small enough.
Consequently, there exists a constant K5 >0 such that
Now let us take an arbitrary x^x0. There exists anm^O such that
We obtain from the three preceding inequalities
n=0
Since xn+1/xn tends to 1, the last expression is bounded. In other words, x~d\y/(x)
forx<x0. Now, a'(x) = 7" VW= -a" VW= -fl"^1" VW' Hence
which is equivalent to G.18).и
If we assume that f(x) is analytic, we obtain a stronger result. It is more con-
convenient to assume this time that/(x)>x. Then we may apply the previous results
to f~i(x). Note that if a function a(x) satisfies the equation <x[f~1(x)]—<x(x) = \,
then a(x)= — a(x) satisfies equation G.14), and conversely.
Theorem 7.3. Let f(x) be analytic for x^O and suppose that f(x)>x, f'(x)>0
for x>0 and
G.27) f(x)=x + ax2 + ..., a>0.
Then the principal solution a(x) of the equation
G.28) a[/-'(x)] = «(x)-l
is analytic for x>0 and fulfils the conditions
G.29) limx^V^xM-iy^fcla-1 for fc=l,2,...
For the proof of the above theorem the reader is referred to the original works
of G. Szekeres [1], [3]. Note that the existence of the principal solution of equation
G.28) and relation G.29) for fc=l results from theorem 7.2. The function a(x)
evidently satisfies equation G.14) in @, oo).

3. Convex solutions 171
§ 3. Convex solutions. Another condition that allows us to choose a single family
(with an additive parameter) of solutions of equation G.1) is that of the convexity
of a(x).
Theorem 7.4. (Kuczma [31], [3].) If feR\[T], ?,ф1,/'(х)фО in I and fulfils
conditional Л), then equation G.1) may possess at most one solution (up to an additive
constant) belonging to М1[Г\. If such a solution exists, it is of class С1[Г\.
Proof. Let aeM'[/] be a solution of G.1), and denote by <pi(x) and <p2(x)
the left and the right derivative of a(x), respectively. Functions <pi{x) and <p2(x)
are defined and monotonic in the interior of /, are equal except for an at most denu-
merable set, and both satisfy equation G.3). On account of theorem 5.4 and in view
of G.7) we must have
where x0 e / is an arbitrarily fixed point. Since px and q>2 coincide almost everywhere,
3»! = i>2 and <р1(х)ш<р2(х) is determined up to a multiplicative constant. Thus a e Cl[I\
and by lemma 7.2 is unique up to an additive constant. ¦
Theorem 7.5.OLet feR\[I\riMl[I\,t$I> and suppose thatf'(x)jtO in I and
fulfils condition G.7). Then equation G.14) has a solution a e М1[Г\ unique up to an
additive constant. This solution is of class C1 [/] and is given by the Levy algorithm
G.13).
Proof. Let ?, be the left endpoint of / and suppose that e.g. fe Ml[J]. Then
feMl[I] for every и^1, and the limit G.13), if it exists, belongs to M+[I]. To
prove the existence of G.13) let us note that the difference quotient is a
У-Уо
decreasing function of y. Hence we get for x e </(x0), x0) (setting yo—f(xo) and
f+ 2(*o) ~/"+ 1fa)>/" +1 (*) -/"+ W
/"+1(*о)-/"(хо) " Л*)-/"(*о) '
whence
f"(x)-f"(x0) ^Г+\х)~Г
This proves the convergence of G.13) for x e </(x0), x0). Convergence for other
values of x results from the identity
fix) -f(x0) fn+\x) -f(x)
G-30) Г+\х0)-Г(х0) /"+1(x0)-/"(x0)+f+1(x0)-f(x0)
and from lemma 7.3.
C1) Kuczma [31], Coifman [4]. The theorem remains true if /e ЩЩпМ1^, except that
then a need not be of class C1 [I].

172 CHAPTER VII. Abel's equation
Consequently, <x(x) given by G.13) is a convex solution of equation G.14).
Its uniqueness and its class of regularity follow from theorem 7.4.¦
Theorem 7.6. (Szekeres [3].) Let f(x) be analytic for x^O and suppose that
f(x)>x, /'(x)>0, G.27) holds and/-1(x) is completely monotonic. Then the principal
solution of equation G.14) is also completely monotonic.
Proof. Put h{x)=f~1{x), and let <x(x) be the principal solution of equation
G.28). Thus
G.31) a[A(x)]=a(x)-l
and by theorem 7.3 <x(x) fulfils condition G.29) for k-1, 2, ... Differentiating G.31)
we get
a'[A (*)]*'(*)=«'(*)•
Suppose that there is an xo>0 such that a'(xo)<0. Since h'(x)>0 for x>0, this
shows that a'[hn(xo)]<0 for every n. Now, since f(x)>x, we have h(x)<x and
h"(xo)->0. Thus the inequality obtained contradicts G.29) for к = 1.
Differentiating a [h(x)] к times (&>2) we obtain
G.32) ~ a{h{x))-a«\h{x))\h'{x)]k~ "% Ak(x)a«\h(x)),
ax i=i
where the coefficients Акг{х) are of the form
G.33) 4(*)
Supposing a certain product Y\ CJ-(A(j)(x))r-/ to be of a constant sign in @, oo), we
see that its derivative also has a constant, in fact the converse, sign. For, differentiating
this product, we obtain a sum of products differring from the original one in that
for a certain j the factor C0J\x))rj is replaced by Cj rj(h(f>(x))r'-1hiJ+1\x). But
CJ{h(i\x))r'-C3rJ{h(i\x)f-1h(i+1\x) = C2j rj(h(J\x)Jr'-1hu+1\x)<0,
since h{x) is completely monotonic.
For k=2 we have A21(x)=h"(x)<0 in @, oo), so sgn Л?(х)=(-1J+1. We
affirm that generally
G.34) sgn4O0=(-l)*+i for fc=2,3, ... and i = l,...,fc-l.
We have verified G.34) for k=2. Let us assume that it is valid for a &>2. We have
G.35)
Since A\(x)=h"(x), we deduce from G.33) and from what has been proved about
the products\[С}{па\х))г' that sgnD(x))' = -sgnAfrc). So sgn Ak+1(x) = (- l)k+i+1
results from G.35), G.34) and from the inequalities h'(x)>0, h"(x)<0 in @, oo).

3. Convex solutions 173
Now we are going to complete the proof that <x(x) is completely mono tonic.
We have already proved that a'O)>0 for x e @, oo). Assume that
) for i = l,...,fc-l.
Then by G.32), G.34) and G.31)
G.36) (-l)k+i{oc(k\x)-oc(k)(h(x))lh'(x)f}>O for xe@,oo).
Suppose that there exists an xo>0 such that (- l)k+1a,(k\xo)<0. Then by G.36)
(-l)k+1a(k)[h(xo)]<0 and by induction (-l)k+1aik)[h\xo)]<0 for и = 1,2, ...
Since A"(xo)-»0, this contradicts relation G.29). ¦
§ 4. A condition for the effectiveness of the Levy algorithm. Theorems 7.2 and 7.5
give some sufficient conditions for the existence of the principal solution of equation
G.14). Here we shall give another condition for the convergence of sequence G.13).
Theorem 7.7. (Szekeres [2].) Suppose that feR^I], lim/'(x) = l and f (x)-\
x->co
is of bounded variation in an interval <a, со) с I. Then formula G.13) defines a con-
continuous and strictly increasing solution of equation G.14) in I.
Proof. Let us fix an xo>a and write
We shall prove that sn(x,y) uniformly converges for xo<y</(xo),
We have
f"+1(y)-f4y)
with fn(y)<un<f\xHf+\y),fn(y)<vn<fn+\y). Fory^x^f(y)wehaveO<^< 1
and for ХоО^/Ы wy\y)]lw[fn+\y)] = llfXzn) <J\y)<zn<fn+\y)) tends
uniformly to 1. The series ? \w'(un) — w'(vn)\ uniformly converges, since w'(x)
is of bounded variation, and therefore the sequence sn(x, y) converges uniformly
in the domain considered.
Consequently, the function a(x) is defined by G.13) and is continuous in
<xo,/(xo)>. The convergence of G.13) for other x e I results from identity G.30)
and lemma 7.3.
The series ? |(я„+1Ау„)-1| = Х \(sn+1-sn)lsn\ converges for xo<y</(xo), y<x
)- Consequently, the same is true about the product Y[(sn+ilsn) and hence
,7W ,. fX*)fXy) x-y ™sn+1(x,y)
G.37) hm ——j = П 7^0 for хФу.
n-*co t

174 CHAPTER VII. Abel's equation
Now let us take хфу from <xo,/(xo)>. Then we have for function G.13)
j\x)-j\y)
a(x)-a(y)= lim
A*)-/"O0
by G.8) and G.37). Consequently, <x(x) is strictly monotonic (and in fact strictly
increasing for f\x) are increasing) in <xo,/(xo)>.
a(x) satisfies equation G.14) in / and is continuous and strictly increasing in
<xo,/(xo)>. Hence it follows that <x(x) is continuous and strictly increasing in the
whole of/(cf. theorem 2.1 and lemma 5.1).и
§ 5. Exponentially growing functions^1) If f{x) has exponential growth, the result
of the preceding section is not applicable. In fact, the principal solution of equation
G.1) need not then exist, and the results obtained so far give us no means of choosing a
solution behaving best at infinity. One of the reasons for this fact is that there is
no elementary asymptotics available for such functions, i.e. there is no elementary
function a(x) such that lim(a(x)/a(x))^0, oo.
X-* CO
We are in a particularly favourable situation in the case of the function e{x)
= ex— 1. The principal solution of the equation a[e(x)] = a(x) —1 is, by theorem
7.2, the best behaved solution of a [e (x)] = a (x) +1 near x = 0. Since e ~1 (x) = log A + x)
is completely monotonic, the same is true about <x(x) in virtue of theorem 7.6 and
so it may be regarded as the best behaved solution of <x[e(x)] = a(x) + l at infinityB).
In the sequel we shall denote this solution, normalized to vanish at x = l, by a(x).
Thus the function a(x) is characterized by the properties:
G.38) a[e(x)] = a(x)+l,
G.39) a(l) = 0, limxV(x) = 2.
x->0
a(x) is defined in @, oo) and by theorems 7.3 and 7.6 is analytic and completely
monotonic there. It tends to infinity more slowly than any finite iterate of log x
(and its inverse ax) grows more rapidly than any iterate of ex) and so it con-
constitutes a new standard order of infinity C).
Lemma 7.4. Suppose that lim (//(x)/x)=0. Then there exists a constant K>0
x-»co
such that
G-40)
a(x)
holds for x sufficiently large.
a'(x+rj(x))
-1
tj(x)
0) Szekeres [3]. Cf. also Reichenbacher [1], [2], P. L6vy [l]-[4], and also Kuczma [24].
B) It was P. Levy [3] who first suggested that the same solution might be best behaved at
zero and at infinity, but G. Szekeres [3] gave more substance to this conjecture.
C) Seven place tables of a(x) and a'(x) have been published by K. Morris and G. Szekeres
[1]. Q. also Reichenbacher [2].

5. Exponentially growing functions 175
Proof. Differentiating G.38) we obtain
whence, by division, we obtain for p(x)= —a"(x)la'(x)
By lemma 2.2 and theorem 2.8 lim p(x)=0; but we need a better estimation.
Choosing an arbitrary xo>0 and writing xn = en(x0) we obtain by induction
n-l n-1 n-1
/?(х„)=[ехр(- ? xi)]p(x0)+ E exp(- X x,).
>=0 j=0 i = j
Having fixed an interval /=<jc, e(x)> and writing M=sup p(x), we obtain hence
for xQeJ the estimation
xn
for n sufficiently large, say for n^-N. Then for x^-eN(x) we have 0<p(x)<2/x.
Now, we have
a'(x+ц (x))=a'(x)+г1(х)а"(х+в (x) r\ (x)\ 0 < в (x) < 1,
whence, if t](x)>0, we obtain for large x
G.41) a'(x+n (x))=a'(x) + в1(х)г1 (х) a"(x)
where O<0!(x)<l and O<02(x)<l, since a"(x) is negative and increasing and
0<p(x)<2/x; and if r](x)<0, we obtain for large x
G.42) a'(x+r, (x))=a'(x) + 63(x) r, (x) a"(x+ц (х))
= a'(x) *-L r, (x) a'(x+r, (x)) ,
x + t](x)
O<03(x)< 1, O<04(x)< 1. G.40) results from G.41) and G.42). ¦
For other functions f(x) with exponential growth we can choose a particular
solution <x(x) of equation G.14) by the requirement of the asymptotic comparability
of a'(x) and a'(x). Namely we have the following

176 CHAPTER VII. Abel's equation
Theorem 7.8. Suppose that fe Я^Ш, f(x)>x, f'(x)>0 in I. Then there may
exist at most one solution аеС1[Г\ {up to an additive constant), of equation G.1) in I
for which there exists a limit
G.43) lim-^
Proof. If a(x) is а С1 solution of equation G.1) in /fulfilling G.43), then <p{x)
= a'(x)fa'(x) is a continuous solution of the equation
?> [/(*)]=/ (*) ~^r ?M
and lim <р{х) = уФ0. By theorem 2.2 we must have
X-* CO
<p(x)=ya'(x) lim (o'[/"(x)]"n
n-»oo >=0
i.e.
G.44) a'(x) = y-i lim (a'[/"(x) )
п-»аэ ( = 0
and the uniqueness of a(x) results from lemma 7.2.¦
The following theorem shows that condition G.43) is effective for a large class
of exponentially growing functions.
Theorem 7.9. (Szekeres [3].) Suppose that fe Rlo[I],f'(x)>0 in I and we have
G.45) f(x) = e\x + w(x)),
where r is a positive integer and w(x) is bounded and for x sufficiently large
M
G.46) |W(jc)|<-
1 ' x
holds with a constant M>0. Then there exists a unique solution аеС1[Г\, up to an
additive constant, of equation G.14) in I fulfilling the condition
G.47) lim^=r.
This solution is given by
G.48) a (x)=- lim {a [/"(*)] - a [/"(x0)]},
Г п-*со
where x0 is an arbitrarily fixed point from I.
Proof. We have by G.38) а[ег(*)]=а@+/>, whence

5. Exponentially growing functions 177
and
a'OO
( v,())
Since w(x) is bounded, w(x)/x tends to zero, and we obtain hence by G.46) and
lemma 7.4
for у sufficiently large, say
Now denote by An{x) the expression under the lim sign in G.44), and let us
fix an arbitrary 0<x0 e /. Then we have for x e <x0, oo) according to G.49)
provided that n^-N, where ./V is chosen so that fN(x0)^y*. Hence we obtain for
xe <x0, oo) and
— I
A(x)
The sequence f(x0) tends to infinity very rapidly, so that the series
obviously converges. Consequently, sequence G.44) converges uniformly in <x0, oo)
and its limit is continuous there. Moreover, by G.49)
a'(x) "-1
tends to one as x->oo. Consequently relation G.43) follows from G.44) in view
of uniform convergence.
X
Now let us put a.(x) = §a{t)dt. Because of the uniform convergence of G.44)
we may integrate term by term, whence
G.50) a(x)=y-1lim J</a[/"(*)]
=г-Мт1{аО[/"(х)]-а[/"(хо)]}.
n-* аэ
The function a(x) satisfies equation G.1) with the constant
G.51) c=r-1lim +1
But by G.38) and G.45)
G.52) a(/[/"(xo)])-
Further, we have
lim [a(x + w(x))- а(х)] = lim w(x) а'(х + в(x) w(x))=0,
x-> аэ х-»аэ
12 Functional equations

178 CHAPTER VII. Abel's equation
since lim a'(x)=0 (*) and w(x) is bounded. Consequently, we obtain from G.51)
and G.52) c=r/y. Thus if we take y=r in G.50), then <x(x) will be a C1 solution of
equation G.14) fulfilling condition G.47), and G.50) reduces to G.48). The unique-
uniqueness follows from theorem 7.8.
The solution has so far been obtained for xe<x0, oo). It can be extended
onto the whole of / by means of theorem 4.1, and formula G.48) is valid in /. (Note
that G.48) is meaningful also for 0>xeI, since for n sufficiently large f(x)>0
and a[fn(x)] is denned.)¦
The above result can be improved further. In fact, the solution fulfilling G.47)
exists if f{x) = ek\h(e~k{x)y\ and h(x) fulfils the conditions of theorem 7.9. This
is a consequence of the following
Lemma 7.5. Let A e /??,[/] and fi(x) be a C1 solution of equation G.2) in I fulfilling
the condition
G.53) lim ±Q
x-
P (X)
and put f{x) = e\h (e~ 1{х)У\. Then equation G.1) has a C1 solution <x(x) in I, unique
up to an additive constant, fulfilling condition G.43).
Proof. By lemma 7.1 <x(x)=fi[e~l(x)] is a C1 solution of G.1); moreover,
<х'(х) = Р'[е-1(х)]1A+х). On the other hand, we have а'(х) = а'[е~1(х)]/(\+х) by
G.38). Hence
o'(jc) а'1е-\хУ]
lim = lim ——— = у
x^ooa(x) x^oo p\_e (x)J
according to G.53). Thus <x(x) fulfils G.43). The uniqueness follows from theo-
theorem 7.8. ¦
Hence the following result (due to G. Szekeres [3]) can be derived.
Theorem 7.10. For every L-functionf(x) of type (r, s, (i) withr>s there exists a C1
solution <x(x) of equation G.14), unique up to an additive constant, defined for x suffi-
sufficiently large and fulfilling the condition
.. o'(x)
hm
rs.
a (x)
We shall give only an outline of a proof of theorem 7.10. Condition @.34)
implies the relations
This follows e.g. from the fact that a'(x) is monotonic and satisfies a'[e(x)]=e~xa'(x).

5. Exponentially growing functions 179
Put h(x)=e~s~2[f(es+2(x)y]. Then it follows from the above relation that h(x)
= er~s(x+w(x)), where w(x) is bounded. But w(x) = e~r~2[f(es+2(x))']-x belongs
to H, and so does w'(x). Consequently, w'(x) is monotonic for large x, and so it
fulfils condition G.46). Now the existence of a(x) results from theorem 7.9 and
lemma 7.5 applied s+2 times, and the uniqueness of <x(x) is a consequence of the-
theorem 7.8. ¦

CHAPTER VIII
ANALYTIC SOLUTIONS
§ 1. Special homogeneous equation. In the present chapter we shall present some
results concerning analytic solutions of equations of form @.49) for complex-valued
functions of a complex variable. (Thus throughout this chapter x will denote a complex
variable.) Some of the simplest cases (the Schroder equation, the conjugacy equa-
equation) were already dealt with in Chapter VI. However, we cannot develop the theory
too far, since deeper results require more advanced methods (while we wish to
keep this book as elementary as possible) and proofs are often too long to be present-
presented here (!).
As a preparation before discussing the general case we shall deal with some
linear equations. The adjective "special" in the section heading refers to the fact
that we are concerned with the case/(x)=,yx. Thus we are studying th; equation B)
(8.1) 9(sx)=g(x)<p(x).
Equation (8.1), just as the more general one
(8.2) ?[/(*)]=0
C1) Concerning these and furthsr equations of form @.49) in thj complex case cf. Cayley
[1], [3], Muhll [1], Moret-Blanc [1], Koenigs [3], [4], [5], [6], Rausenberger [4], [5], Appell [5],
Holder [2], Bertrand [1], Furle [1], Wirtinger [1], МгШп [3], Nielse.i [1], [2], Moore [2], Lemeray
[2], [5], Lerch [4], Bottcher [4], [6], Hardy [1], Tietze [1], Fatoa [1], [4], [5], [11], [15], Stridsberg
{1], Feyer [1], Browne [1], Carmichael [2], Valiron [1], [4], Hvialkovskii [1], Mason [1], Pincherle
[10], Nalli [1], [2], Julia [4], [5], [8], [9], [12], [14], [15], [17], [18], Weddeburn [1], Ritt [11], [12],
[13], [15], [16], Spiess [4], Andrsoli [3], Raclis [1], Picard [10], Badescu [l]-[10], Gsppert [1], Myr-
berg [4], [7], [11], [13], [14], Levin [1], J. M. Whittaker [2], Popovici [7], [8], [10], Sheffer [2],
Bernstein [1], Touchard [1], Birkhoff [6], Wolff [11], Tdpfer [1], [3], Bouzitat [1], Nehari [1],
Pastid6s [1], [3], Wittich [1], Read [1], Hsrve [2], Tanaka [1], Ganapathy Iyer [3], [5], [6], J. Moser
[1], Vaida [1], Valcovici, Vaida [1], Baker [3], [9], W. E. Williams [1], Dsacev [1], Schneider [1],
Leonov [1], Korobeinik [1], Pranger [1], Alblas [1], A. Renyi, С Reiyi [1], Gross [2], [3], Kuczma
[35], [36], A. Smajdor, W. Smajdor [1], W. Smajdor [1], [3], Baker, Gross [11, Matkowski [1].
Here belong also investigations of the Riemann boundary-valus problem with displacements,
which usually involve the use of singular integral equations. In this connection cf. Haseman [1 ],
Carleman [1], Buharinov [1], Vekua [1]-[12], Kveselava [1], Cakvetadze [1], Aleksandrija [l]-[4],
Hvedelidze [1], Mandzavidze, Hvedelidze [1], Litvincuk [l]-[5], Litvincuk, Hasabov [l]-[5], Mel-
nik [1], Bedoeva [1], Berkovic [1], Zverovic-Litvincuk [1], [2], Hasabov [1], Pyhteev [1], Zverovic
[1], [2], [3], Sabitov [1], Dracinskii [1], Hou [1], Zarkowski [l]-[5].
B) Ganapathy Iyer [3], Schweizer [1], Vaida [1], Valcovici, Vaida [1], Myrberg [13].

1. Special homogeneous equation 181
always has the trivial solution (p(x)=0, but this is often the only regular solution
of (8.2). Namely, we have the following
Lemma 8.1. Let f and g be analytic in a neighbourhood ofx=0, /@)=0, and let
(р{х)фО be a solution of equation (8.2), analytic or having a pole at zero. Then g(Q)
= [f'@)]k, where к is an integer.
Proof. If <p{x) has at x=0 a zero of order & or a pole of order — k, then y/{x)
= x~k(p{x) is regular at zero, i//@)^0, and it satisfies the equation y/ [f(x)]=д^(x)^(x),
where gl(x) = (f(x)/x)~kg(x) is regular at x = 0. (If k>0 and/has at x=0 a zero
of order m, then it follows from (8.2) that g(x) must have at x=0 a zero of
order k(m— 1).) On setting x=0 we obtain gi@) = l, since ^@)^0, whence
the lemma follows. ¦
Lemma 8.2. Let f and g be analytic in a set E, submodulus for the function f(x),
and suppose that for every xeE lim/"(x) = 0 (i.e. EczAf@)). If equation (8.2) has
n-> со
a solution (р(х)фО regular in E except possibly at zero, where it may have a pole,
then g(x)^0 in E.
Proof. Suppose that д(хо) = 0, x0 e E. Then ?>[/"(*(>)] = 0 for n = \,2, ..., and
consequently <p(x) = Q in E, contrary to the assumptions
Theorem 8.1. Let 0< \s\< 1, and let g(x) be analytic in a domain E submodulus
with respect to f(x)=sx, g@)=sk(k an integer), g(x)^0 in E. Then equation (8.1)
has a unique solution, up to a multiplicative constant, regular or having a pole at zero.
This solution is analytic in E— {0} and is given by the formula
sk
(8.3) ^^П ,„.
n=o д (s x)
where r\ is an arbitrary constant.
Proof. Let (р(х)фО be a solution of equation (8.1) having the required prop-
properties. We may assume that <p(x) is analytic in E, (p(Q) = t]^O, for otherwise we
replace <p(x) by x~k<p(x) and g(x) by s~kg(x), as in the proof of lemma 8.1. Then
in view of (8.1) and lemma 8.2
whence (8.3) follows on letting т-юэ. (We have k=0 in this case.) On the other
hand, the function sk/g(x) is analytic in E and equals 1 at x=0. Consequently, the
product in (8.3) uniformly converges in every closed disc contained in E, and thus
(8.3) defines an analytic function in E, with a pole at x = 0 if k<0. It can easily be
verified that this function satisfies equation (8.1).и
The case where \s\>l may be reduced to the preceding one by writing equa-
equation (8.1) in the form p(s~1x) = [g(x)]~ iq>(x). But it may also be handled directly,
and the assumption д{х)Ф0 is found to be superfluous. The solution obtained in
this case has the form
(8.4) 9{x) = rixkX[s-kg{S-nx).

182 CHAPTER VIII. Analytic solutions
Let us note that if the function g(x) is entire and k^O, functions (8.3) and
(8.4) are also entire (Ganapathy Iyer [3]).
Now suppose that s is a primitive pth root of unity. Then we obtain from (8.1)
g(x) g(sx)... g(sp~1x) <p(x) = <p(x), whence, if>(x)#0,
(8.5) g{x)g{sx)...g{sp-lx) = \,
which implies that g cannot have zeros. Writing g(x) = exp h(x), where h{x) is an
analytic function, we have
h(x)+h{sx) + ...+h(sp~1x)=2mni,
where m is a suitably chosen integer. If
(8.6) *(*)=?*„*",
л=0
it follows that а„=0 for n being a multiple of p. The function
(8-7) ?(*)=? ^*">
n= 1 S — 1
where the coefficients must be taken equal to 0 if и is a multiple of p, is regular
in a neighbourhood of zero (series (8.7) has the same radius of convergence as (8.6))
and satisfies the equation
Consequently, the function po(x)=x'exp y(x), where j is chosen according to the
relation
(8.8) flf(O)=s'
(it follows from (8.5) that g@) is a pth root of unity), is a regular solution of (8.1).
Now, we can prove
Theorem 8.2. (Ganapathy Iyer [3].) Let s be a primitive p-th root of unity and
let g (x) be analytic in a domain E submodulus for the function f(x) = sx. Then equa-
equation (8.1) has solutions (p{x) regular or having a pole at x = 0 if and only if condition
(8.5) is fulfilled. When this is the case, the solution is given by
(8.9) (P{x)=xiW{xp)txpy{x),
where y(x) is defined by (8.7) and j by (8.8) and V(x) is an arbitrary function analytic
in E except for a possible pole at zero.
Proof. It remains to prove formula (8.9). Let (p{x) Ьг a solution of (8.1) with
the required property. Then q>(x)/(po(x) is an automorphic function for/(x)=5ix.
хх1х2 means that x: =skx2, к an integer, i.e. either л^ =х2=0 or xjx2 is a pth root
of unity. This is equivalent to the condition x\=x. Consequently, by theorem

1. Special homogeneous equation 183
1.7, <p(x)/<po(x)=4'(xp), where !f(x) is a function defined on E. Since (po(x) is ana-
analytic and without zeros in E (except for the origin), W(x) must have the same regu-
regularity properties as (p(x).m
Here again, if g(x) is an entire function, so are functions (8.9).
Finally, if \s\ = 1 but s is not a root of unity (irrational case), we may proceed
similarly and we arrive at series (8.7). But now the convergence of series (8.6) does
not imply the convergence of (8.7). Some necessary and sufficient conditions in
order that series (8.7) have a positive radius of convergence (which is equivalent
to the existence of a regular solution of equation (8.1) in a neighbourhood of the
origin) have been established by D. Vaida [1] with the aid of the theory of chain
fractions. But even if series (8.7) converges, its radius of convergence may be smaller
than that of series (8.6). In particular, if the function g(x) is entire, then series (8.7)
may still have a finite radius of convergence (cf. Ganapathy Iyer [3]).
§ 2. Special inhomogeneuos equation. In considering the special inhomogeneous
equation
(8.10) q,(sx)=g(x)q,(x)+F(x),
we shall confine ourselves to the case 0<|s|<l (*). We assume that g(x) and F(x)
are analytic at zero:
л=0 л=0
and we shall seek a solution (p{x) of (8.10) regular at zero:
<p{x)= ? cnxn.
n=0
Inserting these expansions into (8.10) and equating the coefficients, we obtain the
infinite system of equations
(8.12) sncn=aocB+a1cn^1 + ... + anco + bn, n=0,l,2,...,
from which cn can be calculated successively provided that either
(8.13) sVa0 for n=0,l,2,...,
or for a certain integer к >0
(8.14) sk = a0 and д1с4_1 + .
where c0, ..., ck_l have been determined from the first к equations (8.12). Thus
conditions (8.13) or (8.14) are necessary for the existence of a regular solution <p(x)
of equation (8.10). We shall prove that they are also sufficient.
The case |s| = l is dealt with in A. Smajdor, W. Smajdor [1].

184 CHAPTER VIII. Analytic solutions
Theorem 8.3. (Myrberg [13].) Let 0<\s\< I and let the functions g(x) and F(x)
be analytic (with expansions (8.11) at zero) in a domain E submodulus with respect
to f(x)=sx, g(x)jt=Q in E. Suppose further that condition (8.13) or (8.14) is fulfilled.
Then equation (8.10) has a solution <p(x) analytic in E. This solution is given by
(8.15)
m-l
where P(x)= ? ct x\ ct being calculated from (8.12), m is chosen so that
(8.16) ' ° |S|m<|ao| = |0(O)|,
and
(8.17) <po(x) = - ? F\snx) (П 9 (s'*))~ *,
л=0 }=0
(8.18) F*(x) = F(x) + g (x) P(x)-P (sx).
Solution (8.15) is unique in case (8.13) and contains an arbitrary parameter (*) in
case (8.14).
Proof. First we observe that if <po(x) is a regular solution of the equation
(8.19) (po(sx)=g(x)(po(x)+F\x),
then (p(x) given by (8.15) is a regular solution of equation (8.10). Thus it is enough
to prove that (8.17) actually defines a regular solution of equation (8.19).
00
By (8.18), F*(x) is analytic in E. So we have F*(x) = Y, dnx" in a neighbourhood
n = 0
of the origin, and it follows from (8.18) and (8.12) that do = ...=dm_t~O. The
function l/g(x) is analytic in E and takes the value uq1 at the origin. Let us fix
a finite closed disc К centred at the origin and contained in E. We may find positive
constants А, В such that
(8.20) |F*(x)|<B|x|m, I^Wp^lao1! for xeK.
Moreover, we shall have by (8.16)
(8.21) ^|ao||s|m<l
if the disc К is chosen sufficiently small.
Now let D be any finite closed region contained in E. There is an N such that
N
for n ^ N and x e D, s"x e K. Denoting by M the maximum over D of [ \\ g(s]x) \ ~ *,
we have by (8.20) for x e D and
j=0
| |S| )
(') In case (8.14) the coefficient ck is quite arbitrary. This parameter enters also into the for-
formulae for all cn With n>k.

2. Special inhomogeneous equation 185
with a suitable constant C, which proves in view of (8.21) that series (8.17) uni-
uniformly converges in D. Consequently, q>0(x) is analytic in E. It is easily verified that
expression (8.17) satisfies equation (8.19).
The uniqueness statement follows from the uniqueness of the solution of system
(8.12). It might as well be obtained from the results of the preceding section.¦
Closing this section we shall show how the case of an arbitrary / may be reduced
to that off{x)=sx. Let F{x, y, z) be an arbitrary function of three complex variables
and consider equation @.49). We shall prove the following
Theorem 8.4. Let f(x) be an analytic function in a neighbourhood of a fixed point
? and suppose that the Poincare equation
(8.22) уф-VH/tXx)],
where s~l =/'(?), has in a neighbourhood of the origin a regular solution w(x) such
that v^(O) = f, ^'(°)^°- Then the functional equations @.49) and
(8.23) F(^(sx),?(sx),p(x))=0,
considered in a neighbourhood of ? and of the origin, respectively, are equivalent,
their solutions being related by
?'W = 9[v(*)] . 9(x) = q>\_y/~\x)~] .
Proof. Since ^'@)^0, y/(x) has an inverse y/~l(x) regular in a neighbourhood
of ? and satisfying the Schroder equation
(8.24) v-l\_f{x)-\ = S-\-\x).
Let <p(x) be a regular solution of @.49). Replacing in @.49) x by y/{sx) we obtain
according to (8.22) F(y/(sx),(p[y/(sx)], <p[v(x)]) = 0, i.e. the function (/>(x) = (p[y/(x)]
is a regular solution of equation (8.23). On the other hand, if <p(x) is a regular solu-
solution of equation (8.23), then replacing x by s~ly/~l{x) and taking into account
(8.24), we obtain
i.e. the function q>(x) = (p[y/~1(x)] is a regular solution of equation @.49).¦
§ 3. Entire solutions of the homogeneous equation. We shall keep the reader's
attention fixed on the problem of entire solutions of equation (8.2). It was dealt
with by V. Ganapathy Iyer [2], [3], [5], [6]. The case where f(x) is a first degree
polynomial may be reduced either to f(x) = x+b (then (8.2) becomes a difference
equation; in this connection cf. Ganapathy Iyer [2], Guichard [1], Hurwitz [3],
[4], Picard [10], J. M. Whittaker [1], [3], [4], [5], [6], Scott [1], Naftalevic [1], [4], [8],
[10], Rajagopal, Reddy [1]) or to f(x)=sx (this is the case considered in § 1). Now
we shall piove the following
Theorem 8.5. (Ganapathy Iyer [5].) Suppose that (8.2) holds for entire functions
f,g,(p.If<p and g are polynomials, then so isf. Ifg is a polynomial and f is not a first

186 CHAPTER VIII. Analytic solutions
degree polynomial, then <p is a polynomial and consequently f must also be a poly-
polynomial.
Proof. Let (p and д be polynomials and let xr, \xr\ = r, be the point at which
\f{xr)\=Mf{r). For any polynomial P{x) of degree n the ratio |P(x)|/|x"| tends
to a finite non-zero limit as |x|->oo. Hence \ip[f(xr)]\ = K\f(xr)\nCr = K[Mf(r)]"Cr,
where n is the degree of (p and Cr->1 as r-»oo. (We may assume that/ is not constant,
and hence Mf{r) tends to infinity with r.) Since (p and д are polynomials, we have
Mv(r) Mg(r)^Lrm, where m^n, whence we get the estimation K[Mf(r)]"Cr^Lrm.
This proves that / is a polynomial.
Now suppose that g is a polynomial and / is not a first degree polynomial.
Then we have for suitable constants K>0 and d>l Mf(r)~^Krd. By lemma 0.15
there exists a constant 0, О<0<1, such that тах^[/(х)]^Л/^(Л/Д0/-)), whence
\x\-r
in view of (8.2) M<p(K6drdL:Hr'>M<p{r\ where Я is a constant and n is the degree
of the polynomial g. Putting R=K9drd we get for suitable constants L and С
(8.25)
By a repeated use of (8.25) we obtain
where
wlogC
We choose A: depending on R so that log log R— 1 Ot<log log R. Then the above
relations lead to the estimation log M^Rtys^A log R (where A is a constant), which
proves that <p is a polynomial. ¦
Polynomial solutions of equation (8.2) in the case where f(x)=x2 and g(x)
is a polynomial have been studied by V. Ganapathy Iyer [5]. The case where /(x)
=x2 and g(x) is a regular function without zeros has been dealt with by V. Gana-
Ganapathy Iyer [6].
Lemma 8.3. If g{x) and <p{x) are regular in K={x: \x\<R), R^\,g{x) has no
zeros in K, and the relation
(8.26) (p{x2)
holds in K, then (p(x) has no zeros in K.
Proof. Let «(/•) be the number of zeros of <p(x) in {x: \x\<r}, 1 </•</?. If
xo^O is a zero of q>(x\ then ±^/х0 are zeros of <p(x2). If xo=O is a zero of order
к of (p{x\ then x0 is a zero of order 2k of (p{x2). Thus, since g{x)i=0, 2n(r)^n(r),
i.e. n(r)=0.m

3. Entire solutions of the homogeneous equation 187
Thus, putting p(x)=exp y(x), g(x) = exp h(x), we see that, for functions reg-
regular in a sufficiently large set, equation (8.26) may be replaced by the equation
(8.27) y(.x2)—y(x)=h(x),
00
where necessarily h@)=0. Let /г(х) = ?йглх"; we seek a solution in the form y(x)
00 П= 1
= YJcnxn. Inserting these series into (8.27) and equating the coefficients we obtain
л=0
c2n_1 = -a2n_1, сл-с2л=а2л. Hence
(8.28) cn=-aq-a2q-...-an,
where n=2pq, q odd; c0 may be arbitrary. This shows that the solution, if it exists,
is unique up to an additive constant.
00
Theorem 8.6. (Ganapathy Iyer [6].) Let h(x)= ? а„х" be regular in the disc
00 Л= 1
\x\ < R. Then y(x) = ? cnx", where c0 is arbitrary and for n ^ 1 cn are defined by (8.28),
is a solution of equation (8.27) regular in the disc |x|</" = min(l, R). This solution
is unique up to an additive constant.
Proof. In view of the preceding remarks it is enough to prove the convergence
of the series for y{x). We distinguish two cases.
I. R^l. Given ?>0, we can find M=M(e) such that \an\^MR\, where Rt
= R~1+b>1. Hence by (8.28) we get |с„|< M{p + \)R\, and since for a suitable
constant К we have p^Klogn, we see that limsup ^„l1'"^/?!. Since e has been
arbitrary, the radius of convergence of the series for y(x) is not less than R.
II. R> 1. Choose ?>0 so small that Rl = R~1 +s< 1 and then choose M=M{e)
so that \an\^MR\ for all n. Hence (8.28) gives \cn\^M{p + \)R\^M{p + \), whence
limsup [сл|1/л<1. Thus the radius of convergence of the series for y(x) is not less
Л-* 00
than l.B
Corollary. Ifg{x) is an entire function without zeros, g@) = 1, then equation (8.26)
has a regular solution <p(x), unique up to a multiplicative constant, which, however,
need not be entire. This solution is given by p(x) = exp y{x), where y(x) is the regular
solution of equation (8.27), exp h{x) = g (x), and is regular at least for \x\ < 1.
A more detailed study of the solutions of equation (8.26) in the case where
д (х) is an entire function of finite order having only a finite number of zeros is to
be found in V. Ganapathy Iyer [6].
§ 4. General equation. By a small modification of the proof of theorem 4.5 we can
establish the existence of local analytic solutions of equation A.2) in a neighbour-
neighbourhood of a ? ? (&i\f].
Let
(8-29) /(*) = ?+?Ьл(*-О"
л=1

188 CHAPTER VIII. Analytic solutions
be an analytic function in a neighbourhood of ?, series (8.29) being convergent
for \x—?\<<70- Moreover,
(8.30) 0< jfoij < 1.
Further, let
(8.31) h(x,y)= ? anm(x-Z)n(j-ri)m> aoo=n,
n,m = 0
be an analytic function in a neighbourhood of (f, tf) fulfilling A (?,//)=//. We as-
assume that series (8.31) converges for \x—i\<a0, \y—n\<x. We want to find solu-
solutions f{x) of A.2) in the form
(8.32) <p (x)=n + ? cn{x- f)" •
л=1
Inserting (8.29), (8.31) and (8.32) into A.2) and equating the coefficients we obtain
relations
(8.33) {\-b\aol)cn=Fn{Cl,...,cn^), и = 1,2,...,
where Fn is a polynomial in c1( ...,cn_1 with the coefiicients depending on ai}
and bk. We assume that either
(8.34) b\aol^\ for и = 1,2, ...,
or for a certain integer /^1
(8.35) blaol = l and F^Cj, ..., c1_1)=0,
where ct, ...jC^x have been determined from the first /—1 equations (8.33). (It
follows from (8.30) that the equality b[aol = l may hold at most for one value
of /.) Under these conditions there exists a formal solution of equation A.2) of
form (8.32): unique in case (8.34) and depending on one parameter in case (8.35).
We define functions Hk(x,y,y1, ...,yk) by relations D.1). For every positive
integer к the function Hk{x,y,ylt ...,yk) is an analytic function for \x—?\<a0,
\y—t]\<x and arbitrary (complex) ylt ...,yk. Moreover, relations D.5)-D.7) hold,
where Rk(x, y,ylf ...,yk_1) is an analytic function for \x—?\<a0, \y—rj\<x,
Ji.---.Jt-i arbitrary.
For any solution (actual or formal) of form (8.32) of equation A.2) relation
D.3) holds with any k. Hence it follows that the numbers
(8.36) f?i = ci, fh = 2!c2, .--, Пк=к\ск, ...
fulfil system D.21) for every positive integer r.
Theorem 8.7. (W. Smajdor [1].) Let fix) be an analytic function for \x—i\<a0
with expansion (8.29) fulfilling (8.30), and let h(x,y) with expansion (8.31) be ana-
analytic for \x — ?\<o0, \y — t]\<r. Further, let condition (8.34) or (8.35) be fulfilled,
and let (8.32) be a formal power series satisfying formally equation A.2). Then this
series converges in a neighbourhood of ? providing a local analytic solution of equation

4. General equation 189
A.2) in a neighbourhood of ?. The solution is unique in case (8.34) and contains a
parameter in case (8.35).
Proof. By (8.30) there exists an integer r^\ such that inequality D.23) holds.
We may find positive numbers S< 1, at, and d<x such that inequality D.42) holds
for |x—?|<ffi, \y—n\^d. Further, to arbitrary positive numbers Ml,...,Mr we
may find numbers L0,Llt ..., Lr_1,Lr = 9 such that the inequality
(8.37) \Hr(x, y', y\, ..., y'T)-HJ(x, у", У1,..., y'r')\
<L0\y'-y"\+J:Lk\y'k-yk'\
k = l
is fulfilled for arbitrary {x,y',y[, ...,y'r),{x,y",y'{, ...,y") belonging to the set
<8.38) Z = K(Z,ol)xK(ri,d)xK(.t}1,M1)x...xK(rir,Mr),
where K(xo,a) = {x: \x—xo|<a} is the closed disc with radius a around x0 and
rjk are given by (8.36). This is a consequence of the analyticity of the functions Ht
and of relations D.5)-D.7) and D.42).
We put
and we fix a positive constant К such that
1 — 9
|Яг(х,//, z/i, ...,nr)-Hr(?, i.tlt, ...,tjr)\^——K for
We may find a a2 >0 such that
<8.39) °^
for \x—?|<G<G2. We define numbers Mk occurring in (8.38) by D.48).
By (8.30) there is a number a3 such that
<8.40) |/(*)|<|*| for |дс|«т3.
Now we fix positive numbers a and 9 < 1 such that
(8.41)
and inequalities D.50), D.51) and D.52) hold.
We define s? as the space of functions <p(x) which are analytic for
and have expansions of the form
x—$\<a
? и„{х — |)",
their rth derivative <pM(x) is continuous in the closed disc |x—?|<<7 and

190 CHAPTER VIII. Analytic solutions
fulfils the condition
(8.42) \<pir\x)-rir\^K for
Thus we have, in particular, P ел/, which shows that s/^0. The space stf endowed
with the metric
(8.43) P|>i,?>2]= «up \<plp(x)-q>l?(x)\
|*-4|<»
is a complete metric space.
For an arbitrary function y{x) which is analytic in \x—?\<o, its derivative
yil\x) being continuous in \x — ? |<<7 and
where / is a positive integer, we have
(8.44) У(х)=——г
whence
a1
(8.45) sup |y(x)|<- sup
|*-{|<ff '!|X-{|<
Applying (8.45) for y(x) = <p?Xx)-<p<?)(x), where <р1,<р2ея/, and l=r-k,k^0,
we obtain according to (8.43)
(8.46) sup \<P<?\x)-<piZXx)\^——pi<Pl,<p2-\.
We consider transform C.21) for <p ел/. Ax first we shall show that this trans-
transform is defined in j/ and maps j/ into itself. For \x — $\^a and <p eл/ we have
\д>(х)-п\ф(х)-Р(х)\ + \Р(х)-п\,
whence, making use of (8.46), (8.43), (8.42) and (8.39), we have
И*)-Ч|<^т" «up \<pirXx)-n\+ sup \P(x)-n\<d.
Г1 \x-S\<a |*-«|<<r
Thus y/(x)=h(x, <p[f(x)]) is defined and analytic for \x—^|<c. The function
у/м(х)=Нг(х, <p[f(x)],..., #>(r)[/(x)]) is continuous for |x—?|<G on account of
the regularity of Hr, <p[f{x)], ..., 9>(r)[/(x)] and in virtue of relations (8.40) and
(8.41). The form of the expansion of y/(x) results from (8.29), (8.31) and from the
fact that numbers (8.36) fulfil system D.21). We must prove that
(8.47) \i/-r\x)-tiT\^K for

4. General equation 191
By (8.44) applied to
and l=r—k,0^k^r—l, we have
whence in view of (8.42)
(8.48) \<Pik\x)-\
t=l
Hence it follows in view of D.48), (8.40) and (8.42) that (x,<p[f(x)], ?>'[/(*)]>
• ••, Vir)[f(x)]) eZ> so we с&ъ make use of inequality (8.37). Thus
= \Hr(x, ч>[/(x)], ?'
<\Hr(x, <p[/(x)], ?'[
and (8.47) results in view of (8.48), (8.42) and D.51), just as in the proof of theo-
theorem 4.5.
Now, making use of (8.37), (8.46) and D.52) we show in the same way as in
the proof of theorem 4.5 that for arbitrary (pt, (p2 es& and y/1{x) = h(x, <Pi[f(x)]),
y/2(x) = h(x, q>2[f(x)]) we have
i.e. C.21) is a contraction map. The existence of a solution of form (8.32) now
follows from Banach's fixed point theorem. This proves also the convergence of
series (8.32) at least for \x—?\<o.
This solution can be obtained as limit C.16), where the sequence fn{x) is de-
defined by C.11) and as the first approximation we may take <ро(х) = Р(х).ш
§ 5. The Gamma function in a complex variable.. Let
/ 1 1
C=lim 1+—+ ... + logn
V 2 "
be the Euler constant. The product

192 CHAPTER VIII. Analytic solutions
(8.49) F(x) = eCxxf[ (l+^
i\ n)
n
converges on the whole complex plane. The function F(x) is thus an entire function
with simple zeros at x=0, — 1, — 2, ... We have, moreover, for positive real x
Г(х)
where Г(х) is the function defined in Chapter V, § 10 (cf. in particular formula
E.70)). Consequently, the meromorphic function l/.F(x) (with simple poles at
0, — 1, —2, ...) is an extension of the Gamma function for complex values of
the argument (').
This function can also be characterized by equation E.68) B). The characteriza-
characterization below is due to H. Schmidt [2].
In the sequel we shall freely use the notation x=u + iv.
Lemma 8.4. Let y/(x) be an entire function, periodic of period \, and suppose that
there exist positive constants M and fi such that
(8.50) ||
holds for \v\ sufficiently large and uo^u^uo + l. Then y/(x) is a trigonometric poly-
polynomial of degree at most fi.
+ 00 +00
Proof. Let (у(л)=^4еЫм=Х4^ where z=e2nix. Then by Cauchy's
— 00 — 00
inequalities
(8.51)
Now, \z\ = e~2nv, whence |u|-»oo for r tending to zero or to infinity. More exactly,
we have \v\=logrll2n for r>\ and \v\=logr~ll2n for r<\. Hence,in view of (8.51)
and (8.50), we obtain
(8.52) |Л„| ^Mr~*~n for r sufficiently small,
(8.53) \An\ ^Мг»~" for r sufficiently large.
Letting r-*0 in (8.52) and r->oo in (8.53) we obtain An=0 for |и
(') One of the striking properties of the Gamma function is that it does not satisfy any al-
algebraic differential equation (Holder [1], Ostrowski [1]). Other functions with similar properties,
defined by functional equations, were investigated by Holder [3], Moore [2], Barnes [4], Tietze [1],
Stridsberg [1], Bottcher [10], Carmichael [2J, Mason [1], Ritt [14], Levin [1], Wittich [1], Mesch-
kowski [1].
B) Equation E.68) and related equations have been dealt with in the complex domain by
Prym [1], Scheeffer [1], Mellin [l]-[4], Alekseevskil [1], [2], Jensen [1], Barnes [l]-[4], Gode-
froy [1], Jackson [1], Burkhardt [1], K. P. Williams [1], Post [1], Picard [10], Rowe [1], Rasch [1],
Bendersky [1], Dtvies [1], Touchard [1], H. Schmidt [2], Meschkowski [1], Knobloch [1],
Ghermanescu [22], Alblas [1], and others.

5. The Gamma function in a complex variable 193
Theorem 8.8. (H. Schmidt [2].) Let <p(x) be regular in a strip 0<u0^u^u0 + l,
and let it satisfy equation E.68) whenever it is defined and condition E.69). Further,
suppose that there exist positive constants L and a<\n such that
(8.54)
holds for \v\ sufficiently large and uo^u^uo + l. Then /р(х) = Г(х) (=1/F(x)).
Proof. Using relation E.68) we may continue cp{x) over the whole complex
plane introducing only simple poles at 0, — 1, —2, ... (*). Thus (p(x) may be regarded
as a meromorphic function with the poles just mentioned. Consequently, y/(x)
= F(x) cp{x) is an entire function and, since F{x) fulfils F{x+ \) = x~lF{x) B), y/(x)
is periodic of period 1. Further,
But
v2 \ sinh v
п
л=1
whence we get for v\ sufficiently large and
with an arbitrary e>0. This together with (8.54) gives
Choosing s<\n—a we obtain on account of lemma 8.4 F(x) q>(x)=const. By
E.69) the constant must be !.¦
§ 6. Riemann's Zeta function. The formula
(8.55) C(*)= Z «"*
л=1
defines the function ?(x) for Re x > 1. This function can be continued onto the whole
complex plane except x= 1, where it has a simple pole; thus it is a meromorphic func-
function. ?(x), known as Riemann's Zeta function, satisfies the functional equation C)
(!) For more details cf. Picard [10], pp. 50-52.
B) This may be seen as follows: F(x+\) F(x) is an entire function and vanishes for
real positive x. Hence it must vanish identically. *
C) The first satisfactory proof of (8.56) was given by B. Riemann 1859, but the equation
itself (called the Riemann equation) together with a motivation was already given by Euler 1768.
Other proofs of (8.56) may be found in Wirtinger [2], Landau [1], Hardy [3], Mordell [1], E. T.
Whittaker, Watson [1], Rademacher [1], Ingham [1], de Bruijn [1], Titchmarsh [1], Denjoy [2], [3].
13 Functional equations

194 CHAPTER VIII. Analytic solutions
(8.56) С A - x)=2 Bя) ~x cos j Г(Х) С (x),
where Г(х) is the function considered in the preceding section('). In view of rela-
relations A1.16) and A1.18), with x replaced by (l-x)/2, (8.56) is equivalent to
The function C(x) plays an important part in the number theory B).
Equation (8.56) is to a great extent characteristic for function (8.55). Namely,
we have the following
Theorem 8.9. C) Let <p(x) be analytic in every finite domain except at most a finite
number of poles and suppose that there exist positive constants r and ft such that
(8.57) И*)|<<?|х|" for |x|>r.
Further, let <p(x) be expansible for Re x> 1 in an absolutely convergent Dirichlet series
(8.58) <p(x)= f^ann~x,
and let the function y/(x) defined by
(8.59) Ц1 A - x) = 2 Bn)~x cos {\ nx) Г{х) <р (x)
be expansible in a Dirichlet series
(8.60) у/(х)=^Ь„п~х
л=1
absolutely convergent on a vertical line Re х = мо>0. Then ip(x) = i//(x) = cC(x), where
с is a constant.
(!) Concering (8.56) and related equations cf. Euler [2], Riemann [1], Hurwitz [1], Cahen
[1], Epstein [1], Hecke [l]-[10], Landau [2], [3], Beeger [1], Hamburger [l]-[6], Siegel [1], [2], [6],
Schnee [1], Hutchinson [1], [2], Mordell [2], [3], Mandelbrojt [1], [2], [3], F. K. Schmid [1],
Lichtenbaum [1], Hasse [1], Kober [1], [2], [3], Ferrar [1], Marke [1], [2], Deuring [1], Heilbronn
[1], Weissinger [1], Peterson [1], Speiser [1], Rankin [1], Taylor [1], Anfertieva [1], Schmidt, Teich-
muller [1], MaaB [l]-[4], Hardy [5], Bellman [1], Bochner [l]-[4], Koecher [1], Tamagawa [1], [2],
Leptin [1], Carlitz [2], Bochner, Chandrasekharan [1], Chandrasekharan, Mandelbrojt [1], [2], [3],
Kahane, Mandelbrojt [1], Chandrasekharan [1], Ehrenpreis, Mautner [1], Knobloch [1], Chandra-
Chandrasekharan, Narasimhan [l]-[8], Ghermanescu [22], Bronstein [l]-[3], Ayoub [1], Soni [1], Dwork [1].
B) A number of other functional equations also occur in the number theory. Cf. Lerch [1],
[2], [3], [5], Hansen [1], Appell [9], [10], [11], Bell [1], [2], [3], [5], Holder [4], Maier [1], [2], [3],
Rademacher [2], Mahler [1], Guinand [1], Min [1], Nehari [1], Siegel [4], [5], Bateman [1], W. Fischer
[1], Apostol [1], [2], Carlitz [1], Eichler [1], Berg [1], Bellman [3], [5], Iseki [1], [2], [3], Lambek,
L. Moser [1], Rieger [1], Maier, Krat el [1], Kratzel [1], [2], Froberg [1].
C) Hamburger [4]. The short proof given here is due to E. Hecke [4].

6. Riemann's Zeta function 195
Proof. In Mellin's formula
—¦ f
2ni J
1/2
for Rez>0 we shift the path of integration over the pole x=0 and, taking into
account the residuum 1, we obtain
(8.61) —. Г z~xr(x)dx = ez-l.
-3/4
Formula (8.61) remains valid also for Re z=0, z#0, since the integral is still abso-
absolutely convergent then. Putting in (8.61) z=se±in/2= ±is (s>0) we obtain
| s~xe±inxl2r{x) dx = 2ni{e+is-1),
-3/4.
whence, by taking the mean of the expression with + and with —,
(8.62) J s~x cos Qttx) Г(х) dx = 2tu(cos s- 1).
-3/4
Further,
f s~x cos апх)Г(х) f
KJ ' K dx= s~x cos Qnx)r(x-2) dx
J (x-l)(x-2) J ^ ) к )
5/4- 5/4
1 Г
= j-5 x cos (%пх)Г(х) dx
s J
-3/4
and according to (8.62)
Cs хсоьЦпх)Г(х) .cos 5-1
(8.63) dx=-2nt ^—
J (l)(*2) 2
J
5/4
Since series (8.60) absolutely converges for Rex=M0>0, we have for every t>0
(8.64) B(t)u У bn(t-n)=-— 4L^'dx=~ — -dx.
dx
2ni] x(x + l\ 2ni
u0 ' 1 -uo
Instead of integrating over Re x= 1 — u0, we may integrate over an arbitrary vertical
line Rex = M with 1 — uo^u^ provided that none of the (finitely many) poles of
the integrand lies on this line. Of course, the residua must be taken into account.
Let x=u+iv and |t>|->oo. By (8.60) y/{\ — x) is bounded on u=l — u0 and by (8.58)
a-fioo
For real a we denote by J the integral J over the line Rsx=a.
a a-i<x>
13*

196 CHAPTER VIII. Analytic solutions
and (8.59) |d|~3/V(x) 1S bounded on м=|. Hence by (8.57) in virtue of the
Phragmen-Lindelof theorem, we infer that |f|~3/4 y/(x) is uniformly bounded for
| and
1-uo 5/4.
where R(t) is the sum of the residua of the integrand in the strip 1— мо|
and so it is a finite sum of terms of the form ?°(log t)b. Hence R e C°° [@, oo)]. Now
we have by (8.64), (8.65), (8.59) and (8.58) after changing the succession of summa-
summation and integration
f
Bnnt)~* cos Цпх)Г(х)
5/4
whence, having applied (8.63) with *=
=-2 f
Bтгп)
Hence
-B(t) = R(t+l)-R(t) ,
and consequently 5(?+ l)-B(t) is of class C00 in @, oo). But for m = [t] (the great-
greatest integer not exceeding t)
? bn,
л=1
and thus we must have b1=b2 = -..=c, i.e. y/(x)=cC(x). The equality g>(x) = \f/(x)
results from (8.59) and (8.56).и

CHAPTER IX
ITERATION
§ 1. Iteration groups. Let/(x) be a function defined and invertible in a modulus
set?.
Definition (Cf. e.g. Michel [1]). A one parameter family of functions F(x,u),
x e E, ме(— со, со) A), F(x, и) eE, is called an iteration group of the function
(9.1) F(F(x,u),v) =
(9.2) F(x,l)=f(x)
hold for x e E, u, v e (— со, со).
It follows from (9.1) and (9.2) that for integral n we have F(x, n)=f\x). There-
Therefore the functions F{x, u) may be regarded as an extension of the notion of an
iterate of f(x) onto arbitrary real iterative exponents. In the sequel we shall fre-
frequently write fu(x) instead of F{x, u). Relation (9.2) then reduces to @.4) and (9.1)
appears as a generalization of @.5).
Such a definition of an iteration group, however, proves inadequate when E is a
submodulus set for/(x). Then negative iterates of/(x) are not defined in the whole
of E and the common part of the domains of definition of the integral iterates f\x)
may happen to be empty. This is the case e.g. for f{x)=ex, E=(—co, со), where
/^"(x)=log ... log x is defined for x>exp... exp 0. Therefore in such cases it seems
reasonable to replace the above definition of a global character by a local one.
Let E be a subset of a metric space (e.g. the space of real or complex numbers)
submodulus for an invertible function f{x). Further, let ? e ©i[/] be a fixed point
of f(x) lying in E or on the boundary of E.
Definition (Cf. A. Smajdor [2], Coifman [4].) A one-parameter family of
functions F(x, u) is called an iteration group of the function f(x) with respect to ?
if for every и e(-co, со) the function F(x, u) is defined in VunE, where Vu is a
neighbourhood of ?, which may depend on u, relation (9.1) is fulfilled for every u,
v e (— со, со) and x e Vuv n E, where Vuv is a neighbourhood of ?, which may again
depend on и and v, and relation (9.2) holds for xe E. An iteration group (local or
(i) Неге и is a (real) parameter. One may also consider functions F(x, u) with и ranging
over the set of complex numbers. We shall touch upon such problems in the last section.

198 CHAPTER IX. Iteration
global) is called continuous if F{x, u), regarded as a function of two variables, is
continuous with respect to each variable.
An iteration group with respect to ?, as defined above, is not a group under
the operation of the composition of functions in the strict sense. It will become a
group if we identify functions identical in a neighbourhood of ?.
Relation (9.1) is a functional equation in three variables, and thus not of the
kind dealt with in this book. It is fulfilled by functions of the form
(9.3) F(x,m)=«-1[«(x) + M],
or
(9.4) F(x,u)=v-l[sy(xj\,
where a(x), resp. (p{x), is an arbitrary invertible function defined on E and taking
real or complex values. On the other hand, it can be proved (*) that under some
reasonable conditions (9.3) or (9.4) forms the general solution of equation (9.1); in
particular, (9.3) or (9.4) is the general form of a continuous iteration group. The two
formulas are in fact equivalent: condition (9.4) may be obtained from (9.3) by setting
g>(x)=sxM.
Two different functions a^(x) and a.2{x) generate the same iteration group if and
only if <*! (x) = tx2 (x) + c. For, if
then we obtain for a(u)=xl [aJ* (и)], x=aJ* (v) the equation
which on setting v=0, <r@) = c, immediately yields the solution a(u) = u+c. Simil-
Similarly, two different functions (p±{x) and (p2{x) gen3rate the sams iteration group if
and only if q>i(x) = c<p2(x).
If we insert (9.3) or (9.4) into condition (9.2), we obtain a[f(x)]=<x(x)+ 1, or
p[f(x)]=s<p(x), respectively, i.e. the Abel or the Schroder equation. Thus the prob-
problem of constructing iteration groups of a function/(x) reduces to finding invertible
solutions of the corresponding Abel or Schroder equation. On the other hand, if
<x(x), resp. (p{x), is an invertible real or complex valued solution of the Abel, resp.
Schroder, equation in E, then (9.3), resp. (9.4), actually defines an iteration group
(global, or local with respect to a fixed point of/, depending on whether /maps E
onto or into itself) of the function f{x). In the case of a complex-valued function
a or (p, (9.3) and (9.4) allow one to define the iterates f(x)=F(x, u) even for com-
complex iterative exponents и B).
(') Cf. e.g. Ward, Fuller [1], Bajraktarevic [6], Aczel [5], [7], Aczel, Gol^b [1], and also
Hosszu [1], Coifman [4].
B) Formula (9.3) was already used by С G. J. Jacobi 1825 (cf. Biermann [1]) but only for
integral values of u. E. Schroder [2] seams to Ьз the first who attempted to extend the notion of
iterates onto non-integral iterative exponents. Cf. also Schroder [1], Formenti [1], Korkine [1],

1. Iteration groups 199
In the sequel we shall only deal with continuous iteration groups and we shall
omit the adjective "continuous".
The results of Chapter VI, §§ 3, 7 imply in particular the following
Theorem 9.1. Let f{x) be an analytic function in a neighbourhood o/x=0,/@)=0,
f'@) = s, and suppose that s fulfils condition F.18) or F.43). Then the function f{x)
has an iteration group with respect to 0 such that f(x) are analytic at the origin.
This group is unique and is given by formula (9.4), where (p(x) is an analytic solution
of the Schroder equation F.1).
After a suitable choice of the branches of the function s" formula (9.4) defines
the iterates f(x) even for complex и (*). .
The existence of analytic iterates f\x) in the cases not covered by the above
theorem is a very difficult problem, not yet completely solved. We shall discuss in
§ 6 the particularly interesting case s = 1.
Now we turn our attention to iteration groups of functions of a real variable.
Note that if/"(x) is an iteration group of a function/(x), then hu(x)=g(f[g~1(x)])
is an iteration group of the function h(x)=g(f[g~1(x)]). Since every finite fixed
point ? can be transformed to the origin by a linear transform g, we may confine
ourselves to the study of two cases: ?=0 and ?= со.
§ 2. Regular iteration. Let/(x) be a function of class R%[I] @ may belong to /
or not), and suppose that the limit
(9.5) lim =s
x-*0 X
exists. At first we consider the case where
(9.6) 0<5<l .
In view of the fact that the Schroder equation F.1) has in / a continuous and strictly
monotonic solution depending on an arbitrary function (theorems 2.10, 3.7, 5.7)
and that non-proportional solutions generate different iteration groups, the function
f{x) has an infinity of iteration groups with respect to zero. Now we want to in-
Farkas [1], Koenigs [5], Bottcher [l]-[4], [10], Bourlet [2], [3], Lemeray [1], Aristov [1], Spiess
[1], Pincherle [3], [4], Mattson [1], van Uven [1], Bennett [l]-[3], [5], Bouton [1], Fatou [13],
Handt.Kneser [1 ], Tchebotarev [l],Barba [l],Montel [4], [7], Lewis [2], vanKuik [l],Hadamard [2],
Silberstein [1], Topfer [2], Jabotinsky [2]-[5], Fine, Kostant [1], Lojasiewicz [2], Szekeres [1],
[4], de Bruijn [3], Sarsanov [l]-[4], Lewin [1], [2], Erdos, Jabotinsky [1], Sternberg [5], Baker [4],
[6], [11], Kuczma [13], [24], Th. E. Harris [1], Schubert [1], Aczel, Karteszi [1], Ran [1].
Concerning iterations of functions of a real variable cf. in particular Reichenbacher [1], [2],
P. Levy[l]-[4], Ward [1], Ward, Fuller [1], Bodewadt [1], Walker [1], Batty, Walker [1], Bradley
[1], [2], Batty [1], Bradley, Walker [1], Wright [1], Bajraktarevic [4], [5], [6], M. K. Fort [1], Sze-
Szekeres [l]-[3], Berg [3], Sternberg [5], Kuczma [18], [20], [31], [24], Michel [1], Lundberg [1],
Aczel [6], Coifman [l]-[5], A. Smajdor [2], [4], Hyllengren [1].
C1) Cf. in particular Topfer [2].

200 CHAPTER IX. Iteration
vestigate whether there exists among those groups a single one distinguished by an
important feature.
In some cases we are able to indicate a particular iteration group.
Definition. If <p(x) is a continuous and strictly monotonic principal solution
of equation F.1) in /, then (9.4) is called the principal iteration group of the func-
function feR%[I] fulfilling (9.5) with (9.6).
Since any two principal solutions of the Schroder equation differ by a multi-
multiplicative constant, the principal iteration group, if it exists, is unique. But it need
not exist, even if F.1) has a principal solution, since this solution is not necessarily
continuous or strictly monotonic. Such is the case for the function
f(x)=$x + — x2sin— for x>0, /@)=0, /=@,oo).
For this function there exists a principal solution (p{x)= lim 2"f(x) of the corres-
Л->00
ponding Schroder equation, but q>{x) is constant in every interval <2~m, 2~m+|-2"m)
(Szekeres [1]).
According to the distinguished position of the principal solution in the theory
of the Schroder equation we may expect that the principal iteration group is charac-
characterized by a particularly good behaviour near zero. To make this idea more precise,
we introduce, following G. Szekeres [1], the notion of a regular iteration group.
Definition. An iteration group f(x) of a function /e R% [/] with (9.5) and (9.6)
is called regular if for every и e (— со, со)
(9.7) Um^-=s".
The following theorem of basic importance is due to A. Lundberg [1] (cf. also
Coifman [1], [3], [4]).
Theorem 9.2. Let f(x) be in R°o [Ц and fulfil (9.5) with (9.6). An iteration group of
the function f(x) with respect to zero is regular if and only if it is principal (*).
Proof. We may assume that 0 is the left endpoint of the interval /.
(i) Suppose that formula F.26) defines a continuous and strictly increasing
function (p{x). The convergence must then be uniform in every compact subinterval
of /; hence also the sequence f(x)lf(y) converges to <p(x)/<p(y) uniformly in every
rectangle 0<a^x^/?e/, 0<y^y^3eI.
Let us fix а ме(-со, со) and put fu(x)=q>~1(s"q>(x)). In order to prove (9.7)
it is enough to show that for every sequence xm>0, xm->0, xm e I we have
(') The "only if" part of this theorem was already proved by G. Szekeres [1]. Theorem
9.2 implies in particular that the regular iteration group is unique. The uniqueness of the regular
iteration group was also proved by H. Michel [1].

2. Regular iteration
201
lim (fu(xm)/xm)=su. In the sequel we consider a fixed sequence xm with the
above properties.
Let R={(x, y):fu+1(xo)^x^fu(xo),f(xo)^y^xo}A). Given e>0, we can find
an N such that
f\x)
fb)
<e for (x,y)eR, n>N.
For every m there exists an index nm such that ?m=/~"m(xm)e</(x0), *o>- Since
xm->0, wm->oo as m->co, and consequently we can find an M such that nm^N for
. Now, we have for m^M
/"[/"- (Q] /"m
But the point (/"(O> *m) belongs to R for every m, whence
<e for
Since <plf(tm)y<p(tm)=su, we get hence
¦ —s
<e for
which was to be proved.
(ii) Let/"(x) be the regular iteration group of f(x), and let us fix an xo>O,
x0 e I. f(x0) is a continuous and strictly monotonic function of и (the strict mono-
tonicity results from the continuity; cf. Aczel [5], [7]), which approaches 0 as и-юэ
and the right end-point of / as и decreases. Consequently, for every x e I, x ф О,
there is a unique ux such that fx(x0) = x. Hence
<p{x)= hm — = lim
л-»ооУ (^o) n-*co f \xo)
] ..
= lim
и
= sx
f
exists in virtue of (9.7). ux depends on x in a continuous and strictly monotonic
manner, and so does also ip(x). Thus <p(x) is a continuous and strictly monotonic
principal solution of the Schr6der equation. Next we have
whence
fu(x)=<p-l[su<p(x)]M
From theorems 6.3, 6.7 and 6.8 we obtain in view of theorem 9.2 the following
C) Here xo refers to the point occurring in F.26) and is independent of the sequence xm.
But, on the other hand, there is no loss of generality in assuming that the sequence xm starts just
with this term xo.

202 CHAPTER IX. Iteration
Theorem 9.3. Let fix) be in R%[I\ and fulfil (9.5) with (9.6). Each of the following
conditions is sufficient for fix) to possess a regular iteration group:
(a) fe Cl[I\ and fulfils condition F.13) (*).
(b) fix)/x is increasing in I B).
§ 3. Multiplier zero. Now we replace conditions (9.5) and (9.6) by
fix)
lim^
(9.8)
for a certain constant ц> 1. (9.8) implies (9.5) with s = 0.
We shall confine ourselves to the case where /t=<0, 1). We may do so without
loss of generality, since the behaviour of / for large values of x is irrelevant to the
problem of the choice of a particular iteration group, and the possible part of /
placed in the negative part of the x-axis must be treated separately and can then be
easily transformed into the positive 'part by reflexion (i.e. one replaces fix) by
-/(-*) = <?(/[<T1(*)]) with g(x)=-x).
By a slight modification of the argument in the proof of theorem 6.9 it may be
shown that if fe R%[I] fulfils (9.8), then
(9.9) hix) = )
where
([-logx] for x>0,
(9.10) g(x) = \
[0 for x=0,
belongs to Ro[gil)] and fulfils (9.5) with s=jx~1.
Definition. An iteration group fix) of an feR%[I] fulfilling (9.8) is called
principal if it is of the form
where g{x) is given by (9.10) and h\x) is the principal iteration group of func-
function (9.9).
Following G. Szekeres [1] we shall also define a regular iteration group.
Definition. An iteration group fix) of an fe R%[I] fulfilling (9.8) is called
regular if for every и e (— oo, oo)
fix) ""+1
(9.11) iim^y. = eM-i.
x-0 XT
(!) Szekeres [1], Lundberg [1].
B) Lundberg [1], Kuczma [21]; strangely enough, the condition that/(л;)/* be decreasing is
not sufficient, ф (x) still exists then and is continuous in / but not necessarily strictly monotonic.
C) M. K. Fort [1], Kuczma [20], [21], Lundberg [1].

3. Multiplier zero
203
We shall be confined to a weaker analogue of theorem 9.2.
Theorem 9.4. (Szekeres [1], Lundberg [1], Michel [1].) Let f{x) be in R°0[I],
/c<0, 1), and fulfil (9.8). A regular iteration group f{x) of f(x) with respect to zero
must be principal and thus is unique.
Proof. Let/"(x) be an iteration group whose members fulfil (9.11) and put
hu(x) = g(f[g~л(х)]). It is easily seen that h\x) form an iteration group of h(x). In
view of the remark at the beginning of this section we have, moreover, lim (h"(x)/x) =
x->0
fi~". Thus h"(x) is the regular iteration group of h{x) and by theorem 9.2 it is also
the principal iteration group of h(x), which means that/"(x) is the principal itera-
iteration group of f(x)M
Theorem 9.5. (Szekeres [1].) Let f{x) fulfil the hypotheses of theorem F.9).
Then the formula
(9.12) Пх) = ср~\ци(р{х)\ ,
where <p(x) is given by F.35), defines the regular iteration group of fix) with respect
to zero.
Proof. The thing to show is that functions (9.12) fulfil (9.11). For this purpose
we shall prove that for every positive integer n we have
(9.13)
/i-l
log а-у? log x
/Л-1
1 /t-1
provided that x is sufficiently small, where Kx is the constant occurring in F.37).
For n=\ (9.13) is identical with F.37). Assuming the validity of (9.13) for n, we get
(9.14)
log/"
log а-ц"
/t"-l
/t-1
Multiplying F.37) by /*" gives
(9.15) \n"logf(x)-nnloga-nn+1 logx|
and adding (9.14) and (9.15) we obtain
+ i
At—1
/t-1
log a—fi" logx
x5,
i.e. (9.13) for n +1. Now, (9.13) and F.35) (where we take rj = \) give for x sufficiently
small
(9.16)
1
q>{x) + loga + logx
1
-l

204 CHAPTER IX. Iteration
whence
(9.17) if'w-fl'^'V'Kb"
for у sufficiently large. Relation (9.11) now results from (9.12), (9.16) and (9.17).и
§ 4. Levy iterates. Now we turn to the case of functions of multiplier one.
We assume that f(x) belongs to R%[I], /=@, c), and fulfils
(9.18) Iim^^-)
X
for a certain constant fi>0. (9.18) implies (9.5) with s=l. In this case it is more
convenient to operate with formula (9.3). In order that (9.3) define an iteration
group of f(x), <x(x) must be a solution of the Abel equation G.14).
Definition. An iteration group f(x) of anfeRg[r\ fulfilling (9.18) is called
principal if it is given by (9.3), where <x(x) is a continuous and strictly mono tonic
principal solution of equation G.14).
Again we define a regular iteration group (Szekeres [1]).
Definition. An iteration group f\x) of an feR°0[I] fulfilling (9.19) is called
regular if for every и e (— oo, oo)
(9.19) ^^L
x->0 X
Theorem 9.6. (*) Let fix) be in R%[I\, /=@, c), and fulfil (9.18). An iteration
group fix) of the function f(x) with respect to zero is regular if and only if it is prin-
principal.
Proof, (i) Suppose that (9.3) defines the principal iteration group of f{x).
Then a(x) must be given by G.6) and must be continuous and strictly monotonic
in /. Therefore the convergence in G.6) must be uniform in every compact subinterval
of /. Further, we have
,. /"+100-/0 ,. ffn+\y)-f\x0) Пу)-Пхо)\ _,, ч_ , ч .
hm = lim ) = a[/O0]-aO0=l
а а а )
uniformly in every compact subinterval of /, and consequently
/*0 ~f(y)
lZf"+\y)-fn(y)
.. (Пх)-Пх0) dn f\y)-f\xo)
= hm I • -^ • —q
= cc(x)-cc(y)
and the convergence is uniform in every rectangle
(i) Coifman ?l], [3], [4], A. Smajdor [1]. Cf. also Szekeres [1], Lundberg [1], Michel [1].

4. Levy iterates
205
We fix а м e (— oo, oo) and a decreasing sequence xmel tending to zero. To a
given ? > 0 we find an N such that
f\x)-f\y)
r\y)-f\y)
-[«Й-ФЯ
<s for (x,y)eR,
where R={(x, j):/u+1(*<№^/"(*o), /W^^Xo}, and/"(*) is given by (9.3).
For every m we choose an nm such that tm = f ~"m(xm) e </(x0), xoy. Since lim nm=co,
we have nm^N for m^M. Now we have "¦*°°
<9.20)
_f"(xj-xm
f(xj-xm '
and, on the other hand, since (f(tm), tm) e R,
<s for
But/"(x) = a ^а^ + м), whence а[/"(л:)]-а(л:) = м for every x. Consequently, it
follows from (9.20) and (9.21) that
<9.22)
xm-f\xm)
— u
<e for
xm~f(xj
The sequence xm has been arbitrary, and thus (9.22) proves that
<9.23)
lim • = и .
о x—f(x)
(9.19) results from (9.23) and (9.18).
(ii) Now let fu{x) = a,~i{a{x) + u) be a regular iteration group, i.e. f\x) fulfils
(9.19). Since solutions of the Abel equation differring only by an additive constant
generate the same iteration group, we may assume that а(хо) = 0 (where xo is an
arbitrarily fixed point from /). Replacing in (9.19) the dummy variable x by t, we
obtain the formula
lim Г"[^-а( )
t-o
validfor all u. Now we fix an arbitrary x e /and set u = a(x). By G.14) lim оГ^и)
= 0, whence "¦>co
<9.24)
But a
lim
n-*co
and (x~1(n + a(x))=f(x); so (9.24) may be

206 CHAPTER IX. Iteration
written as
<x(x)= hm
where we have set dn= — a[/"(xo)]"+1. This means that a.{x) is the principal solu-
solution of equation G.14). The continuity of <x(x) may be proved in the same way as
the continuity of q>{x) in theorem 9.2.¦
As an immediate consequence of theorems 9.6 and 7.2 we obtain the following
Theorem 9.7. (Szekeres [1].) Let f{x) fulfil the hypotheses of theorem 7.2. Then
formula (9.3), where a(x) is given by the Levy algorithm G.13), defines the regular
iteration group of fix) with respect to zero.
Similarly, theorems 9.6 and 7.5 (cf. the remark in the footnote) imply the fol-
following
Theorem 9.8. (Kuczma [31].) Let fe R°0[I]nM1 [I], /=@, c), and fulfil' (9.18).
Then formula (9.3), where a{x) is given by the Levy algorithm G.13), defines the
regular iteration group of f{x) with respect to zero.
§ 5. Regular iteration at infinity. We shall consider functions feR°x[I] (or
even fe R^ [/]). We start with the case where
(9.25) lim ^^
х-* со X
Then (under certain conditions) we have for integral n
lim =n
*^co f(x)-X
(cf. formula G.11)). Therefore it is natural to ask whether/(x) has an iteration group
(with respect to infinity)
(9.26) f\x)
such that for every и e (— oo, oo)
(9.27) li
Quite similarly to theorem 9.6 one can prove the following
Lemma 9.1. LetfeR°JJ]. An iteration group (9.26) of f(x) fulfils (9.27) if and'
only if<x(x) is a continuous and strictly monotonic principal solution of equation G.14).
The following theorem, resulting from lemma 9.1 and theorem 7.7, gives a
sufficient condition for the existence of an iteration group fulfilling (9.27).
Theorem 9.9. (Szekeres [2].) If f{x) fulfils conditions of theorem 7.7, then it has
a unique iteration group f{x) with respect to infinity fulfilling condition (9.27). This:
group is given by (9.26), where a(x) is given by G.13).

5. Regular iteration at infinity 207
Now we pass to a more general situation. We assume that
(9.28) lim -^ -Ш— = 0 ,
jc-»co X
where a{x) is the function defined by G.38) and G.39). First we show that this case
contains (9.25).
Lemma 9.2. Letfe 2?^[7] be of the form
(9.29) fix)
where r is an integer and lim (\д(х)-х]/х)=0. Then f(x) fulfils (9.28).
x-*co
Proof. Put <р(х)=ха'(х)/й(х). According to G.38) (p{x) satisfies the equation
(9.30) «,[еМ>1^!^
and is continuous in @, oo). Since the coefficient of <p(x) in (9.30) tends to zero
as х-юо, we have by lemma 2.2 and theorem 2.8 lim <p(x)=0, i.e.
(9.31) ^U ""
lim^U.
*^со а(х)
Now we write g{x) — x = q{x). By hypothesis
(9.32) \\m {q(x)jx) = Q
x-*co
Moreover, we have according to G.38)
=х+а'[а-\х-г) + в(х) q(a-\x-r))]q(a-\x-r)) ,
0<6(x)<l. Hence we obtain, since lim a~1(x—r) = oo,
~1(
X-* CO
,. a[f(a\x))]-x q(y) а'
lim = lim *
o'OO oOO + r
by (9.32), (9.31) and lemma 7.4.И
Lemma 9.3. Let feR^lI] be of form (9.29), where r is an integer, and suppose
that lim ([g(x) — x]/x) = 0 and g{x) has an iteration group д\х) fulfilling
(9.33) lim
*^co g(x)-x

208 CHAPTER IX. Iteration
Then fu(x) = er[gu(e~r(x))] is an iteration group of f{x) fulfilling
(9.34) lim
Proof. Write q\x) = g\x) — x, q (x) = q1 (x) = g (x) — x. As in the proof of lemma
9.2, we obtain
<»'[> +0*00 9 GO] '
and this equals м in virtue of (9.33) and of lemma 7.4.¦
Definition. An iteration group f(x) of an /e j?° [/] fulfilling (9.28) is called
principal if а(/"[а~1(л:)])=а~1[а(л:) + м] and a(x) is a continuous and strictly mo-
notonic principal solution of the equation
The following definition is due to G. Szekeres [3].
Definition. An iteration group f(x) of an fe Д° [/] fulfilling (9.28) is called
regular (with respect to infinity) if it fulfils condition (9.34).
Theorem 9.10. Let f belong to Д° [/] and fufill (9.28). An iteration group f(x)
of f(x) with respect to infinity is regular if and only if it is principal.
Proof. The proof follows immediately from lemma 9.1.И
Theorem 9.11. Let f(x) =er[g{e~r(x))] be in R^JI], where r is an integer,
lim g'{x)= 1 and g\x)—\ is of bounded variation in an interval <a, oo)ce~r(/). Then
f{x) has a unique regular iteration group with respect to infinity.
Proof. The proof follows from theorem 9.9 and lemmas 9.2 and 9.3.¦
Theorem 9.12. (Szekeres [3].) Let fe R^ll] and let a(x) be a C1 solution of equa-
equation G.14) in I fulfilling condition G.43). Then f"(x) = a~1[a(x) + u] is the regular
iteration group of f(x) with respect to infinity.
Proof. Put fi(x) = a[x~\x)]. Then
(9.35) 1тЯ«(*)]=Кт^=?.
x->co x->co (X (X)
Moreover,

5. Regular iteration at infinity 209
But according to (9.35)
lim F(<x[a~\x)] + e(x) u)= lim J?'[aO) + 0*O) «]= lim j8'[a(z)]=y ,
since z=or1[a(y)+0*QO и]-юо as j>->oo. Hence (9.34) follows.B
Hence we immediately obtain
Theorem 9.13. (Szekeres [3].) Iff(x) fulfils the hypotheses of theorem 7.9, then
it has a unique regular iteration group with respect to infinity.
Theorem 9.14.A) Every L-function f{x) which tends to infinity as x-*ao has
a unique regular iteration group with respect to infinity.
Proof. If/(x) is of a type (r,s,fi) with r>s, the theorem results from theorems
7.10 and 9.12. If f(x) is of a type (r,s,(i) with r<s, then f~1(x) is of a type
(r',s',fif) with r'>s' (Hardy [2], p. 86) and again theorems 7.10 and 9.12 are ap-
applicable. Lastly, if f{x) is of a type (r, r, fi), then we have (cf. the last passages of
Chapter VII)
where w{x) is bounded and w'{x) is mono tonic for large x. Consequently, w'(x) is
of bounded variation and the theorem results from theorem 9.11 provided that
w(x)>0. If w(x)<0, we replace/(x) by f~\x).m
§ 6. Analytic iteration. Now we are going to deal with the iteration of
analytic functions of the form
CO
(9.36) /(*)=*+ I anxn, an^0, m>2,
where x is a complex variable. Following P. Erdos and E. Jabotinsky [1] we shall
call a one-parameter family of functions F(x, и) (и complex) an analytic iterate of
f(x) if the following conditions are fulfilled:
A) For every complex и the function F{x, u) is analytic in a neighbourhood of the
origin and has an expansion of the form
(9.37) F(x,u) = x+ f аЛи)лГ,
л = 2
convergent for \x\<r(u), r(u)>0. The coefficients а„(и) are polynomials in и and
F{x, u) is analytic in u.
B) The relation
F{F(x,u),v) = F{x,u + v\
is fulfilled for every complex u, v in a neighbourhood of the origin.
(!) Szekeres [3]. Iteration of exponentially growing functions was investigated also by Reichen-
bacher [1], [2].
14 Functional equations

210 CHAPTER IX. Iteration,
C) The relation
F(x,\)=f(x)
is fulfilled in a neighbourhood of the origin.
As we see, an analytic iterate of f{x) forms an iteration group of f(x) with
respect to the origin, in the sense of § 1. We may note that for integral и conditions
A), B), C) (except for the analyticity in u) are automatically fulfilled. The know-
knowledge of an(u) for integral и and the condition that aB(u) are polynomials allows one
to determine а„(и) uniquely for every complex u. Hence it follows that the analytic
iterate of a function f(x) is unique, provided it exists.
With every function/(x) of form (9.36) we can associate a formal power series
Lf(x) of the form
(9.38) L/x) = amxm+ ? bnxn
n = m+l
statisfying formally equation F.87).
Lemma 9.4. Each of series(9.36) and (9.38) is uniquely determined by the other
and relation F.87).
Proof. Let us fix a k^m. Equating the coefficients of xk+m in F.87) we obtain
(9.39) bk+m+bk+i.(k+\)am + Tl{k)+bmma1l+1
= bk+m+(k+l)ak+1bm+T2(k) + mambk+1,
where Т^{к) and T2(k) are expressions built of at's and b/s with ij^k. (In (9.39)
bm=am.) Since k+\>m, we can find from (9.39) Ai+1 knowingbj with j^ к and аг,
and conversely, we can find ak+1 knowing at with i^k and bj.m
Theorem 9.15. (Erdos, Jabotinsky [1].) A function fix) of form (9.36) has an
analytic iterate F(x, u) if and only if its associated series Lf(x) has a positive radius
of convergence.
Proof. I. Let series (9.38) have the radius of convergence p>0. We consider
the differential equation
(9.40) ^i=iz^..
dx Lf(x)
@, 0) is a singular point for (9.40). In order to overcome this difficulty we write
z = x+yxm, y=y(x). Equation (9.40) then becomes
dу ЬЛх+yxm) — [1 + myxm ~x] LAx)
(9.41) — = _ .
dx xTLjix)
All the coefficients of x" for n<2m in the nominator of the expression on the
right-hand side of (9.41) are now zero, and consequently the right-hand side of
(9.41) is analytic in a neighbourhood of any point @, y0); more exactly, it is analytic
in x and in у for x, у such that \x\ <p and \x+yx™\ <p. Thus for every complex y0
equation (9.41) has a unique solution y(x), analytic in a neighbourhood of the

6. Analytic iteration 211
origin and fulfilling the initial condition
9.42) y@)=y0.
Let y{x, u) be the solution of (9.41) fulfilling condition (9.42) with уй=иат.
CO
Thus y{x,u) = uam+ ? с„(и)х". The function z=F(x, u)=x+xmy(x, u) is the
n= 1
unique solution (or rather a one-parameter family of solutions) of equation (9.40),
analytic in a neighbourhood of zero. This solution has an expansion of form (9.37)
with а„(м)=0 for л=2, ..., m—\, am{u) = uam, am+i{u) = ci(u). Now we shall prove
that F{x, u) is the analytic iterate of function (9.36).
Let us fix constants u, v and consider the functions F{x, u + v) and F(x,u).
They satisfy the differential equations
dF(x,u + v) Lf[F(x,u + vy] dF(x,u) Lf\J(x,u)~\
= and =
dx Lf(x) dx Lf(x)
respectively, whence by division
dF{x,u) ~ Lf[F(x,u)-]
Taking in (9.43) t = F(x, u) as an independent variable and writing ? = F(x, u + v)r
we obtain df/dT=L/@/L/(T), an equation of type (9.40), whose solution, as we have
just seen, must be of the form f =F(x, w) with a suitable constant w. Reverting to
the old variables, we obtain hence the relation
F(x,u + v) = F(F(x,u),w),
valid in a neighbourhood of x = 0. Equating the coefficients of xm yields w = v, and
consequently F{x, u) fulfils condition B).
For n<,m, а„(и) are polynomials in u. If we assume а„(и) to be polynomials
for я<&, к^т, then inserting (9.37) into (9.1) and equating the coefficients of
xk+1 we obtain
(9.44) ak+1(u + v) = ak+1(u) + ak+1(v)+Pk(u, v),
where Pk(u, v) is a polynomial in u, v. From the continuous dependence of solutions
of differential equations on initial values it follows that ak + t{u) is regular. Differentiat-
Differentiating (9.44) with respect to v and then setting v = 0 we obtain
dPk
Bv
i.e. ak+1(u) is a polynomial, and so is also ak+1(u).
F(x, u), as a sum of an absolutely convergent series of polynomials, is analytic
in u. Consequently, condition A) is fulfilled.
It remains to prove (9.2). For this purpose put g(x)=F(x, 1). Thus we obtain
14*

212 CHAPTER IX. Iteration
from (9.40) with z = F{x, 1)
i.e. Lf(x)=Lg{x). (Note that f{x) and g(x) have equal coefficients of xm\). By lemma
9.4 it follows that f(x) = g (x) and condition C) is also fulfilled.
II. Now suppose that F{x, u) is the analytic iterate off(x) and put
8F(x,u)
=0
ди
Thus L(x) is analytic in a neighbourhood of the origin. Differentiating (9.1) with
respect to и we obtain
F'B(x, u + v) = F'x{F{x, u), v)F'u{x, u),
whence, setting м = 0, we obtain in view of the fact, resulting from (9.1), that
F(x,0) = x
(9.45) F'u(x, v) = F'x(x, v) L (x).
Differentiating (9.1) with respect to v we obtain
F'u(x, u + v) = F'u(F(x, u), v),
whence, setting v = 0 and writing v in place of u, we obtain
(9.46) F'u(x,v)
Equations (9.45) and (9.46) together give
whence putting v = l (cf. (9.2)) we obtain F,87).
Since for integral и ат(и) = иат and am(u) is a polynomial, the latter relation
holds for all u. Similarly, а„(и) = 0 for l<n<m, u^eI. Since F(x, u) is analytic
in x and in u, we have
д
lim x mL (x) = lim x mF'u(x, 0) = — lim x mF (x, u)
x->0 x->0 OU x->0
д
= а„
i=0
It follows by lemma 9.4 that L(x)=Lj(x), and consequently series (9.38) has a posi-
positive radius of convergence.¦
P. Erdos and E. Jabotinsky [1] have also proved that in the case where the radius
of convergence of series (9.38) is zero series (9.37) (which are uniquely determined
by conditions A), B), C) for every function fix) of form (9.36)) may converge only
for real м belonging to a set U of a linear measure zero, or for complex и belonging
to a set U of a planar measure zero. A better characterization of the set U may be
obtained from theorem 10.12.

CHAPTER X
COMMUTING FUNCTIONS
§ 1. The real case. Two functions q>(x) and/(x) are called commuting or permutable
if the superpositions <p[f(x)] and f[<p(x)] are both possible on a common set E and
(lo.i) ?[/(*)] =/!>(*)]
holds for x e E. If fix) is given, A0.1) is a particular case of equation A.1). From
the results of Chapter III we may deduce the following theorem, due to J. Lipinski.
Theorem 10.1. (Lipinski [2].) Let fix) be a continuous and strictly monotonic
function on a submodulus interval I. Then equation A0.1) has in I a continuous
solution q>{x) depending on an arbitrary function.
Proof. If f{x) = x in /, then q>{x) may be quite arbitrary, so we assume that
f(x)^x. We shall assume that/(x) is increasing; the case where/(x) is decreasing
may be handled in a similar manner.
Let (?, b) be an interval contained in /such that/e R°[(?, b)] and either f(b) = b
or b is the right end-point of/; in the latter case (?, b) may also be replaced by (?, b}.
By theorem 3.1 equation A0.1) has in (?, b) a continuous solution q>{x) depending
on an arbitrary function and such that t;<(p(x)<b for x e (?, b). Taking arbitrarily
xoe{^, b) and denoting by m and M the infimum and supremum, respectively, of
<p(x) on </(x0), x0}, we get by A0.1)
f>(x) e </"(«),/"(Af)> for хб</"+1(х0),/"(х0)>, 11 = 0,1,2,...,
whence lim q>{x) = t;, and, if f{b) = b, Urn q>{x) = b. Thus we may extend (p{x)
onto <?,&) by putting <p(g) = ? (onto <?, by by putting <p(g) = ?> <p(P) = b) and
(p(x) is then continuous in <?, ?) (in <?, ?» and satisfies equation A0.1). We may
proceed similarly if fe R°[(a, ?)].
Now, / may be represented as a sum of at most denumerably many intervals /v
(open, except possibly in the case where they have a common end with /) and of a
setF:
such that f(x)^x in /v a.ndf(x) = x in F. Since / is submodulus for/, feR°v[Iv]
for every v, where ^v is an end of /v (suitably chosen). If we construct a continuous
solution р(х) of A0.1) such that cp{x) e /v for x e /v, independently in each /v, and
put (p{x) = x for x e F, we shall obtain a function continuous in the whole of /.and,

214 CHAPTER X. Commuting functions
satisfying A0.1) in /. This solution depends on an arbitrary function, since it may
be prescribed arbitrarily in suitable subintervals of every /v.b
The situation changes if we require <p(x) to be of class C1. As an example we
shall prove the following
Theorem 10.2. (*) Suppose that f e R^I], Oel, and that F.13) holds. Then the
only functions that are defined, positive for positive x, and of class C1 in a neighbourhood
of the origin, assume values from I and commute with f{x) are the regular iterates
of fix).
Proof. The regular iterates off(x) exist according to theorem 9.3 and evidently
commute with f{x). Now, let cp{x) be a function fulfilling the conditions of the
theorem and such that A0.1) holds. But we have f(x) = y/~1[si//(x)], where y/(x)
is a principal solution of equation F.1). Hence <p[f(x)] = y/~1[sy/(<p(x))], i.e. writing
co(x) = y/(<p(x)) we get
«[/(*)] = «»(x).
But co(x), like (p{x) and y/(x), is of class C1 in a neighbourhood of zero, and hence,
by theorem 6.3, it must be a principal solution of equation F.1) and thus a constant
multiple of y/(x):
co(x) = r]y/(x).
y/(x), being strictly monotonic, has a constant sign for positive x. Hence r\ must
be positive and we may write rj=s" with a suitable u. Finally, we obtain
<p{x) = y/~i[suif/{x)\, i.e. (p(x) is a regular iterate of/(x).B
If we drop the assumption that cp(x) is positive for positive x, we obtain also
the solution q>(x) = 0 and, if 0 is an inner point of /, a dual family of solutions cp{x)
=y/~1[—suy/{x)]. Thus e.g. the only C1 functions that commute with f{x)=sx,
Q<s<\, are q>(x) = ax with arbitrary a. For a>0 they are the regular iterates of
f{x).
In spite of the fact that permutable functions have been extensively studiedB),
there are still many open questions, some of them seeming very simple. Such is e.g.
the following one, known as Isbell problem C):
Suppose that f(x) and cp{x) are commuting continuous mappings o/<0, 1> onto
itself. Does there exist a ?, e <0,1> such that <p(Q=f(g) = ?l
A) Kuczma [20]. Cf. also Hadamard [2], M. K. Fort [1], Berg [3], Kuczma [18], Hosszu
tl], Coifman [4] concerning related results.
B) Koenigs [5], [6],Demeczky [1], Lemeray [2], [5], Giraud [2], Julia [8], [9], [12], Fatou [7],
I8],[13], Ritt [7]-[10], Sheffer[l], Hadamard [2], Walker [1], [2], Batty, Walker [1], Silberstein [1],
Bradley [1], [2], Bradley,Walker Ш, Rosenbloom [1], Pastides [1], [3], Block, Thielman [1], Drazin
Ш, Jacobsthal [1], [2], M. K. Fort [1], Baker [2], [4], [5], Ganapathy Iyer [4], Kuczma [18],
Hosszu [1], Chen [1], Lipinski [2], Schubert [1], Goralcik [1] Aczel, Karteszi [1].
C) Isbell [1]. Only partial solutions to this problem are known (cf. Drazin [1], Tarski [1],
Hedrlin [1], [2], Pultr [1], De Marr [1], [2], [3], Baxter, Joichi [1], [2],[3], Shields [1], Cohen [1],
Schwartz [1].) Other problems may be found in De Marr [4], Mioduszewski [1].
Added in proof: Recently the Isbell problem has been settled in the negative
by W. M. Boyce [1[ and J. P. Huneke [1].

2. Semipermutable polynomials 215
§ 2. Semipermutable polynomials. Besides commuting continuous or differentiable
functions, also the permutability of functions belonging to some narrower function
classes has been extensively studied. Thus P. Fatou [8], [13], G. Julia [8], [9], [12],
and J. Ritt [8], [9], [10] studied permutable rational functions and J. Ritt [7], I.M.
Sheffer [1], H. D. Block and H. P. Thielman [1] and E. Jacobsthal [1] investigated
permutable polynomials. In particular, the latter, using number theoretical methods,
determined all couples of commuting polynomials. It would take to much space to
reproduce his considerations here; we shall restrict ourselves to determining families
of commuting polynomials which contain exactly one member of every degree.
Instead of considering permutable polynomials, we shall study a more general
relation, first considered by G. af Hallstrom [1 ] and called by him semipermutabilityf}).
Two functions,/(x) and (p{x), are called semipermutable if there exists a homographic
function F(x) such that
A0.2) ?[/(*)]=* №(*)]).
However, in the case where/and q> in A0.2) are polynomials, F must also be a
polynomial:
A0.3) F(x)=Kx+L, K^O,
for otherwise the left-hand side of A0.2) would be an entire function whereas the
right-hand side would have a pole B).
Definition. A family P of polynomials is called an SP-chain resp. T-chain, if P
contains exactly one polynomial of every degree and every two members of P are
semipermutable resp. permutable.
If P={fn(x)} is an SP-chain (resp. P-chain) and g{x) = Ax + B, A=?0, then
g~1Pg = {g~1[fn(g(xj)]} also is an SP-chain (resp. P-chain). Chains P and g~*Pg
will be called equivalent.
At first we establish some results concerning the semipermutability of poly-
polynomials with some of the simplest polynomials of the first and second degree. These
result are due to G. af Hallstrom [2].
Theorem 10.3. f{x) = x+b, ЬфО, is not semipermutable with any polynomial
of a degree и ^2.
Proof. Suppose that /(x) is semipermutable with a polynomial (p{x) = cxn +
+dx"~1 + ..., сфО, n>2. Inserting this into A0.2) with A0.3) and comparing the
coefficients of x" and x", we obtain c=Kc and d=K(cnb+d), whence cnb = 0,
contrary to the assumptions
A) Cf. also Julia [9]. Semipermutable functions were studied by af Hallstr6m [2], [3], [4]
and J. Nikolaus [1].
B) This argument is independent of whether we consider real or complex polynomials. For
if A0.2) with polynomials/, q> and a homographic function Fholds on a real interval, then it remains
valid on the whole complex plane.

216 CHAPTER X. Commuting functions
Theorem 10.4. Let f(x)=ax, a^O, 1. If a is not a root of unity, then the only
polynomials semipermutable with fare
A0.4) <p(x)=Cx"+D,
С and D being arbitrary. If a is a primitive p-th root of unity (p>l), then the only
polynomials semipermutable with fare those of the form
A0.5) 4>(x)=tcixip+n + D,
>=o
C; and D being arbitrary.
Proof. It is easily verified that/is semipermutable with every function A0.4)
or A0.5); in both cases F(x)=a"~1x+D(l — a"). Now suppose that / is semiper-
semipermutable with a polynomial
A0.6) Hx)=ilcJxJ, cN*0.
Inserting / and <p into A0.2) with A0.3) and comparing the coefficients of Xs and
xm, 0<m<N, we obtain cN aN = KacN and cmam=Kacm, whence either cm=0 or
aN~m=l. Hence A0.4) follows in the case where a is not a root of unity. If a is a
primitive ptb root of unity, then the only possible non-vanishing coefficients in A0.6)
are those for which N—j=kp with a non-negative integer k. Writing N=rp+n,
0^n<p, and i=r—k, we obtain A0.5).и
Theorem 10.5. If f{x) = x2+b and f{x) is semipermutable with a third degree
polynomialcp(x), then b=0or b= — 2 and(p{x) = cxi or ср{х)=с{хъ — 3x), respectively.
Proof. This results from inserting/(x) and (p(x) = cx3+dx2+ex+finto A0.2)
and equating the coefficients. ¦
Let Tn{x) = cos (n arc cos x) be the nth Cebysev polynomial. Since every two
Cebysev polynomials commute, {Т„(х)} forms a P-chain. In the equivalent chain
{{п(х)}> where tn(x)=2Tn(^x), all the polynomials except to(x) have the coefficient
of x in the highest power equal to one (cf. Jacobsthal [1], p. 246). In particular,
to(x) = 2, tt(x) = x, t2(x)=x2-2, t3(x) = x3-3x, tA.(x)=x*-4x+2. For even n,
tn{x) is an even function; for odd n, tn(x) is odd.
Now we shall determine all SP-chains.
Theorem 10.6. (G. af Hallstrom [2].) Every SP-chain is equivalent either to
{а„ х„}, а„ being arbitrary, or to {а„ tn(x)}, a\ = \ for n>0, a0 being arbitrary.
Proof. Every two functions of the form an x" are evidently semipermutable.
Now let f{x) = antn(x), <p(x) = amtm{x), a\ = a2m=\. Then <p[f(x)]=etm[tn(x)] and
f[<p(x)]=ntn[tm(x)] = ritm[tn(x)] = rie<p[f(x)], where ?2 = ?/2 = l, and hence/and cp are
semipermutable.
Now let P={pn(x)} be an SP-chain. If p2(x)=cx2+2dx+e, then for g{x)
= {x—d)jc we get g~ 1[p2(g(x))] = x2 + ce+d-d2. Thus, replacing, if necessary, P by
an equivalent chain, we may assume that p2(x) = x2+b. Hence by theorem 10.5
either p2{x)=x2 or p2(x) = x2-2.

2. Semipermutable polynomials 217
First we assume p2(x) = x2, and let р„(х)=а„ x" + bxm+ terms of lower degrees,
0, а„фО, ЬфО. Then p2[pn(x)] = c^x2n+2an foe"+m+terms of lower degrees,
whereas F(j>n[p2{x)'\)=Kan xln + Kbx2m+terms of lower degrees, which is impossible,
since 2m<n + m. Thus necessarily рп(х)=а„ x" and P is of the form {а„ х"}.
Now suppose that p2{x)=x2 — 2=t2(x). For «>0 we may write pn{x) in the
form
>=0
and suppose that ЬтФ0. We have
(Ю.7) Р2Ы*У] = а2[.Ф)? + 2апф)^Ь1ф)+(^ Ь,ф)J-2,
i=0
A0.8) F (р„|>2(*)]) = Кап ф2(хУ] +к?ь{ Г,[Г2(х)] + L.
Equating coefficients of x2n yields #= а„. Moreover,
а2[Ф)У = ЙФ)] + 2а„2 = а„2?„[?2(х)] +2а„2.
So equating A0.7) and A0.8) we obtain
(f )?
1 = 0 i=0 >=0
The degree of the polynomial on the left-hand side is n + m, while the polynomial
on the right-hand side has degree 2m<n + m. This contradiction shows that bm=0,
and consequently р„(х) = an tn(x).
The condition that p3(x) is semipermutable with Pi(x) and with p^{x) gives
(after direct, though rather long calculations) а2 = 1, а\=\, c%=\. Hence for
= а„ а ф3(х)] = а„ а ф„(хУ]
= ana[tn(x)Y-3anan3tn(x),
and the semipermutability of p3 andpn gives
A0.9) a3 а3п[ф)У -За3ап ф)=Кап аЦф)У - ЪКап ап3 ф)+L.
Equating the coefficients of x3n we obtain K=a2/an3~1, and consequently A0.9)
reduces to
За3а„ф)=3а3а3ф)-Ь,
whence a2 = 1. ¦
Let Po denote the chain consisting of 0, x, x2, ..., x", ..., Pt the chain {l,x, x2,

218 CHAPTER X. Commuting functions
...,*", ...}and Г the chain {Tn(x)} = {l,x, 2x2-\, ...} of the Cebysev polynomials.
From theorem 10.6 we derive the following result, due to H. D. Block and H. P.
Thielman [1] (cf. also Jacobsthal [1], [2]).
Theorem 10.7. Every T-chain P is equivalent to Po or Pt or T.
Proof. Since every P-chain is also an SP-chain, P must be equivalent either
to {а„х"} or to {а„ tn(x)}, where, moreover, we may assume az = 1. But an SP-chain
equivalent to a P-chain must itself be a P-chain. Hence we obtain for и>1, т^\,
an =am .
On setting m = 2 we obtain hence an = 1 for n ^ 1. It remains to determine the constant
member of P. Putting (p{x)=t; = const in A0.1) we obtain /(?) = ?• The common
fixed points of pn{x) = x" are 0 and 1, and these lead to Po and Pt, respectively.
The only common fixed point of tn(x) is 2. But the chain {tn(x)} may be replaced
by the equivalent one, T={Tn(x)} = {$tnBx)}.m
§ 3. Permutable entire functions. We start with two lemmas, which will prove
useful later. They are both due to I. N. Baker [2].
Lemma 10.1. Let f{x) and (p{x) be permutable transcendental entire functions.
If fix) has Fatou's exceptional value ?, then ? is also Fatou's exceptional value
for (p(x).
Proof. At first we assume that v in @.35) is positive and, for an indirect proof,
we suppose that there exists an цф^ such that (p(y\) = t,. Since пф^, there exists
infinitely many points ? such that/@ = 77. For such points we have by A0.1) f[<p @]
= ф[/(.0] = ф(я) = ?> whence (p{Q=^. Similarly, for every n there are infinitely
many со such that f(co)=r] and for all such со we have <p(co) = ?.
We choose an xoe 3f[/L хофсуо. By theorem 0.6, (iv), there exist sequences
com, nm such that com-*x0, /"m(a>m)=ц. On account of what has just been said <p (com) = ?,
whence g>(x) = ^, contrary to the assumption that q> is transcendental. Consequently,
the equation (p{x)=?, has either no solution, or ? as the only solution, i.e. ? is Fatou's
exceptional value for (p.
Now assume that v in @.35) equals zero, i.e. f(x)=? has no solution, and
suppose that q>{rf)=^, цф?,. There exists а С such that/@ = 77. Then we have for
«=<P(O'- f(<x)=f[<P(O]=<p[f(O] = <P(ri)> again a contradictions
Lemma 10.2. Let f(x) and cp(x) be permutable transcendental entire functions.
Then there exists a positive integer n such that Mp,(r)>MJ/) for r sufficiently large.
Proof. We choose an х0фсо in 5t/]; iff(x) has Fatou's exceptional value ?,
then we choose x0 so that \x0 — ?|>2 (theorem 0.6, (i)). Let K={x: \x—xo\^l}.
In virtue of lemma 10.1 ? $ cp{K) and hence, by theorem 0.6, (iii), there exists an m
such that
A0.10) <p(K)<=fm'(intK) for m'^m.
By theorem 0.6, (ii), (iii), there exist a z and an integer p^m such that/p(z)=z,

3. Permutable entire functions 219
\z— xo\<%, and
A0.11) Kczf(intK)<=fp(K).
According to A0.10) we have also
A0.12) q>{K)c:f>{K).
Let K'={x: \x—z\^r0} be the largest disc centred at z and contained in K.
Evidently
xoeK'<=K.
Put
A0.13) y = X~
and D = {x: \x—z\^%r0}. D is transformed by A0.13) onto {y: |j|^|}. Further,
put Д, = тах |/"p(x)-z| and F(y) = B;\f\z+roy)-z]. F(y) is regular for \y\^\,
F@) = 0, max |F(_y)| = l. By a theorem of H. Bohr [1] there exists a constant c>0
|y|<3/4
(independent of n) such that the domain on which the unit disc is mapped by F
contains a circle (circular circumference) centred at the origin, of a diameter greater
than с Thus f"\K') (and hence also f\K)) contains a circle centred at z, of a di-
diameter at least сВ„. By theorem 0.6, (iii), cBn tends to infinity with n, since x0 e D.
Thus for n sufficiently large fp(K) contains the greatest circle Cn centred at z and
such that the distance An of С„ from the origin exceeds the number
A0.14) pn=iCmax|/"p(x)|.
xeD
Resorting once more to theorem 0.6, (iii), we infer that there exists an a such that
A0.15) Kczfap(D),
whence
A0.16) /Р(С„) c/p[/"p(K)] c/(n+a+1)p(?>)
for n sufficiently large. Moreover, putting ?„=тт |х|, we have in virtue of Liou-
ville's theorem
A0.17) max|/p(x)|>4c-2Ln
1*1 =?«
for n sufficiently large. Also all the following relations are valid for n sufficiently
large. So we have by A0.16), A0.17) and A0.14)
<10.18) тах|/(п+а+1)р(х)Ьтах|/р(х)|> max |/p(x)|
xeD хеС„ \x\ =?„
>4c~2 min |x| >4c~2pn=2c-' max |/"p
xeCn xeD

220 CHAPTER X. Commuting functions
Replacing in A0.18) n by n — a— 1 we get
A0.19) Pn> max |/<»"-- "\x)\>mn \/^2а- 1)p(x)| = Rn.
xeD xeK
The sequence Rn increases:
|/(a1)p(/''(x))|= max \
xeK xeff(K)
Ss max |/(n2a1)p(x)||/()| n
xefP(iat K) xeK
in view of A0.11), since /p(int K) is an open set containing the closed set K. More-
Moreover, by theorem 0.6, (iii), Rn tends to infinity.
Now, we have for Rn^r<Rn+1
A0.20) М„(г)<М„(Я„+1)^ max|p(x)k max L(x)|,
xeCn+l x?/<» + «)j.(jf)
since An+1>Rn+1 and Cn+1<=/(n+1)p(iT). On account of A0.12) and A0.1)
A0.21) р[/("+1)р(К)]=/("+1)р[р(Ю] = /("+2)р(Х).
Let ^„ denote the disc {x: |x|^jR,,}. By A0.15) and A0.19) /("-2а-1)р(^)<=^„, whence
This together with A0.20) and A0.21) gives for Rn^r<Rn+1
M9(r)< max |p(x)|< max |||B)|
Since n (sufficiently large) has been arbitrary, the inequality obtained,
holds for all r sufficiently large.¦
The possibility of an entire transcendental function being commutable with a
polynomial is to a great extent restricted by the following
Theorem 10.8. (Baker [2], Ganapathy Iyer [4].) Let f(x) be an entire trans-
transcendental function commuting with a polynomial (p(x). Then (p(x) = a.x-\-P, a a root
of unity, or(p{x)=?,=const, /(?) = ?.
Proof. Let <р(х) = ао+а1х + ... + а„х", апф0, л>0, and put
A0.22) F(x)=f(ao + a1x + ...+anxn)=ao + a1f(x) + ... + alf(x)Y
(cf. A0.1)). We have |?>(х)|>||а„| r" for |x|>r provided r is sufficiently large. Let В
be the domain on which the function cp{x) maps the disc {x: \x\^r}. В contains.

3. Permutable entire functions 221
the disc {у. \у\^Ца„\ r"}, whence
A0.23) M^r)=max/[>(jc)]=max|/O0|^ max \f(j)\
\x\<r уев ||<*||"
=МД\а„\г").
On the other hand, there exists a K>0 such that for
A0.24)
Now, let xr be a point on \x\ = r at which the maximum of \F(x)\ is realized. Then,
according to A0.22), we have
A0.25) MF(r) = \F(xr)\ \ \
Since MF(r) tends to infinity as /-->oo, the same must be true about \f(xr)\, and
consequently \f{xr)\>K for r sufficiently large. Hence we get by A0.25) and A0.24)
for r sufficiently large
A0.26) MF(r)<2\an\ \f(xr)\n^2\an\[Mf(r)]n.
A comparison of A0.23) and A0.26) gives
Since/(x) is not a polynomial, we must have, in virtue of lemma 0.14, « = 0 or
n = 1. In the former case (p(x) is a constant, <p(x) = ?, and by A0.1) ^ must be a fixed
point of f{x). If и=1, then (p(x) = a.x+{l, a^O. We shall distinguish three cases.
I. a = 1, ^=0. Then A0.1) holds with every entire function/(x).
II. a = 1, РФП. Then A0.1) becomes/(jc+^)=/(x)+^andputtingh(x)=f(x)-x
we obtain h{x+p)-h{x). Thus A0.1) holds for an entire transcendental function
f(x) (x) if and only if f{x) is of the form
f(x)=x+ ? cme2nimxW.
m= — <x>
III. аф\. Put g(x) = x+f3/(l-a). Then the functions y/(x) = g~1(<p[g(x)]) = ax
and h(x) = g~1(f[g(x)\) also commute, and h(x) is an entire transcendental function.
The commutativity relation of h and y/ becomes
A0.27) й(ох) = ай(х).
Writing h{x) = YJhmxm, inserting into A0.27) and equating the coefficients, we
m = 0
obtain
A0.28) /гт(ат-а) = 0, w = 0,l,2,...
(') Let us note that in this case A0.1) cannot hold for a polynomials f(x) other than x+ B';
cf. theorem 10.3.

222 CHAPTER X. Commuting functions
If a were not a root of unity, then we would have hm=0 for m>2, contrary to the
assumption that h(x) is transcendental A). Consequently a is a root of unity, say a
primitive pth root of unity. Then it follows from A0.28) that hm=0 for all m except
those of the form m=kp + l. Thus we come to the conclusion that h(x) must have
the form
A0.29) h(x) = x ? ckxkp
1=0
(ck=hkp+1), and conversely, every function of form A0.29) satisfies A0.27). Hence
it follows that in this case A0.1) holds for an entire transcendental function f(x)
if and only if f(x) is of the form
\ \—olJ 1 — <x
where h(x) is of form A0.29).¦
If f(x) is a polynomial of a degree greater than one, then equation A0.1) has
(necessarily polynomial) entire solutions, their number, however, is denumerable
(Jacobsthal [1 ]). I. N. Baker [5] expressed the following conjecture.
If f{x) is an entire function, other than a polynomial of degree less than two,
then equation A0.1) has denumerably many entire solutions <p(x).
In favour of this conjecture he proved
Theorem 10.9. If fix) is an entire function, other than a polynomial of degree
less than two, and if fix) has a fixed point of some order which is either strongly repulsive
or of multiplier +1, then equation A0.1) has denumerably many entire solutions.
For the proof of this theorem the reader is referred to Baker's original paper
(Baker [5]).
This problem is closely connected with the question raised by P. Fatou [15],
whether every entire transcendental function has a strongly repulsive fixed point of
some order. An affirmative answer to Fatou's problem would establish the truth
of Baker's conjecture.
§ 4. Exponential function. If f(x) is an entire transcendental function, then
equation A0.1) has entire transcendental solutions: the iterates of f(x). But it may
happen that there are not other non-trivial entire solutions. A typical example is
provided by the exponential functions.
Theorem 10.10. (Baker [2].) Let f(x)=aebx + c, aby^O. If <p{x) is an entire
function commuting with f{x), then either <p(x) = const (a fixed point of f(x)),
or P(jc) =/"(*), «^0-
Proof. Suppose that #>(x)#const is an entire function satisfying A0.1). If <p(x)
is a polynomial, then by theorem 10.8 (p{x)=xx+P and A0.1) becomes
A0.30)
(•) It follows from theorem 10.4 that the only polynomials h(x) commuting with i/s(x)=ax
are those of the form h(x)=a.'x.

4. Exponential function 223
Expanding the functions in relation A0.30) in power series and equating the coef-
coefficients of x and x2 we obtain aab=aabebp, %aab2=%aa2b2ebfi, whence <x=l and
ebp=l. Equating the constant terms in A0.30) yields aa+ac+fi=aebl>+c, i.e. j8=0.
Thus (p(x)=x=f°(x).
Now we assume that cp{x) is transcendental, с is Fatou's exceptional value for
f(x) and by lemma 10.1 it must also be Fatou's exceptional value for <p(x). Thus
A0.31) <р(х)=с+(х-сГе"'м,
where j«>0 is an integer and y/(x) is an entire function. We shall distinguish two
cases.
I. ц>0. We obtain from A0.1)
whence
B7ii\ 2nmi
B7ii\
X+T)=
where m is an integer, m cannot depend on x, for
Ъ V f 27iiN
is an integral-valued continuous function of x and so it must be constant, m cannot
be equal to zero, since cp{x) = c if and only if x=c. Thus с—2тапЦЪфс, and con-
consequently (p{x) = c—2nmijb for infinitely many x, say x=x1,x2, ¦¦¦ But then for
x=Xi +2m/b, x2+2ni/b, ... cp(x) is equal to с on account of A0.32), which is impos-
impossible. Consequently, fi cannot be positive.
II. ju=O. Then A0.31) may be written in the form (p(x) = c+aeb<PlM=f[(p1(x)],
where (pi(x) is an entire function. From A0.1)
A0.33)
b
Again m is constant and may be assumed to be equal to zero, since otherwise we
replace ^(jc) by q>X{x) = q>l{x) + 2nmilb. (This is possible, since e*«"(x) = e»«>I(x)). Then
A0.33) expresses the permutability of/(x) and iptix): <p1[f(x)]=f[<p1(x)].
Now we may repeat the above argument: either <pt(x)=x and <p(x)=f(x)=f1(x),
or (p^x) =f [<p2(x)], where (p2{x) is an entire function commuting with/(x). Continuing
this procedure we arrive at a sequence of functions (pm{x) such that <pm(x) =f [<pm+x(x)]
and every (pm{x) is permutable with/(x). If for an integer n, cpn{x)=x, then cp {x) =f(x).
Otherwise the sequence (pm(x) is infinite and, for every m, (pm(x) is an entire trans-
transcendental function commuting with f(x). Moreover, <p(x)=fm[ipm(x)], whence by
lemma 0.15
A0.34)
holds for а в, О<0<1, and for r sufficiently large. Since <pm{x) is transcendental,

224 CHAPTER X. Commuting functions
MvJ6r)>r for large r. Thus we obtain from A0.34) М„(г)>М/т(г) for all m and
for r sufficiently large, contrary to lemma 10.2. Consequently, <p(x) =f(x) for some n.m
Forf(x)=e(x)=ex — 1 the result obtained may still be improved. Put
A0.35) <p(x) = <pu(x) = x+$ux2+ ? amxm
m=3
and insert cp(x) into A0.1) with f(x) = e(x):
A0.36) P[e(x)] =
Then the coefficients am=am(u) can be calculated successively (they are real for
real u), and thus there exists exactly one formal power series of form A0.35) (or
rather a one-parameter family of formal power series, и being the parameter) com-
commuting with e{x). We shall prove the following
Theorem 10.11. (Baker [2], [4].) Let (p(x) = (pu(x) be the formal power series
of form A0.35) commuting with e(x). <pu(x) has a positive radius of convergence if
and only if и is an integer.
Proof. If u=n is an integer, then <pu(x) is the expansion of e"(x) (this results
from the uniqueness of #>„) and has therefore a positive radius of convergence.
Now suppose that и is not integral and series A0.35) has a positive radius of
convergence r. By theorem 10.10 we have r< oo.
At first we show that <p{x) can be continued onto the halfplane H={x: Re x
>0}. Supposing the contrary, let x0 be a singular point of cp(x) such that Re xo>0
and <p(x) is regular for |x|<|xo|, Re x>0. For >>0 = Log(l +x0) we have |j>0|<|x0|,
Reyo>0A). Consequently, the function
A0.37) ?(x) = e(<plLoga+x)-])=e(<ple-1(xy\)
is regular at x=x0. But for |x| sufficiently small we have by A0.37) and A0.36)
y/(x) = <p(x), and thus y/(x) is an analytic continuation of cp(x) over x=x0.
The function e(x) maps H onto the domain D = {x: |jc+1|>1}. Let Do be
the domain obtained from D by cutting along the segment —oo<Rex^—2. Then
Log A +x) is regular in Do and assumes values in H. Consequently, function A0.37)
is regular in Do and yields an analytic continuation of cp(x) onto Do.
Now we put y_1 = n, yo = l, yn+1=sin у„ for n>0. The sequence у„ decreases
to zero. Write
Dnl = {x=-l+peie:
Xq xq Xo
(!) In fact, |Log(l+x0)|= < d\t\=x0, where denotes the integral over the
oo о
segment joining the origin with xo- Futher, Re Log(l +*o)=log|l + xo\ >0. Here Log(l + x) denotes
the branch of the logarithm which is regular on the plane cut along the segment — oo <Re x< — 1
and vanishes at the origin.

4. Exponential function
225
(cf. fig. 8). For x e Dn resp. x e D12 we have Im Log A +x)> 1 resp. Im Log A +x)
< -1. So for x e Z>nuD12 we have Log A +x) e Do and thus y/(x) yields an ana-
analytic continuation of cp{x) onto DouZ)uuiI2. Generally, for x e Dnl resp. x e Dn2
we have ImLog(l+х)>у„_1 resp. ImLog(l+x)^ -yn_1; whence
Log(l+xND0u"u Dklu"\J Dk2O-
k=i k=i
Thus we may continue successively cp(x) with the aid of function A0.37) onto the
whole complex plane cut along the infinite segment -oo<Rex<0. Hence it fol-
follows that on the circle \x\ = r there exists a unique singularity of <p(x) and this must
occur at x= — r. In the sequel we must distinguish two cases.
Do
H
--To
Fig. 8
I. и is real. If <pu(x) converges for |jc|<r, then cpu 1(x)hasthe expansion cpu x(jc)
= x—%ux2 + ..., which converges for |jc|<r', r'>0. Moreover, g>^1(x) commutes
with e(x). Therefore 9Ui(x) = <p_u(x). Thus we may assume that м>0.
For real x> — r, cp(x) is real. We shall show that cp(x)>x. Since [<p(x)—x]/x2
approaches ^м>0 as x->0,<p(x)>x for x sufficiently near the origin. If <p(x)>x
did not hold for — r<x<0, there would exist an xo<0 such that (p(xo)=xo and
<p(x)>x for xo<x<0. Put x1=e(x0) e (x0, 0). We have by A0.36) <p(x1) = e[<p(xo)]
= e(xo)=x1, contrary to the supposition that <p(x)>x for x e (x0, 0).
Thus <p(x)>x for —r<x<0. Now, e{—r)>—r and e{—r)> — 1, whence
it follows that <p[e(x)] is regular at x=—r and <p[e(—r)]>e(—r)> — 1. Conse-
Consequently, also the function
0) Note that for x e Dn\, x— — 1 +peW, we have lmx=p sin 0 and thus Imx<yn_v Similarly,
for x e Dn2, Imx>-yn_x.
15 Functional equations

226 CHAPTER X. Commuting functions
A0.38) V(x)=\og(\ + (p[e(x)] = e-\<p[e(x)])
is regular at x= — r and yields an analytic continuation of cp(x) along the negative
real axis over x=—r. This contradicts the supposition that x=—r is a singular
point of <p(x).
П. и is not real. At first we shall show that cp{x) is not real for real xe{—r, 0).
Supposing that <p(x0) is real for an x0 e(-r,0), we have by A0.36) that<p[e(x0)]
= e[<p(x0)] is also real and hence <p[e"(x0)] is real for и=0, 1, 2, ... But х„=еп(х0)
tends to zero as n->oo and thus the sequence [<р(х„)—х„]/х* of real values would
tend to \u, which is impossible.
Thus ?>[e(x)] is regular at x= — r and takes no real values on < —r, 0). As in
the preceding case, function A0.38) yields an analytic continuation of <p(x) over
x= — r, which contradicts the fact that x= — r is a singularity of ср(х).ш
The above theorem is a particular case of the following, more general situa-
situation (!). Let
CO
/(*) = *+ I anx", ат+1Ф0, т^\
n — m+ 1
be a formal power series. To every complex и there exists a unique formal power
series <pu{x) of the form
A0.39) <pu(x) = x + uam+1xm+1 + ? bn(u)x"
n = m+2
which formally satisfies A0.1) B). Let U be the set of those и for which series A0.39)
has a positive radius of convergence. I. N. Baker [4] proved the following
Theorem 10.12. The set U of и corresponding to the convergent series A0.39)
has one of the forms:
(i) the point u = 0;
(ii) one-dimensional lattice {nu0}, n = 0, ±1, ±2, ...,иоф0;
(iii) two-dimensional lattice {nuo + mu^, n,m = 0, ±1, ±2, ..., иоф0, и^фО,
uolu1 not real;
(iv) the whole complex plane.
As we have seen, the function f(x)=e(x) provides an example of case (ii).
The function f{x) = xj{ 1-х) yields an example of case (iv); here q>u{x) = xj(\—ux).
An example for case (i) is described in Baker [4]. No example is known so far for
case (iii).
A) Baker [4]. Cf. also Lewis [2], Silberstein [1], and also M. K. Fort [1].
B) In A0.39) the coefficients bn(u) are polynomials in и and series <pu(x) formally satisfy also
the relations <pu[<pv(x)] = (pu+v{x). Thus in case (iv) (cf. theorem 10.12 below) A0.39) forms a complete
analytic iterate off(x).

CHAPTER XI
SIMULTANEOUS EQUATIONS
§ 1. Biperiodic functions. It often happens that a single functional equation can
characterize several unknown functions (this is an especially characteristic feature
of functional equations in several variables; cf. e.g. Aczel [5], [7], Kuczma [24]).
On the other hand, it may happen that several equations are necessary to determine
a single unknown function. In such a case we shall speak about simultaneous equa-
equations, reserving the expression system of equations for the case in which we have
as many equations as there are unknown functions (Chapters XII, XIII).
A systematic theory of simultaneous equations does not exist—not even an
outline of such a theory. In the present chapter we shall treat some very special
functions satisfying simultaneously two or three functional equations of type @.49)A).
Except perhaps for theorem 11.4, the theorems proved here do not have that gen-
general character which we have tried to maintain throughout all the remaining chap-
chapters of this book.
Since the condition
A1.1) <p(x + co) = <p(x)
has form @.49), here belong theorems concerning periodic solutions of functional
equations B). We shall return to this question in the next section and now we turn
our attention to functions fulfilling two equations of form A1.1):
A1.2) <p(x + co1) = <p(x), <p(x + a}2) = <p(x), (О1(о2ф0.
Such functions may be called biperiodic C).
@ For special cases cf. e.g. Mertens [1], Cayley [2], Rausenberger [l]-[5], Guichard [1],
Wirtinger [1], Escherich [1], Moore [1], Bernays [1], Kuylenstierna [1], [2], Fatou [7], Maier [1], [2],
Hua [1], Ghermanescu [6], [16], [22], de Rham [1], [2], [3], Anan^a-Rau [1], [2], Adachi [2], Nafta-
levic [2], [3], [5], [6], [7], [11], Kabaila [1], Howroyd [1], Kuczma [37].
B) Esclangon [1], [2], Robbins [1], Wintner [1], Halanay [1], Kucatia, Szymiczek [1].
Here belong also investigations of so called Picard transcendents. These are functions <pv(x\
v=\,...,n, (of a complex variable) periodic with period o' and satisfying a system of equations
<p,,(x+co) = Rr((p1(x),...,<pn(x)), v=\,...,n,
where Rv are rational functions. In this connection cf. Picard [l]-[6], [8]-[10], Appell [3], [4], [12],
Ritt [6], Lowig [1], [2], Gerst [1], Valiron [6], Ghermanescu [22].
C) We distinguish here biperiodic from doubly periodic functions. Doubly periodic functions
occurring in the elliptic functions theory, are biperiodic functions of a complex variable for which
the ratio 01/02 of periods is not real.

228 CHAPTER XI. Simultaneous equations
Let x be a real variable. If co1lco2=plq is a rational number (p, q relatively
prime), then A1.2) is equivalent to A1.1) with co=(\jq)co2{n.2) results from A1.1)
in virtue of the equalities col=pco, a>2 = qco. On the other hand, since p and q are
relatively prime, there exist integers m,n such that mp+nq=\. Hence ma>1+nco2
= co and A1.1) results from A1.2).
Thus this case can be reduced to that dealt with in the preceding chapters.
Equation A1.1) has in (— oo, oo) a Cx solution depending on an arbitrary function
(theorem 4.1) and even the requirement of the analyticity of (p does not furnish a
unique solution (x). On the other hand, the functions #>(x) = const are the only
monotonic solutions of A1.1) (theorem 5.3) as well as the only solutions for which
the limit lim <p(x) exists (theorem 2.11).
X->OO
If, however, co1lco2 is irrational, then the constant functions are the only con-
continuous solutions of simultaneous equations A1.2). This results from the following,
more general theorem.
Theorem 11.1. (Montel [14], Popoviciu [2].) Ifco1lco2 is irrational and positive,
then the only functions cp(x) continuous at a point and satisfying simultaneously the
inequalities
A1.3)
for x 6 (— oo, oo) are <p(x) = const.
Proof. Let (p(x) be a function continuous at a point x0 and fulfilling A1.3).
Let us fix arbitrarily an x1 and suppose that ^(x1)>^(x0). There exists a <5>0
such that
A1.4) (pix^xpix) for |x-xo|<<5.
The set of numbers of the form nco2 — mco1, where n, m are arbitrary positive inte-
integers, is dense in ( —oo, oo). Consequently, there exist positive integers p, q such
that
A1.5) |x1-x0-p<y2 + ^u;1|<5.
Writing x2 = x1—pco2 + qco1 we have by A1.5) and A1.4) <p(x1)>g>(x2). On the
other hand, x2 +pco2 = xt + qcox, whence in view of A1.3)
which is a contradiction. The supposition ^(хх)<^(хо) similarly leads to a con-
contradiction, and consequently we must have <p(x1) = <p(x0). Since xt has been chosen
arbitrarily,
С1) Е. g. A1.1) is fulfilled by the trigonometric polynomials
V / 2kn 2kn \
CW= h %cos—x+o^sin—x\
k—n \ ы <O I
¦which are all analytic.

1. Biperiodic functions 229
The hypothesis of the continuity of cp at at least one point cannot be altogether
omitted. Let A be the set of points of the form nco2 — mco1, where n, m range over
the set all integers. A is thus denumerable. We define the relation p for real numbers
as follows: x p у if and only if x—у е А. р is an equivalence relation. Let E(x) denote
the set of real numbers equivalent to a given x. Then the general solution of simul-
simultaneous equations A1.2) is evidently given by
where Ф is an arbitrary function defined on the (non-denumerable) quotient space
(—oo, oo)/p. Since every set E(x) is dense in (—oo, oo), this gives another proof
of the fact that constant functions are the only continuous A) solutions of A1.2),
and this proof is valid for functions cp with ranges in an arbitrary metric space.
Now let x be a complex variable. If q}1Iq}2 is real, then we may consider equa-
equations A1.2) independently on every straight line passing through the points xa
and Xo+cox, with an arbitrary complex x0. On every such line L we may introduce
a real parameter and the case reduces to that considered previously. In particular,
we have
Theorem 11.2. If co\\co2 is a real irrationalnumber and (p(x) is a function analytic
on the whole complex plane except isolated singularities and (p{x) satisfies equations
A1.2), then q> (x) = const.
Proof. <p{x) must be constant on every line L and hence, being analytic, must
be constant everywhere.¦
Here the supposition of continuity is not sufficient, as can be seen from the
example of the function <р(х)=ФAтх) (with an arbitrary continuous complex-
valued function Ф of a real variable), which satisfies A1.2) with arbitrary real cot
and a>2-
If со!/со2 is not real, then cp(x) is a doubly periodic function. Thus the whole
theory of elliptic functions belongs here. This, however, goes far beyond the scope
of this book. We close this section with the following
Theorem 11.3. (Montel [3], [5], [8]-[12].) If <p(x) is an entire function satisfying
simultaneously the equations
<p(x + co1) = <p(x), (p(x + co2) = (p{x), <p(x+co3)=<p(x),
and there exist no integers p, q,r such that pco1+qco2 + rco3 = 0, then cp{x) is con-
constant.
Proof. It is impossible for c^, co2, co3 to be all equal to zero. Let e.g.
«з/О. If cojcoi and CO2/CO3 are both real, then at least one of these ratios must
be irrational, and the statement results from theorem 11.2. If one of the ratios is
not real, then cp{x) is a doubly periodic entire function and thus a bounded entire
function, whence it must be constants
(') It is enough to assume continuity at a single point.

230 CHAPTER XI. Simultaneous equations
§ 2. Periodic solutions of functional equations. Here we shall prove a theorem
concerning the uniqueness of continuous periodic solutions of equation A.1). Un-
Unfortunately, there exists no satisfactory theory of periodic solutions of functional
equations; the existence is established only in some particular cases A).
Let/(x) fulfil the following
Hypothesis 11.1. /e R°[f\, /=<?, oo), lim/(x)= oo, and for every a, b, %<a<b,
lim sup [/ " "(b) -f ~ "(a)] >co > 0.
Lemma 11.1. If f(x) fulfils hypothesis 11.1, then for every xe I the set of points
f(x + mco), n, m = 0, 1, 2, ..., is dense in I.
Proof. Let us fix arbitrarily x,a,be I,%<a<b. It follows from hypothesis
11.1 that/""(a)->oo as n->oo; therefore/""(a)>x for n>N1. Further, there exists
an N>NX such that f~N(b)—f~N(a)>co. Consequently, there exists an integer
M>0 such that f~N(a)<x+Mco<f~N(b). Since fN(x) is increasing, just like f(x),
we obtain hence a<fN(x+Mco)<b. Since a and b have been arbitrary, this proves
that the set of points f(x+mco) is dense in (?,oo), and hence also in <<J, оо)=/.и
Theorem 11.4. (Kuczma, Szymiczek [1].) Let f(x) fulfil hypothesis 11.1. Then
for every n fulfilling equation C.24) there may exist at most one function <p(x) con-
continuous in I, satisfying equation A.1) and condition C.15) and periodic with pe-
period CO.
Proof. Let <p(x) be a function fulfilling the conditions of the theorem. By
C.15) and A1.1) we have <р(^+тю)=г] for m=0, 1,2, ... By A.1) we obtain
A1.6) <р[/"(.€ + та>У1 = вА€ + та>,ч), m, n=0, 1, 2, ...,
where д„(х,у) are defined by A1.4) B). According to lemma 11.1 <p(x), being con-
continuous, is by A1.6) uniquely determined in the whole of/. (The periodicity condi-
condition allows us to extend cp(x) uniquely onto the whole real axis.)B
Remark. If the function g{x,y) has a unique inverse with respect to y, then
in the above theorem it is enough to assume that cp(x) is continuous in a (right)
neighbourhood U of ?, instead of continuity in the whole of /. The above argu-
argument then yields the uniqueness of (p in U. Now, for every xe I there exists an и
such that /"(*) e U (theorem 0.4) and we have <p(x)=g_n(f"(x)> ?>[/"(*)])' Con-
Consequently, <p(x) is uniquely determined in the whole of/.
(!) Cf. e.g. Fortet [2], Kac [1], Ciesielski [1], Wintner [1]. One is also led to a study of periodic
solutions of linear functional equations by some classical problems of mathematical analysis, like
the restricted three body problem, the mapping of a circle or of an annulus onto itself, etc. In this
connection cf. Arnold [1], [2], [3], J. Moser [2], [3], [4], Birkhoff [3], [4], Kolmogorov [1], [2],
Moiseev [1]. In all these investigations fairly complicated methods are used; lack of space prevents us
from presenting them here.
B) Note that the existence of cp is here assumed and therefore nothing is supposed concerning
g(x, y). The existence ofgn(Z+mco, if) results directly from the existence of p and from A.1).

2. Periodic solutions of functional equations 231
As an application we prove the following characterization of the cosine.
Theorem 11.5.0 <p(x)=cos x is the only function defined for all x, continuous
in a neighbourhood of x = 0, satisfying equation D.76), periodic with period 2n and
such that
A1.7) (p(x)>0 for -±71<Х<|т1, (р(х)^0 for |
Proof. By A1.7) and the periodicity condition the sign of cp{x) is unambig-
unambiguously determined for all x. Thus cpQx) can be expressed on the basis of D.76) by
<p(x). ц—\ is the only root of D.77) which agrees with conditions A1.7). Theorem
11.4 with the subsequent remark yield the uniqueness of a function fulfilling the
conditions of the present theorem. On the other hand, #>(x) = cosx is known to
fulfil all these conditions. ¦
We do not know how far the continuity condition is essential for the validity
of the above theorem. It may also be proved (Kuczma [26]) that conditions A1.7)
can be replaced by
(П.8) <р(хо) = Уо
with a suitable xoe(— oo, oo) and j>0 = cosx0. However, we are unable to find
effectively such an x0. It is also an open problem for which y0 e < — 1, 1> equation
D.76) has a continuous and periodic solution 0>(x)(with period 2л) fulfilling A1.8).
Such a solution, if it exists, is unique. But it is known that for one dense set of values
y0 e < — 1, 1> such a solution exists and for another dense set of values y0 it does
not.
The sine function can be characterized by similar conditions.
Theorem 11.6. (Dubikajtis [1].) ^(x) = sinx is the only function defined for
all x which is odd, continuous in a neighbourhood of x = 0, satisfying the equation
A1.9)
and such that
A1.10) ?(x)>0 for 0<x<%n.-
Proof. Let y/{x) be a function fulfilling the above conditions. Setting y=— z
=in—-J-x we obtain from A1.9)
Since ? is odd, we have [?(y)]2 = [?(z)]2, whence
A1.11) ?(n-x) = ?(x).
(!) Kuczma [26]. Cf. also Robbins [1] for another characterization of cosx as a continuous
periodic solution of a functional equation.

232 CHAPTER XI. Simultaneous equations
Next
= y/( — к — x) = — у/(к + х)
where A1.11) has been used twice, and twice we have used the fact that y/ is odd.
Consequently, y/(x) is periodic with period 2тс. We also note that in view of A1.9)
y/(x) is continuous in a neighbourhood of x = ^n.
Now, the function <p(x) = y/Qn — x) is continuous in a neighbourhood of
x = 0 and periodic with period 2n, just like y/(x). Further,
2 0 (x)f = 2 О фи - x)]2 = 1 - у, Bх - ^ти) = 1 + у, фи - 2x),
whence
i.e. <p(x) satisfies equation D.76). Relations A1.7) result from A1.10), A1.11) and
from the periodicity of y/, since y/ is odd. Thus in virtue of theorem 11.5 q> (x) = cos x,
whence ^/(x) = sin x.m
Although the periodicity of y/(x) results from A1.9) and from the condition
that y/(x) is odd, the latter condition cannot be replaced by periodicity (counter-
(counterexample: y/(x) = ^). Of course, we may replace the condition that y/(x) is odd and
A1.10) by
for 0^x<7i, y/(x)^0 for
and the condition that y/(x + 2n)=y/(x). If we drop only condition A1.10), then
the functions
y/ (x)=(- if sin (Bk+l)x), k an integer,
fulfil all the remaining conditions of theorem 11.6. It is an open question whether
these are the only odd and continuous solutions of equation A1.9).
§ 3. Further properties of the Gamma function ('). In Chapter V, § 10, we de-
defined Euler's Gamma function Г(х) as the unique logarithmically convex solution
of equation E.68) fulfilling condition E.69). Now set
A1.12)
The function y/(x) is logarithmically convex in @, oo); moreover,
Thus y/(x) may differ from Г(х) only by a multiplicative constant, i.e.
A1.13) 2xrax)r(Hx + l))
The constant d= 2Г(%) will be determined later.
(!) Artin [1], [2]. Cf. also Jensen [1], Godefroy [1], Lerch [4], Nielsen [3], Graf [1], Courant [1],
Anastassiadis [7].

3. Further properties of the Gamma function 233
Now we shall show that properties E.68) and A1.13) characterize the Gamma
function in the class of differentiable functions. For this purpose we prove first
Theorem 11.7. (Artin [1], [2].) The only C1 solution of the equation
A1.14) a^-Ua^^VaW, xe@,oo),
which is periodic of period 1, is a(x) = 0.
Proof. Replacing in A1.14) x by %(x + k) we obtain
whence, summing over k=0, 1 we obtain by A1.14)
By induction
and
As p-+ oo, the left-hand side in the above relation approaches the definite integral
and so
a'(x)= J a'(t) = a(x+l)-a(x) = 0,
X
since a{x) is periodic. Hence a(x) is a constant, and in view of A1.14) the constant
must be zero.B x
The example of the function <x(x) = ? 2"" sin Brx) shows that in theorem
n= 1
11.7 it would not be sufficient to assume that <x(x) is continuous.
Theorem 11.8. (Artin [1], [2].) If a positive function peC'[@, oo)] satisfies
equation E.68) and
then <р(х) = Г(х).
Proof. The function a(x)=log^(x)-logr(x) fulfils the hypotheses of theo-
theorem 11.7.И „
Equation E.68) allows us to extend the function Г(х) onto [J (—и—1, -n).
n = 0
Then also function A1.12) is defined there and A1.13) holds everywhere except
non-positive integers.

234 CHAPTER XI. Simultaneous equations
Set Р(х) = Г(х) ГA — x) sin roc. This function is defined everywhere except at
x= —л, n = 0, 1, 2, ..., and is periodic with period 1:
so that it can be extended onto the whole real axis. Moreover, it is of class C1 in
00
\J(n-l,n). Since
— 00
sin roc sin roc
Г( 1)ГA)
A1.15) р(х) = хГ(х)ГA-х)
X
P(x) is of class C1 in (—00, 00). According to A1.13)
x
~2
whence
x\ /x + l\ /x\ ( x\ roc fx + l\ fl-x\ roc
)P) Л —m1 sin-Г Г cos —
2j \ 2j 2 \ 2 ) \ 2 ) 2
= ~ Г(х) d2x-ir(l-x)$sm nx^p(x) .
4
Consequently, the function a(x)=log —P(x) fulfils the hypotheses of theorem
11.7, whence it follows that P(x) = d2/4. Setting x=0 in A1.15) we obtain by E.69)
/?@)=7i, whence P(x) = n and d=2-Jn. So A1.13) becomes
This formula is due to Legendre.
More generally, one can prove the following relation (due to Gauss and
known as Gauss' multiplication formula)
p = 2, 3, 4, ..., where dp=pll2Bn)(p 1)/2. Whereas each of the equations
A1.17) px П <P\—-\ = dB<p(x), p=2,3,4,...,
j=o \ P )
has continuous solutions satisfying E.68) but different from Г(х), it may be proved
that (р{х) = Г(х) is the only continuous solution of equation E.68) fulfilling all
the equations A1.17) (Artin [1], [2]).

3. Further properties of the Gamma function 235
The equality ji{x) = n also immediately gives the relation
к
A1.18) Г(х) ГA-х)=-.
sin nx
discovered by Euler. A1.18) is sometimes referred to as Euler's functional equa-
equation.
We close this section with deriving Stirling's formula. Put
A1.19) (p(x) = xx-ll2e-x+"(x).
If fie M+[@, oo)], then function A1.19) is logarithmically convex (since [log (p{x)]"
= x~1 +^x~2+fi"(x)>0 in @, oo)) and if ц(х) satisfies the equaton
A1.20)
then <p(x) satisfies equation E.68). Since FeMl[0, oo)] and HmF(x)=0, equation
Х-ЮЭ
A1.20) has a one-parameter family of solutions ц е М+1 in @,oo) (cf. theorem 5.11).
This must be identical with the unique family of monotonic solutions of A1.20)
(cf. theorem 5.3):
A1.21)
n=0
Now, setting y= l/Bx+1) in the expansion
2^logW „e0
(valid for |y\ < 1) gives
00 1 1
 „e1Bx.+ lJn>
x e @, со), which yields the estimation
1 1
12(x + l)
Consequently, the series ? F(x + n) converges and
n = 0
(П.22)
Choosing in A1.21) t]0= ~Y<F(xo+n)> we obtain a convex solution of A1.20)
n = 0
(П-23) /*(*)=- I
n = 0

236 CHAPTER XI. Simultaneous equations
where in view of A1.22)
A1.24) O<0(x)<l.
If we set A1.23) into A1.19), (p{x) will be a logarithmically convex solution of equa-
equation E.68), whence (р(х) = сГ(х), or (with b = c~l)
A1.25) Г(х)=Ьхх-1/2е-х+в(х)П2х.
In order to determine b we insert A1.25) into A1.16):
where by A1.24) гг, r2, r3 tend to zero as x->cc. Hence
A1.26)
X * I *•
The exponent on the right-hand side of A1.26) tends to — | as x->oo. Oh the other
hand, the coefficient of e in A1.26) may be written in the form
1
1+—
X
Hence b = \/2n and we obtain from A1.25) the Stirling formula
ЩТ7Л rv \ /o x—1/2 — jc + 0(jc)/12jc
.z/j i \X) = ^jj.Tix e
where 0(x) fulfils A1.24).
§ 4. A continuous curve filling a square. Here we shall present a simple example
of a continuous curve with a positive area, due to W. Sierpinski [3] (').
Theorem 11.9. There is a unique function (p{t) which is bounded, even and satisfies
the functional equations
A1.28) <p(t) + q>(t + j) = O for every te(— 00,00),
A1.29) 2<p(\i) + <p(t+^) = \ for fe<0,l>.
Proof. Let (p{t) be a function fulfilling the above conditions. Using relation
A1.28) for t and for t+%, we obtain
A1.30) <p(t+\)=<p(t) for fe(-oo,oo).
Replacing t by At in A1.29), we get
A1.31) (p(i)=\—;^(|+4f) for
(!) Another example of a continuous curve filling a square and defined with the aid of simulta-
simultaneous functional equations is discussed by G. Andreoli [5].

4. A continuous curve filling a square
237
Replacing t by ±-t in A1.31), we obtain by A1.30) ?>(i-f) = i-
= 2-2 ^(i-40 for fe <?,?>. Since <p is even, we have by A1.28) <p(t) = q>(-t)
= — 4>(.2~0- Hence
A1.32)
for
Similarly, replacing f by f — | in A1.31) and A1.32) and making use of A1.28) and
A1.30), we obtain
A1.33) q>(t)=-\ + j<pQ + 4t) for
A1.34) g>(t) = j — jq>(^—4t) for fe<|,
respectively. Introducing functions/(f) and g(t) defined by
5@ = 1 for
0@=-1 for
5@=-1 for
5@=1 for
A1.35)
and
A1.36) /0+l)=/@, 5(' + l) = 5@, *e(-oo,oo),
we may write relations A1.31 )-(l 1.34) uniformly as
A1.37)
In virtue of A1.30) and A1.36), A1.37) is valid for all t e (-oo, oo).
We have |^(O(l-j)|=i|l-j|<max A, \s\). Thus by theorem 3.5 the func-
function
A1.38)
n=0
П
i=0
is the unique bounded solution of equation A1.37). (In formula C.11) we take
(po(x) = 0) Now we shall show that function A1.38) actually fulfils all the condi-
conditions of the present theorem.
Function A1.38) is bounded; more exactly, we have
A1.39)
fe(-oo,oo).
<p{t) satisfies equation A1.37). By A1.38) and A1.36) <p{t) fulfils A1.30).
It follows directly from A1.35) and A1.36) (although there are several cases
to consider) that
A1.40)
A1.41)
/(-0=/@ + 4, 5(-0 =
for
for
te(-oo,co),

238 CHAPTER XI. Simultaneous equations
A1.40), A1.30) and A1.37) give A1.28). From A1.41), A1.30), and A1.37) we obtain
A1.42) <p(-i) = <p(i)
for t e @, i)u(i, i). For t = 0 relation A1.42) is trivial.
For t=i, A1.37) gives p(i)=i[l-p(i+i)]=i[l+p(i)] by A1.28). Hence
p(I) = l and ^(i)=-i[l-K|-l)]=-|[1-^I)] = 0-
Similarly,
Thus relation A1.42) has already been established for te<0,%). For fe<?, 1)
we have f-±e <0, i) and by A1.28) and A1.30)
Thus A1.42) is valid in <0, 1), and hence, by A1.30), in (-oo, oo).
A1.37) implies A1.29) for /e<0, 1). For /=1 we have
so A1.29) holds also for t = l.a
Now we shall establish further properties of function A1.38).
Lemma 11.2. The inequality \t-t'\<2~4n implies
|p@-p(Ol<2-"+2, «=1,2,...
Proof. For n=\ we have by A1.39) \<p{t)-<p(t')\^2. Suppose that the lemma
is true for an л>1 and let |f—f'|<2~4ll~4<2~8. We distinguish two cases.
I. There exists an integer к such that t,t'e (~k,~(k+l)). Then for an
integer / we have t, t' e Ql, i A+1)>. According to A1.37) and A1.30)
A1.43) \<p(t)-<p(t')\ = 12kU(t)']-<PU(O']\ = lk^t±l)-<pDt'±l)\.
Now we find that 11 = At±? and t[ = At'±% both belong to an interval (\m,i(m+1)>,
m being an integer. Repeating the above argument gives
(Ц-44) И'1)-ро;)|=*И2)-ро;)|
with t2 = Att±i, tz = At[±\. Since |f2-^| = 4 \^-г[\ = \6 |r-f'|<2"", we have
on account of the induction hypothesis \q>(t2) — ^02)|<2~"+2, whence by A1.43)
and A1.44)
Thus in this case the lemma is valid also for n +1.
II. There exists an integer к such that
к< <&+l r
Then for f and t0, and also for f0 and t', we may apply the first part of the proof,
which gives
Р('о)|<2~" and

4. A continuous curve filling a square 239
whence
and again the lemma is valid for n+ 1. Induction completes the proof.¦
Corollary. Function A1.38) is continuous in (— oo, oo).
Theorem 11.10. (Sierpinski [3].) The equations
A1.45) x = (p(t), y = <p{t-\), fe<0,l>,
where <p{t) is the function occurring in theorem 11.9, define a continuous curve filling
the square g = <-l, 1>х<-1, 1>.
Proof. Equations A1.45) define a continuous curve Чё lying in Q (cf. A1.39)).
Moreover, by A1.30), for every value of te (—00,00) the corresponding point
(x, y) lies on <8.
Let (x, y) e & and let t be the corresponding parameter. Put t1=$ — t,t2 = l — t.
Then by A1.28), A1.30) and A1.42)
i.e. ( — x, у) е &, and similarly
i.e. (x, -у) е <ё. Consequently, <? is symmetric with respect to both the coordinate
axes.
Now we shall show that the point @, 0) lies on <g. We have from A1.29) for
t = 0 p@) = 0, since p(i)=l. Consequently, xo = 9>@) = 0,yo = 9'@-i) = 9'(i) = 0
lies on <%.
Lete1=(-l,l)x(-l,l), ef. = (l_/,2-/)x(l-;, 2-Д/,y=l,2, andlet
A1.46) 2u-l = x, 2d-1 = j.
We shall prove that (x,y) etfnQ1 if and only if (м, г) е
So let (x.jOetfng1, х = р@, У = 9>(*-*), te(fl, 1). If te<0,|), we put
On the other hand, |<i4-|< 1, whence
So (м, r) e ^, and it follows from A1.46) that (u,v)eQ2u.

240
CHAPTER XI. Simultaneous equations
If t e <|, 1), we put s=\ t-~ and, as in the preceding case, we verify that
(u,v)e<enQ211.
Now let (и, г;) e #ng^, u = (p{s), v = (p{s-%), s e <0, 1). Thus м>0, г;>0. From
A1.37) and A1.39) we obtain the inequalities
which are possible only if je@,i). Now we put t=4s-\. We have by A1.30),
A1.28) and A1.31)
Since J+f e(f, 1), we have by A1.42), A1.28), A1.30), A1.34) and again A1.30)
Thus (x,y)e<? and it follows from A1.46) that (x,y)e
1
Fig. 9
This proves that the sets 'enQ1 and ^nQ^ are similar, the similarity being
established by formulae A1.46). Because of the symmetry of (€, the set i^nQ1 is
similar to VnQfj for i,j=l,2. Hence it follows that VnQ1 is similar to

4. A continuous curve filling a square 241
/,У=1, ...,2й, where
Since the centre of Q1 belongs to (€, the centres of all Q"j belong to (€. Thus #
contains a set dense in Q. On the other hand, # may be regarded as a continuous
map of the compact set <0, 1> and hence ^ is a compact set. Consequently Qc<e.m
It can be shown that <ё (sometimes called Sierpinski's carpet) may be obtained
as the limit of a sequence of closed polygonal lines <?„. (The first few members of
the sequence are represented in fig. 9). If x=(pn{t), y=y/n(t) are equations of #„,
where t is the natural parameter (arc length) measured in the positive direction from
the point A where #„ intersects the positive part of the x-axis, then we have y/n(t)
= <pn(t — ^), and the sequence q>n(t) converges to a function <p(t) fulfilling the condi-
conditions of theorem 11.9.
§ 5. Cantor's singular function. The following theorem is also due to W. Sier-
pinski [2], and its proof is similar to that of theorem 11.9.
Theorem 11.11. There is a unique function <p{x) which is defined and bounded
in <0, 1> and satisfies for x e <0, 1> the functional equations
A1.47) q>(jx) = ^q>(x), $9[i(x+l)]=i, <p[^(x + 2)~\ = ^+^q>(x).
This function is continuous, increasing, and possesses a dense set of intervals of con-
constancy.
Proof. Suppose that a function (p(x) fulfils the above conditions. We extend
<p{x) onto <0, oo) by putting (p{x) = \ for x> 1. Let
for 0^x<§,
for §^x,
A1.49) f(x) = 3x-2F(x).
Then <p{x) satisfies for x^O the functional equation
С11-48) П*) = ^ ,__ a.
A1.50) 9(x)
On account of theorem 3.5 the function
A1.51)
f
n=0
is the unique bounded solution of equation A1.50). We must verify that it has the
properties stated in the theorem.
For x>l we have/(x)>l and/"(*)> 1 for n = 0,1,2, ... Hence F[fn(x)] = l
and (p{x)= 1. For j<x<§ we have F(x) = 0,/(x) = 3x> 1, whence by A1.50) <p(x)=-\.
For x = l,F{x)=\ and f(x) = 0, whence <pQ) = ±+q>@). Setting in A1.50) x = O we
obtain <p@) = 0 and so <p{x)=^ for |<x<|. This proves the second relation of
A1.47). The first and the third result immediately from A1.50) and A1.48), A1.49).
16 Functional equations

242 CHAPTER XI. Simultaneous equations
Now we prove that g>(x) is increasing (not strictly increasing, of course!) Let
and write an=F[f"'1(x)], bn = F\Jn'1{y)]. Then according to A1.51)
bi b2 b3
^+22 + 25 + '"
If (р(х)ф<р(у), there exists an index p^\ such that арфЪр, ai = bi for i<p. Since
f(x) = 3f~1 (x) — 2an, /"GO = 3f~1(j)- 2bn, we obtain f\x) <fl{y) for i = 1, ...,
77-I, whence ap=F[fp'\x)]^F[fp''1(y)] = bp. Since ap#fep, we have ap<bp.
But the only possible values of а„, bn are 0 and 1, and thus we must have ap=0,
bp=\. Hence
1 1
i.e. ^(лО^ОО. Thus 9) is increasing from ^@)=0 to g>(l)=l.
Finally, we prove that
A1.52) ,(x)=^ for xe{^,l^),k=0,...,2^~l;
where « = 1,2,3,...,
A1.53) /0 = l, 1к+2т-2=1к+2-Зт~1, к=0,...,2т-2-1; m=2,3,...
It is clear from A1.53) that 0</t<3m for fc=0, ..., 21" -1. Consequently,
3m-1-l for fc=0,...,2m-2-l,
A1.54) m = 2,3, ...
г-З^Ч/^З"-! for fc=2m~2 2"-!,
Now, for m=\, к must be 0, and A1.52) reduces to the relation (p(x)=^ for
xe<?, |>, already proved. Suppose that A1.52) is true for an m>l and fc=0,...,
2m~l-\. Let us fix a fc, 0^fc^2m-l. If k^2m'l-\, then by A1.54)
0</4<3"-l. For xe<lkl3m+1, D+l)/3m+1>c@, i) we have 3xe<4/3m,
D+ l)/3m> с @,1) and by the induction hypothesis ^Cx) = Bfc+ l)/2m. On the other
hand, from A1.47) we obtain ^(x)=^Cx) = Bfc+l)/2m+1, i.e. A1.52) for m + 1.
If k^2m~\ then by A1.54) 2-3m</t<3m+1-l. For
we have
x
and
О 7 О ^ "^ 7 О ^ "^ | ^ T 7 i I / 1 \
/ \ / ~~ ~~ \ _ i /\ 1
з"е\ 3»i+i ' ' irn+i / = \ 3m+1 ' ~ 3m+1 / I ~3 }

5. Cantor's singular function 243
By the induction hypothesis <pCx-2) = Bk-2m+l)/2m=Bk+l)/2m-l and by
A1.47) 4>(x)=i+^Cx-2) = Bk+l)/2m+1, i.e. A1.52) holds also for m+1. Thus
relation A1.52) has been proved.
The intervals </t/3m, (lk+ l)/3m> lie densely in @, 1). Moreover, it follows from
A1.52) that <p(x) assumes all the values of form k/2m, 0^fc<2m, m=l,2,3, ...,
which lie densely in @, 1). Consequently, q>{x), being monotonic, must be continuous
(otherwise it would have jumps). ¦

CHAPTER XII
EQUATIONS OF HIGHER ORDERS AND SYSTEMS OF EQUATIONS
§ 1. Systems of equations. In this chapter and in the next we shall present
some fragments of the theory of continuous solutions of equations of higher orders
and systems of equations. A more complete treatment would cover enough space
to form a separate book. We shall not endeavour to give a construction of the
general solution of equations of higher orders A), nor (except for a simple example
in § 5) shall we deal with differentiable or analytic solutions B).
The systems
A2.1)
can be dealt with in much the same manner as in Chapter III. Denoting by bold-
boldface type the points of an и-dimensional real space 9Г, we may write A2.1) as
A2.2) (Plf(x)]=g{x,(p(x)),
where <p = {q>i q>n) a.ndg=(g1, ...,д„). Besides systemA2.1) we may also consider
a system solved with respect to <Р;(х), which may be written shortly as
A2.3) <p(x)=h(x,<p[f(x)]).
For u, v e 91" we write
A2.4) |e-»l=i>i-»il-
A) An attempt to do this may bs found in Presi6 [2].
B) Concerning equations of higher orders and systems of equations cf. in particular Holder [3],
Bertrand [2], Poincare [3], [4], Grevy [1], Oltramare [1], [2], Picard [3]-[5], [8]-[10], de Brun [1],
Levi [1], Bottcher [7], Lattes [9], [13], Giraud [l]-[3], Sincov [1], Appell [12], Onicessu [1], Colombo
[1], [2], Popovici [2], [3], [10], Lowig [1], [2], Trjitzinsky [3], Touchard [2], Birkhoff [6], van der
Corput [2], Hukuhara [1], Beckenbach [1], Strodt [1], [3], Bochner, Martin [1], Aczel [3], Urabe [1],
Ghermanescu [9], [17], [22], Valiron [6], Samoloff [1], Panov [l]-[8], Bajraktarevic [8], [9], [14],
Sternberg [2]-[4], Azpeitia [1], [2], Hartman [1], [2], Kordylewski, Kuczma [3], Kordylewski [1], [3],
Choczewski [1], PreSic [2], Naftalevic [10], Leonov [1], Mitrinovic, Dokovic [1], SzegS, Kalman [1],
Chen [1], Kuczma [23], [29], [24], Majcher [1], Tauber [l]-[4], W. A. Harris, Sibuya [l]-[4], Artiaga
[1], Mitrinovic, Vasic [1], Vajzovic [1], Kwapisz [1], Reghis, Vuc [1], Adamczyk [1], W. Smajdor
[2], [3], Krzeszowiak [1].

1. Systems of equations 245
Then the results of Chapter III remain valid for equations A2.2) and A2.3) without
any essential changes in the proofs. As an example we are going to formulate anal-
analogues of theorems 3.1, 3.4 and 3.7(x).
Theorem 12.1. LetfeR°[I], ?ф1, and let hypotheses 3.1, 3.4, 3.6 and 3.7 be ful-
fulfilled. Then equation A2.2) has in I a continuous solution <реФ[Г\ depending on an
arbitrary function. More precisely, for any x0 el and an arbitrary continuous function
<po(x) belonging to Ф[10], where I0 = (x0,f(x0)y or <f(xo),xoy, and fulfilling the
condition
<Po [f(x0)] =g{x0, <Po (x0)),
there exists exactly one function q> (x) belonging to the class Ф [I], satisfying equation
A2.2) in I and such that <p(x) = <po(x) in Io. This function is continuous in I.
Theorem 12.2. Suppose that feS°[I], ?,el, and let hypotheses 3.1, 3.2 and 3.5
be fulfilled. Further, let r\be a root of r\=h(^,,r\) such that t\eQ^ and suppose that
the inequality
A2.5) \h(x,yi)-h(x,y2)\^ \У1-у2\,
holds in a neighbourhood of (?, j/). Then there exists exactly one function <p (x) which
is continuous in I, belongs to Ф[Г\, satisfies equation A2.3) in I and fulfils the condi-
condition
A2.6)
This function is given by tp{x)= \imtpk{x), where (pk+1(x) = h(x,(pk[f(x)]) and <po(x)
k-tao
is an arbitrary continuous function belonging to Ф[Г\ and fulfilling condition A2.6).
Theorem 12.3. Let feR°[I], ? el, and let hypotheses 3.1, 3.4, 3.6 and 3.7 be
fulfilled. Further, let ц be a rcot of ц =g(E,,ti) such that ц е Q^ and suppose that the
inequality
A2.7) \g(x,y1)-g(x,y2)\^S\y1-y2\, 0<5<l,
holds in a neighbourhood of{?,, j/). Then equation A2.2) has in I a continuous solution
<p e Ф [I] fulfilling A2.6) and depending on an arbitrary function.
The remarks in the footnotes on pp. 73 and 76 apply also in this case.
In spite of the apparent perfect analogy between the above theorems and those
in Chapter III, the theorems in this section give us less information about the behav-
behaviour of continuous solutions of equations A2.2) and A2.3) than do the correspond-
corresponding theorems in Chapter III relatively to equations A.1) and A.2). This is due to
the fact that we now have more cases which do not come under condition A2.5)
or A2.7). We shall see this better in the next chapter, where the case of a linear
equation with constant coefficients will be discussed in detail.
@ In the hypotheses one must replace у by у, д(х, у) and h(x, y) by g(x, y), h(x, y). The sets
Qx, Гх, Лх , and the function class Ф[У] are-defined formally as in Chapter III, §1, but this time
Qx, Гх, Лх are subsets of 3?n.

246 CHAPTER XII. Equations of higher orders and systems of equations
An important case of a single equation of order n (Tauber [2], Kordylewski [1])
A2.8) Pl№)]=flr(*, ?>(* 1)
may be reduced to system A2.1) with
129
gn(x, y1,...,yn)=g(x,y1,...,yn).
Namely, we have the following evident
Theorem 12.4. Let f(x) be defined in a submodulus interval I. If a function <p{x)
satisfies equation A2.8) in I, thenthe system of functions q>i{x) = (p\fi~l{x)], i=\, ...,n,
satisfies in I the system of equation A2.1) with gt defined by A2.9). Conversely, if a
system of functions <Pi(x), i=\, ...,n, satisfies in I the system of equations A2.1)
with the right-hand sides defined by A2.9), then the function <p{x) = q>1{x) satisfies in
I equation A2.8).
If each of the equations in A2.1) can be solved with respect to any one of the
functions q>i{x), then also system A2.1) may be reduced to an equation of form
A2.8).
§ 2. Equations and systems of equations of order m. Now we are going to deal
with equations of the form
A2.10)
or
(i2.il)
). Here bold-face type denotes points of 9Г(и>1) endowed with metric A2.4).
We assume / to be a fixed real interval and Q to be a fixed connected set contained
in 9Г; for a given interval J we denote by <&[J\ the class of thoss functions <p{x)
which are defined in J, and for each x e J we have <p(x) e Q.
Hypothesis 12.1. The function h{x,ylt ...,ym) is defined and continuous in
the set
A2.12) E={(x,ylt...,yJ: xeI,y^Q, ... ,утеп}
andh(x,ylf ...,ут)еп for(x,yu ...,ym)eE.
Hypothesis 12.2. The function g{x,yx, ...,jVi) is defined and continuous in
set A2.12) and g(x,yt, ...,yJeQ for (x,yu ...,ym) e E.
Hypothesis 12.3. The function g(x,yL, ...,ym) fulfils hypothesis 12.2 and,
moreover, for every fixed (x,yi, -..,Ут) the function v=g(x, u,y2, ...,yj possesses
a unique inverse u=h(x,y2, --^Ут, о), which fulfils hypothesis 12.1.
Hypothesis 12.4. The functions f^x), i=\,...,m, belong to R°[I] and
A2.13) /!(*)<//*) ^/m-!(x)</m(x) for x<?,xel, _^ ^_^
A2.14) Mx)>fj(x)^fm^(x)>fm(x) for x>t,xel, J~'-'m

2. Equations and systems of equations of order m 247
Let ?, be an endpoint of / and ?Ф x0 e I. We put Io = <x0,/m(x0)> or </m(x0),
and J0 = (x0, О or (?, xoy according to whether ?, is the right or the left endpoint
of / (*). Further, we put J* = (/-Jo - {?}) и h ¦
Lemma 12.1. Let Q be a simply connected subset ofW, and let hypotheses 12.2
and 12.4 be fulfilled. Then for any x0 e I, хоф?, and for an arbitrary function<p0e Ф[10]
fulfilling the condition
A2.15) <p0 [/m(Xo)] =g{x0, <Po(xo),
there exists exactly one function <p(x) belonging to the class 0[JO], satisfying equa-
equation A2.10) in Jo and such that
A2.16) (p(x) = (po(x) for xelo.
Moreover, if <Po(x) is continuous in Io, then <p (x) is continuous in Jo.
Proof. We may assume that ?, is the left endpoint of /. Put
A2.17) x1=/m(x0), хк+1=/яУ^-1(хЛ /с=1,2,...
By lemma 0.8 and theorem 0.4 the sequence xk is strictly decreasing and tends to ?,
CO
as fc-»oo. Consequently, Jo= IJ (xk+1, xky. We define <p(x) inductively: for xelo
k = 0
by relation A2.16), and supposing that <p(x) has already been defined for xe(xk, xo>,
fc>l, we put
A2.18) <p (x) =g (/-' (x), <p \_f-' (x)-] , <p \_f.if-' (x))] ,...,ip [/„_!(/„-l (x))])
for xe(xk+1, xky. For such x we have
ЛГ \x) >f~ \xk+1) =f~-\(xk) > xk,
since fm1(x) is increasing, f~\(x)>x, and in virtue of A2.17). Thus f~1(x)e
(xk,xoy. Next
by A2.14) and A2.17). Thus f[f-\x)] e (xt, xo>, /= 1, ..., m-1, and consequently
formula A2.18) actually defines <p(x) in (xk+1, xky. It follows from hypothesis A2.2)
that tp e 0[JO]. Replacing in A2.18) x by fm{x) we obtain A2.10). The uniqueness
of the solution obtained is obvious. x
If <Po(x) is continuous in Io, then^o(x) is continuous in \J(xk+1, xk). The con-
k = 0
tinuity of <p at the points xk can be checked just as in the proof of lemma 3.1. ¦
Similarly one can prove the following
@ In this chapter ? may again be infinite. Cf. the footnote on p. 47.

248 CHAPTER XII. Equations of higher orders and systems of equations
Lemma 12.2. Let Q be a simply connected subset of Wand let hypotheses 12.1
and 12.4 be fulfilled. Then for any хое1,хоф?, and for an arbitrary function <p0 e Ф [Io ]
fulfilling the condition
there exists exactly one function tp{x) belonging to the class <P[J%], satisfying equa-
equation A2.11) in J* and such that A2.16) holds. If, moreover, tpo{x) is continuous in Io,
then <p{x) is continuous in J*.
Lemmas 12.1 and 12.2 imply the following
Theorem 12.5. (Kordylewski, Kuczma [3].) Let Q be a simply connected subset
of 9Г, let hypotheses 12.3 and 12.4 be fulfilled and suppose that ? ф I. Then equation
A2.10) has in I a [continuous] solution <p еФ[Г\ depending on an arbitrary function.
More precisely, for any xoe I and an arbitrary function <роеФ [Io] and fulfilling con-
condition A2.15) there exists exactly one function <p(x) which belongs to Ф[Г\, satisfies
equation A2.10) in I and fulfils condition A2.16). If, moreover, <po(x) is continuous in
Io, then <p(x) is continuous in I.
Although, as in theorem 3.1, we obtain a solution depending on an arbitrary
function, actually the class of continuous solutions obtained is larger than in the
case of equation A.1). It can be proved that, under some conditions, the class of
continuous solutions of equation A2.10) (n = 1) is larger than the class of continuous
solutions of any equation of form @.49) (cf. Kuczma [29]).
§ 3. Uniqueness theorems. Let A=(At hn), y>i={yn, ¦¦¦,yin).
Theorem 12.6. (x) Let Q be a compact, connected subset of Wand let I be an
interval submodulus for each of the functions fix), /= 1 m. Suppose that the func-
function h(x,ylt ..., ym) is defined in set A2.12) and assumes values in Q; moreover, for
(x,y1,...,yJ,(x,yi,...,ym)eEthe inequalities
n m
A2.19) \hi{x,yl,...,yJ-hi{x,~y1,...,}m)\^Y, E eyi|.Vy-
hold, where ai}l are positive constants such that for
A2.20) ,4г
we have
A2.21) S=?4<1.
Then equation A2.11) has in I a unique solution tp{x) belonging to the class Ф[1].
Majcher [1], Bajraktarevic [8], Choczewski [1].

3. Uniqueness theorems 249
This solution is given by
A2.22) <p(x)=lim<pk(x),
where <po(x) is an arbitrary function from the class Ф [I] and
A2.23) (Pk+i(x)=h(x,(pk\_f1(x)-],...,(pk[fm(x)-]), /c=0,l,2,...
If, moreover, the functions f are continuous in I and h is continuous in E, then <p is
continuous in I.
Proof. We define !F as the space of functions <p(x) defined in / and taking
values in Q. For <p(x) = (q>1(x), ...,<pn(x)) and <p(x) = (<ip1(x), ...,^n(x)) in ^ we
define the distance
A2.24) P(<P,V)="
Endowed with metric A2.24), & is a complete metric space.
For <p e IF we define the transform \// = h [<p] by
A2.25) ?(x)=h(x,
For tpe^ also h[<p] e J5"; moreover, A2.25) is a contraction map. In fact, we have
by A2.24), A2.19), A2.20) and A2.21) for«p,?e JF
П
р(й[>],й[?])= E sup \hi(x, <p[ft(x)] ,..., <p[fm(x)~])-
n n m
I E I ^
n n m
i=l J= 1 /= 1
n n
< E ^ E supk/*)-^
i=l j=l /
Hence, by Banach's fixed point theorem, A2.25) has a unique fixed point in !F,
which respresents the required solution <p(x). This fixed point may be obtained as
the limit A2.22) of the sequence A2.23) of successive approximations.
In the case where /; and h are continuous, the same argument applies when
we replace & by the set У of these <p e !F which are continuous in /, showing the
existence of a unique continuous solution of A2.11) in /. Since y^lF, this solu-
solution must be identical with the one previously obtained. ¦
Theorem 12.7. Let I be an interval submodulus for each of the functions f(x),
1=1, ..., m, and let the function h(x,ylt ..., y^) be defined for xel, yly ...,yme 91".

250 CHAPTER XII. Equations of higher orders and systems of equations
Further, suppose that there exists a constant d>0 such that (*)
Щх,У1 Уп)\< max(d, \yt\,..., \yn\)
and that the function h fulfils condition A2.19) with A2.10) and A2.21). Then equa-
equation A2.11) has a unique bounded solution (p(x) in I. This solution is given by formula
A2.22), where the sequence <pk{x) is defined by A2.23) and <po(x) is an arbitrary bound-
bounded function in I. The function <p(x) fulfils the inequality
A2.26) |<p(x)|^ for xel,
and, if the functions f(x) and h(x, yit ..., ym) are continuous, is itself continuous in I.
Proof. Applying theorem 12.6 with Q = {ye 9Г: \y\^M}, M^d, we obtain
the existence of a unique solution <p{x) of equation A2.11) fulfilling \(p{x)\^M.
Letting M->oo we see that <p(x) is the unique bounded solution of A2.11), and tak-
taking M=dv/e obtain relation A2.26). ¦
Let ? e / be a fixed point of all the functions/;(x), /= 1 m, i.e. /t (?) =... =
fm(?) = ?. If equation A2.11) has a solution tp(x) in /, the value ti=<p(?) must fulfil
the equation
A2.27) n=h{t;,ti,...,ti).
Theorem 12.8. (Choczewski [1].) Let Q be an open, simply connected subset
of ЧЯ", let hypotheses 12.1 and 12.4 be fulfilled and let ?,el. Further, let ц be a root
of equation A2.27) such that qeQ and suppose that the function h(x,ylt ...,'y^)
fulfils in a neighbourhood of(?, ц, ..., j/) inequality A2.19) with A2.20) and A2.21)
and with аг11—ап independent of j. Then there exists exactly one function <p(x) which
is continuous for x = ?, belongs to the class Ф[Г\, satisfies equation A2.11) in I and
fulfils condition A2.6). This function is continuous in I and is given by formula A2.22),
where the sequence <Pk(x)/J defined by A2.23) and <po(x) is an arbitrary function from
the class Ф [/], continuous at ? and fulfilling condition A2.6).
Proof. We may assume that ?, is the left endpoint of /. For a given d>0 we
denote by Qd the set
A2.28) Qd
Now we choose c>? and d>0 such that Qd<^Q, eel, inequality A2.19) is fulfilled
in <?, c> x Qd x ... x Qd and, moreover,
A2.29) \h{x,4 *)-*«,„,...,*)|<A-S)d for xe<?,c>.
We have for xe (?, сУ, yteQd, 1=1, ...,m, by A2.27), A2.29), A2.19) and A2.21)
For h=(hi,..., и„)е 9?" we write |и| = ? \щ\.

3. Uniqueness theorems 251
,У1 yJ-4\ = \*(x,yi,...,yJ-HZ,4, -,ч)\
^\h(x,ylt ...,yj-h{x,n, ...,ч)\ + \к(х,ч, ..., ч)~Щ, П, -,ч)\
l
i=l 1=1 i=l
i.e. A(x,yx ym) eQd. By theorem 12.6 equation A2.11) has in <?,c> a unique
solution ^ (x) which fulfils the condition
A2.30) \<p(x)-ti\<d for xe<?,c>.
This solution can be obtained as the limit of a sequence defined by A2.13) and is
continuous in <?, c>.
The value q=<p(?) fulfils equation A2.27). In view of A2.30) i/ e Qd. Hence we
get by A2.19)
|«7-?| = |й(?, j/, ...,j/)
n и m
whence it follows that J/=qi, i.e. <p fulfils condition A2.6).
On account of lemma 12.2 the solution obtained may be uniquely extended
onto the whole / in a continuous manner, and it can be proved, as in theorem 3.3,
that relation A2.22) holds in the whole of /. Thus in order to complete the proof
of the theorem it is enough to show that every solution <p(x) of equation A2.11)
which is continuous at x=? and fulfils condition A2.6) fulfils also A2.30).
We put xq=f\(c), 0=0,1,2,... The sequence xq is strictly decreasing and
tends to ?,. It follows from the continuity of <p and from A2.6) that for a certain
6>0 the inequality \<p(x)—ti\^d holds in <?, xQ}. Let us suppose that
A2.31) \q>(x)-4[^d for
For xe (xq+1,xqy we have by A2.14)/;(x) e <?, x?+1>, 1=1 m, and hence by
A2.31) <p(x) = h(x,<p[f1(x)] <p[fm(x)i) belongs to Qd. This means that \(p(x)-q\
^d for x e (xq+!, xqy; induction now shows that A2.30) holds.¦
In the case of a single equation (n= 1)
A2.32) q>(x) = h (x, q>[/i(x)] , ¦•¦¦,(?[/m(*)])
the conditions of the above theorem regarding the function h will certainly be ful-
fulfilled if A is a C1 function in E taking values in Q, and its partial derivatives at the

252
CHAPTER XII. Equations of higher orders and systems of equations
point (?, r/, ...,rj) fulfil the condition
dh
A2.33)
1=1
Then we may take as ain=ai any numbers such that
8h ..
< at and
7=1
§ 4. Lack of uniqueness. In the present section we shall prove analogues of theo-
theorems 3.6 and 3.7 for equation A2.10). The value ti=<p(^) of a solution of A2.10)
at a common fixed point of all the functions ft(x) must fulfil the equation
A2.34) 4=g(i,1,. ..,*).
A point x0 ?/,xo#?, being fixed, we shall now denote by Io the interval <xo,fm(xo))
or (/mOo)> *o> according as xo<?, or хо>?.
Theorem 12.9. (Choczewski [1].) Let Q be an open, simply connected subset
of 9Г, let hypotheses 12.2 and 12.4 be fulfilled and let ? e I be a common fixed point
of all the functions f(x). Further, let r\be a root of equation A2.34) such that t\eQ
and suppose that the function g(x,yt, ..., ym) fulfils in a neighbourhood of{?,, ц, ...,»/)
the inequalities
A2.35) \gfa,y1,...,ym)-gi(x,y1
where an are positive constants such that
A2.36)
j=l 7=1
Then there exist positive numbers c, d such that for every xoe I, 0<\xo — ?\<c, and
for every solution <p(x) of equation A2.10) in I such that
A2.37)
we have
A2.38)
for xelo,
lira (p{x)=ii.
Proof. We may assume that ?, is the left endpoint of /. We choose c>0, d>0
in such a manner that Qd<^Q (where Qd is defined by A2.28)), i + c el, inequalities
A2.35) are fulfilled in <?, ? + c> x Udx ... x Qd and, moreover,
A2.39) \g(x,t,,...,t,)-g(Z,t,,...,t,)\^(l-S)d for xe<?,?+c>.
Let <p(x) be a solution of equation A2.10) in / and suppose that there exists an
x0, 0<|xo-?|<c, such that A2.37) holds. We define the sequence xk by formulae
A2.17).

4. Lack of uniqueness 253
00
At first we shall prove that <p(x)eQd for xe(^,xo>= \J (xk+1, xk}. For
*=o
x e (xL, xo> this is true in virtue of A2.37). Suppose that \<p(x)—q\ ^dfor x e (xk, x0},
A;> 1, and take an x e (xk+1, xk~). As has been shown in the proof of lemma 12.1
/¦~1(x)e(*o,Xo>c:<?,? + ?> and fl[f~\x)]e(xk,xoy, /=l,...,m-l. Hence in
virtue of A2.10), A2.34), A2.35), A2.36), A2.39) and of the induction hypothesis,
we obtain
\<P(x)-n
= |*(/»~ Ч*). 91У-Ч*)] - 9[/i(/»~ 4*))]. - . 9[/.-i(/.TЧ*))])-*«. 1> - - «)|
4*). ?[/»Ч*)]. ^(лгч*))]. -, «'[л.-^лгч*))])-
Ч*). ч, -, v)|+|*C/"-4*).«. - - «)-*«.«. -. ч)\
Е (
i=1
1=1 / = 1
i.e. ^d(x) e Qd for x e (xt+1, xt>. In virtue of the induction principle we have hence
<12.40) \p(x)-ti\^d for X6(^,xo>.
Now we put Mt=/*(x0), k=0, 1,2, ... The sequence uk is strictly decreasing
and tends to ?,. We fix a number в such that $<в< 1 and choose from t/t a subse-
subsequence yt in such a manner that
for xe(^,Dt>, fc=0,l,2, ...,
which is possible since g{x,q, ...,»/) is continuous at ?, and Mt^^. Evidently for
every к we have
<12.42)
We shall prove that
A2.43) |9(x)-,|<d0* for
For A:=0 A2.43) holds in view of A2.42) and A2.40). Suppose that A2.43) holds for a
fc>0, and take an xe(?,vk+1y. There exists an index q such that vk+1 = uq+1 and,
of course, q^k. We have/-1(x)</~1(j;t+i)=/m1(M«+i)=M«<j;t since v* is a sub-
subsequence of uK and yt+1 = Me+1. Consequently, fin1(x) e (?,vky and hence also
4x)]e (^>yt> f°r /=1 m — l. Hence we obtain, as before, by A2.41) and

254 CHAPTER XII. Equations of higher orders and systems of equations
by the induction hypothesis
Thus A2.43) is valid for every fc>0. A2.38) results from A2.43) in view of the fact
thatO<0<l andrt-»(!;.¦
Hence and from theorem 12.5 we obtain
Theorem 12.10. (Choczewski [1].) Let Q be an open, simply connected subset
of %" and let hypotheses 12.3 and 12.4 be fulfilled. Further, let ?e/, let q be a root
of equation A2.34) such that t\ e Q and suppose that function g(x,yt, ...,ym) fulfils
in a neighbourhood of (?,ц, •••,»/) condition A2.35) with A2.36). Then equation
A2.10) has in I a continuous solution <реФ[Г\ (х) fulfilling A2.6) depending on
an arbitrary function.
Finally, let us note that in the case of a single equation (n= 1)
\=д(х,<р(х),<р[Л(х)],..., <p[/m_i(x)])
conditions A2.35) and A2.36) will certainly be fulfilled if g is a C1 function in E
taking values in Q and if its partial derivatives at the point (?,,r], ...,rf) fulfil the
condition
89 ,v
§ 5. Gaussian normal distribution. We shall prove here a theorem due to
E. Vincze [2] (cf. also Hosszu, Vincze [1], Laha, Lukacs, Renyi [1]), giving a charac-
characterization of the Gaussian normal distribution occurring in the probability theory
(the Gaussian law of errors). This section may be regarded, on one hand, as an
illustration of our preceding considerations and, on the other hand, as an example
of a treatment of cases not covered by theorems of §§ 3-4.
Let / be an interval containing zero and let a, b be positive numbers such that
A2.44) a2+b2=l.
We consider the equation
A2.45) <p{x) = <p{ax)<p{bx)
for xel.
Theorem 12.11. The only solutions in I of equation A2.45) (where a, b fulfil
A2.44)) which are twice differentiable at x=0 are
A2.46) <p(x)=0
(!) More exactly, the class Ф[Г\ should be replaced here by the class of functions p e Ф[Г\
fulfilling A2.37) with *o and d suitably chosen.

5. Gaussian normal distribution 255
and
A2.47) <p(x) = ecx\
with an arbitrary constant с
Proof. Equation A2.45) is a particular case of A2.32); here ?=0. Equation
A2.27) takes the form r\=r\2 and has two roots: t]=0 and ti = l. For r]=0 theorem
12.8 is applicable (we take Q = {—oo, oo)) since condition A2.33) is fulfilled. Con-
Consequently, equation A2.45) has in / a unique solution continuous at zero and ful-
fulfilling the condition ^@)=0. This must be function A2.46).
Now we are going to investigate solutions fulfilling
A2.48) 9>@)=l.
Restricting, if necessary, the interval / to a suitable neighbourhood of 0, we shall
have, in view of A2.48) and of the continuity of <p at zero, q>(x)>0 in /. Thus we
may put A(x)=log <p{x), and equation A2.45) is reduced to
A2.49) X(x)=k{ax)+X(bx).
Condition A2.48) gives A@)=0. Differentiating A2.49) at x=0, we obtain A'(°) =
(a+b) A'@) and, since by A2.44) a+b>l, Г@)=0. Х(х) is, like <p(x), twice differ-
entiable at x=0. Consequently, the function а>(х)=Цх)/х2 for хфО, со@)=А"@),
is continuous at x=0 and satisfies the equation
A2.50) co(x)^a2co(ax) + b2co(bx).
We shall show that the only solutions of equation A2.50) which are continuous at
x=0 are the constant functions. Hence A2.47) results directly (*).
Let us take an arbitrary x0 e /, xo#0. According to A2.50) and A2.44), of the
two values co(ax0) and co(bxo) one must be not more, the other not less than co(x0).
Thus we obtain two points t/l5 »l9 fulfilling |м1|<тах(а, b)\xo\, |i;1|<max(a, b)|xo|
and such that ©(^^©(хо^со^). The same argument may be repeated for их
and Vy in place of x0 and of the four values auy, buy, avx, bvt we may choose two,
м2 and v2, fulfilling |м2|<тах(а, ^>)|мх|, |tf2|^max(a, ^)|^i|. <и(м2)<й)(м1), co(v2)
fyi). Generally, we are led to two sequences, uk and vk, fulfilling
\uk+1\^msix(a,b)\uk\, \vk+1\^ma.x(a,b)\v
k\,
By A2.44) max (a, 6)<1 and hence uk-*0, vk-*0. It follows from the continuity of
со at zero that lim co(uk)= lim co(vk)=co@), whence co(x0)=(o@), i.e. co(x) is con-
к-юо к-юо
stant.H
@ For x belonging to the restricted interval. But then the validity of A2.47) in an arbitrary
larger interval follows from theorem 12.5 (one must take fl=@, oo)).

256 CHAPTER XII. Equations of higher orders and systems of equations
It easily follows from theorem 12.8 (since, by A2.44), a3+63<l) that the
functions k(x) = cx2 are the only solutions of equation A2.49) three times differenti-
able at zero; consequently functions A2.47) are the only solutions of equation A2.45)
fulfilling A2.48) and three times difiFerentiable at zero. Theorem 12.11 gives a stronger
result. However, this result cannot be strengthened further. To see this, let us take
(.2,!, b^. .*
These numbers fulfil A2.44). Let y/(x) by an arbitrary continuous solution of the
equation
A2.52) V(bx) = bV(x).
Then the function
A2.53) <p(x)=exp$y/(t)dt
о
is a C1 solution of equation A2.45). In fact, we have by A2.52) and A2.53)
q> (ax) q> (bx) = q> (Ь2х) <р (bx)
Ъгх bx
=exp[f ^(г)Л+ \v(i)di\
о о
= exp j [b2y/(b2u) + by/(bu)~]du=exp j(b*+b2)y/(u)du
о о
x
= exp \y(u)du = q> (x).
о '
Since b< 1, equation A2.52) has in /a continuous solution depending on an arbitrary
function (theorem 2.10). Consequently, equation A2.45) with a and b given by
A2.51) has in / a C1 solution depending on an arbitrary function (x). This result,
however, could not be obtained from theorem 12.10.
The conclusion of theorem 12.11 remains valid for complex-valued functions
g>(x) of a real variable.
Lemma 12.3. The only complex-valued functions <p(x) of a real variable which
satisfy equation A2.45) (with a, b fulfilling A2.44)) and are twice differentiable at
zero are functions A2.46) and A2.47), where с is an arbitrary complex constant.
Proof. Let <p(x) be a function fulfilling the conditions of the lemma. We write
<p(x) = р(х)еЩх\ with real p(x) and 9(x). Then A2.45) implies that
A2.54) p(x)=p(ax)p(bx),
A2.55) в(х) = в(ах) + в(Ьх) + 2к(х)п,
(') It can be proved that formula A2.53) (with continuous y/ satisfying equation A2.52))
gives the general C1 solution of equation A2.45) with a, b given by A2.51); cf. Chapter XIII, §2.

5. Gaussian normal distribution 257
where k(x) is an integral-valued function. The function p(x) must be continuous at
zero, and if p@)#0 it must be twice differentiable at zero.
Now, if <p @) = 0, then p @)=0 and by theorem 12.8 p (x)=0. This leads to function
A2.46) independently of в(х). If g>@)^0, then necessarily
A2.56) p@) = l and p@) = l
and by theorem 12.11 p(x) = e°ix2 with a real constant cx. Further we have, according
to A2.56), for small |x|,
в (x) = arg <p (x) = i log p (x) - Hog q> (x),
vhere log denotes an arbitrary branch of the logarithm on the complex plane cut
ilong the negative real axis. Hence it follows that 6{x) is also twice differentiable
it zero. Consequently, k{x) in A2.55) must be continuous at zero, and hence constant
in a neighbourhood of zero. Thus A2.55) becomes 6(x) = 6(ax) + 6(bx) + 2kn, i.e.
the function 1(х) = в(х)+2кп satisfies (in a neighbourhood of zero) equation A2.49)
and is twice differentiable at zero. The argument in the proof of theorem 12.11
shows that X(x) = c2x2 with a real constant c2. Consequently,
with c=c1 + ic2- Formula A2.47), established so far for small |x|, must be valid
everywhere because of A2.45).¦
From theorem 12.11 and lemma 12.3 we obtain the following
Theorem 12.12. (x) The only function y(x)B) which is defined for all x, twice
differentiable at x=0, satisfies the equation
A2.57) y{x)
with positive a, b fulfilling A2.44) and а афО, and such that
A2.58) fy(x)dx=l,
-oo
is the Gaussian normal distribution
A2.59) y(L*\
Proof. Let y(x) fulfil the conditions of the theorem. Then the function <p(x)
= yjln cry(x) satisfies equation A2.45) and is twice differentiable at x=0. By theorem
12.11 resp. lemma 12.3 <p(x) must be of form A2.46) or A2.47); the former, however,
is impossible in view of A2.58).
0) Vincze [2]. Another characterization of the Gaussian distribution may be found in Gun-
ten [1].
B) Real or complex valued.
17 Functional equations

258 CHAPTER XII. Equations of higher orders and systems of equations
Since J ecxldx=— (c must be negative, since otherwise the integral is
-co V ~c
divergent), condition A2.58) gives
\l2na
whence c= —г—г • Thus we obtain
y(x)=—j=—q>(x)=—.
i.e. A2.59).и'

CHAPTER XIII
LINEAR EQUATIONS OF HIGHER ORDERS
§ 1. Reduction of order. The present chapter contains some remarks con-
concerning linear equations of the formA)
A3.1) an(x)<plfn(x)-] + an-1(x)<p\:fn-1(x)l + ... + a1(x)<plf(x)-] +
+ ao(x)q>(x) = F(x).
As in Chapter II, x is a real variable, whereas the values of g> and of the coefficients
at(x) may lie in the field of reals or that of complex numbers.
We are going to see if it is possible to reduce the order of equation A3.1) by a
substitution of the form
A3.2) V(x)=<pU(x)-\-X{x)(p{x).
From A3.2) we obtain by induction for i> 1
1 k~
A3.3) <p[f(xy]=?[/'-\xy]+? [p[A*)]vUk~4*)]
k=lj=k j=0
Inserting A3.3) into A3.1) and ordering the terms according to increasing exponents
in y/ [fm(x)], we obtain
A3.4) an(x)^[/n-1(x)] + "i;2(am+1(x)+ ? afyt) 'f\ A [/'(x)]V [/m(x)] +
0 J +2 '+l
n
j=0
(i) Guichard [1], Grevy [1], [2], [3], Bourlet [4], Pincherle, Amaldi [1], Tornquist [1], Spiess [21,
Lattes [3], Fulco [1], Bottcher [7], HUb [2], Myrberg [1], [2], Urysohn [1], Popovici [2], [3], [6], [10],
[12], Badescu [5], Lowig [1], Walsh [1], [2], Sheffer [2], Biikhoff [6], John [1], [2], Hamilton [1],
Ghermanescu [l]-[4], [9], [10], [11], [13], [14], [15], [17], [19]-[24], Valeiras [1], Beckenbach [1],
Aczel [2], [3], H. Schmidt [1], Bellman [2], Kordylewski, Kuczma [2], [4], Aczel, Ghermanescu,
Hosszu [1], Naftalevic [10], Kucharzewski, Kuczma [1], Bajraktarevic [12], [16], [18], Presic [3], [4],
Mitrinovic [1], Majcher [l]-[4], W. A. Harris, Sibuya [2], Hirche [1], Hosszu [2], Vuc, Boro? [1],
Crstici [1], Czerwik [1], [2].
Here belong also linear q-difference equations, or geometric difference equations, which corre-
correspond to the case/(x) = ?x. In this cennection cf. Thomae [1], Jackson [2], Carmichael [1], Birkhoff
[1]. [2], [7], Mason [2], Ryde [1], Adams [l]-[10], Carman [1], Trjitzinsky [1], [2], [3], Starcher [1],
Birkhoff, Guenther [1], Le Caine [1], Strodt [2], Hahn [l]-[8], Meschkowski [1], Ghermanescu [22L
Abdi [1], Carlitz [3], Tauber [4], [5].

260 CHAPTER XIII. Linear equations of higher orders
Now, if X{x) in A3.2) is chosen so as to satisfy the equation
A3.5) an1 2
then A3.4) becomes
A3.6) I
where
A3.7)
bm(x)=am+1
bn-1(x)=an(
(*)] + ...4
n
(*)+ E a
l = m + 2
b,{x)vi
i-i
i\x) 11
j = m+l
= 0 n-2,
A3.6) is again an equation of form A3.1) but of order n— 1.
If we can find a solution k{x) of equation A3.5), then equation A3.1) reduces
to a system consisting of equations A3.2) and A3.6). Namely, we have the following
evident
Theorem 13.1. Let f(x) be defined in a submodulus interval I and let X{x) be
a solution of equation A3.5) in I. If a function g>(x) satisfies equation A3.1) in I, then
the functions <p{x) and y(x)=<p[f(x)]—X{x)(p{x) satisfy in I the system of equations
A3.2) and A3.6). Conversely, if functions <p(x) and y/(x) satisfy in I equations A3.2)
and A3.6), then q>{x) satisfies equation A3.1) in I.
Of the two equations A3.2) and A3.6) only the latter is difficult; A3.2) is a
linear equation of order one and we know a great deal about such equations (cf.
Chapters II and V).
However, equation A3.5) is very difficult. It is of a lower order than A3.1),
but is not linear. The theorems obtained in Chapters XII and III do not apply in
this case. Equation A3.5) need not have a solution, at least a solution of a prescribed
regularity, though equation A3.1) may have one. The following example is taken
from the paper Kuczma, Vopenka [1], where the case n = 2 is discussed in detail.
Example 1. Let л=2, аг(х) = 1, ai(x)=Q, aa(x)=—eb+h'~x), where h (x) is real-valued and
continuous on /=(— 1,iy, h@)=0, b is real, and the series
A3.8)
diverges. Further, let f(x)=\x. Equation A3.5) becomes
A3.9) Щх)Цх)-е"+"^ =0.
It is clear from A3.9) that X (x) ф 0 in /. So if X(x) is a continuous (complex-valued) solution of A3.9)
in /, then p(x)=\og \X(x)\ is a continuous (real-valued) solution of the equation
A3.10) p(
But on account of theorem 2.11 and in view of the divergence of series A3.8), equation A3.10)
has no continuous solution in /. It should be noted that the corresponding linear equation of order 2
A3.11) <p(ix)-e»+bM<p(x)=F(x)

1. Reduction of order 261
has a continuous solution (unique or depending on an arbitrary function, according as b >0 or 6<0)
provided that ЬфО and F(x) is continuous in /; in fact, equation A3.11) may as well be regarded
as an equation of order one and the existence of its continuous solutions results from theorems.
2.7 and 2.10.
In spite of the above difficulties, the method of the reduction of order just
described can successfully be used in many cases. An important point here is that
we need not solve equation A3.5) completely. It is enough to know a single particular
solution. We shall discuss one more example.
Example 2. Consider in /=(—i,i) the equation
Equation A3.5) takes the form
I x\ 3x + 19x10 x + x6
A3.13) X\ — ]X(,x) X(x) + =0.
K ' \2/ 2x^ + \lx+% 2X2 + 9X+4
Leaving aside the problem of solving equation A3.13) completely, we may search for a solution
of a particular form. Since the coefficients and the function f(x)=ix in equation A3.13) are ra-
rational, we may suspect the existence of a rational solution. If we try the simplest possibility X(x) =
(Ax + B)/(Cx + D), we find X(x) = (x + 3)/(x + 4) as the only solution of A3.13) of this form. In virtue
of theorem 13.1 equation A3.12) is equivalent to the system
!x\ x+3
\2/ x+4
<13Л4> /- ,-2 5,
4x + 2
If (p(x) is continuous or differentiable in /, then so is v(x). Now, by theorem 2.7, the second equa-
equation of A3.14) has a unique continuous solution in /. Obviously this is the solution i/r(x) = x. Con-
Consequently, <p(x) is a continuous solution of A3.12) in / if and only if it is a continuous solution
of the equation
A3.15)
in /. By theorem 2.10, A3.15) has in / a continuous solution depending on an arbitrary function.
More precisely, every solution of equation A3.15) in / which is continuous in (—?, 0)u@,?) is
continuous in / (theorem 2.9). The same is not true about equation A3.12). This equation has solu-
solutions which are continuous in (—i, 0)u@, i) but not continuous at x=0. Such are the solutions
/x\ x+3
for which q> I — 1 ц>{х)Фх. Since (theorem 12.5) a solution continuous in ( —\, 0)u@, \)
\2/ x+4
of equation A3.12) can be prescribed almost arbitrarily on ^/?,i/?}>u^ia, oiy with — ^</?<0<a
<i, there exist solutions which do not fulfil A3.15), and they depend on an arbitrary function.
On the other hand, according to theorem 4.5 equation A3.15) has a unique solution of
class СЧП--
ro v 2~fx+4
A3.16) „w^E2-*n .
p-0 i-0
Consequently, function A3.16) is the only О solution of equation A3.12) in /.

262 CHAPTER XIII. Linear equations of higher orders
§ 2. Equation with constant coefficients. If attempts to find a solution of equa-
equation A3.5)—or a solution with a desired regularity—fail, then the method described
in the preceding section cannot be used. This may indeed happen, as can be
seen from example 1 in § 1. Anyhow, there is an important class of equations A3.1)
for which equation A3.5) is always solvable^). They are equations with constant
coefficients B)
A3.17) an<p[f"(x)-] + an-1<p[f"-\x)-] + ...+a1<plf(x)-] + ao<p(x) = F(x).
Then we may expect to find a solution of the corresponding equation A3.5) inde-
independent of x. If we assume X(x)=X in A3.5), we obtain
A3.18) апЛ" + ап-.1Л"-1+...+а1Л+ао = 0.
Equation A3.18) is called the characteristic equation of equation A3.17).
If X=XQ is a root of A3.18) and <p{x) satisfies A3.17), then the function y/(x)
=<p[f(x)] — X0 (p{x) is a solution of the equation
A3.19) bn^W[r-
where the coefficients bt and at are connected by the relation
(cf. formulae A3.7)). Consequently, every root of the characteristic equation of
equation A3.19) is a root of equation A3.18), and as a consequence of theorem
13.1 we obtain the following
Theorem 13.2. (Kordylewski, Kuczma [4].) Let f(x) be defined in a submodulus
interval I and let Xl; ..., Xn be the full sequence of roots of equation A3.18). Then
equation A3.17) is equivalent to the system
]-AiP C>0 =
A3.20) У. [/(*)]
Having reduced A3.17) to system A3.20), we may proceed further as in example
2 in § 1. As an example we shall determine the behaviour of the continuous solutions
of equation A3.17). Here the constants at may be real or complex, а„ф§, ао^0 C),
(') By this we mean here that a solution always exists, without entering into the difficulties
of the practical determination of the exact value of the solution.
B) Guichard [1], Bourlet [4], Spiess [2], Urysohn [1], Walsh [1], [2], Sheffer [2], John [1], [2],
Hamilton [1], Beckenbach [1], Aczel [2], [3], H. Schmidt [1], Kordylewski, Kuczma [2], [4], Baj-
raktarevit [12], [16], [18], Kucharzewski, Kuczma [1], Ghermanescu [22], Hirche [1], Czerwik
11], Ш.
C) The assumption апф0 is justified by the requirement that the order of A3.17) be effective.
Similarly, if ao#0, we may reduce the order of A3.17) by setting ?(x)=p [/(*)].

2. Equation with constant coefficients 263
and (p{x), ^j(x), F{x) are complex-valued functions of a real variable./(x) belongs
to Щ[1], ? e 7, and F{x) is continuous in /.
The behaviour of the continuous solutions of equations in system A3.20)
depends on the values of |A,-|. Since the order of roots of A3.18) in A3.20) is of no
importance, we may assume that |Я1|^...^|Я„|. Let e.g. |Ai|^...^|At|<l, |At+1|
= ... = |A*+,| = 1, l<jAt+1+1[^...^[At+1+m[, k+l+m=n (including the possibility
that some of к, I, m equal zero). Then the last equation of A3.20) has a unique
continuous solution in /, and the same is true of all the preceding ones as far as
the equation
The equation
has either a unique continuous solution or a unique continuous solution up to an
additive constant (this may happen only if At+1 = 1) or no continuous solution in /.
Whether it has a continuous solution or not, depends on the nature of y/k+i(x),
and thus in fact on F(x).
This argument may be repeated. If k<i^k+l and у/г(х) exists, unique or
determined up to an additive constant, then we have three possibilities regarding
the equation
A3.21) щ_ t[/(x)] - А; щ.,(x) = Vi(x).
It may happen that A3.21) has no continuous solution in /. If it has one, it is unique
if А;ф 1 and if/iix) is unique. If кгф 1, but y/?x) is determined up to an additive con-
constant, say y/i(x) = ^j(x) + c, then also y/t- t(x) is determined up to an additive constant:
^,-_1(x) = ^i_1(x) + c/(l-Ai), where ipt-^x) satisfies ^-,[/"(x)]-A, ipi-l(x) = y/i(x).
If A,-=l, y/i(x) is unique and A3.21) has a solution, it is determined up to an ad-
additive constant. Finally, if A;= 1 and y/i{x) is determined up to an additive constant,
A3.21) may have a continuous solution only for that member y/^x) of the family
(with an additive parameter) which fulfils ^;(<^)=0. If A3.21) has a continuous
solution, this is again unique up to an additive constant.
Thus, step after step, we arrive at the equation
A3.22) ^-.[/W]-At^_1(x)=^(x)
provided that all the preceding equations had continuous solutions in /. If one of
them fails, then A3.17) has no continuous solution in /, either. Leaving this case
aside, let us suppose that a continuous y/k(x) exists. It is then unique or determined
up to an additive constant. Now, A3.22) has a continuous solution depending on
an arbitrary function. This gives rise to a family of continuous solutions depending
on an arbitrary function of the equation
У*-2[/М] -4-1 ?k-i(x)=y/k- i(x).

264 CHAPTER XIII. Linear equations of higher orders
We thus start a "chain reaction" of creating continuous solutions. The situation
becomes simpler if we eliminate ?l, ..., ?k_t from the first к equations of A3.20)
obtaining
A3.23) <p [/*(*)] + ck.l9 [/*- '(*)] +... + c, <p[/(x)] + c0<p(x) = ?k(x),
where the coefficients ct are determined by
A3.24) Af[ + ct_1A'[~1 + ... + c1A + c0=(A-A1)...(A-At).
In virtue of theorem 2.9 every solution of equation A3.22) which is continuous in
/— {?} is continuous in the whole of /. On the other hand, a solution continuous
in /— {?} may be prescribed almost arbitrarily on an interval of the form <x0 ,fk(x0)y
or </Vo), *o> (theorem 12.5).
Similarly, ?k{x) is a solution of the equation
A3.25) ?i
where the coefficients dj are determined by
A3.26) rkk
Equation A3.25) has a unique continuous solution in / in the case where |
for j=k+l, ..., n. If 7^0, i.e. some A/s have the absolute value 1, then equation
A3.25) either has no continuous solution in / or has a unique such solution (this
is possible only if Х^ф\ for/=fc+l, ...,л), or a one-parameter family (with an
additive parameter) of such solutions (this is possible if at least one of A/s is equal
to +1).
It may be noted that if the coefficients a-t are real, then the coefficients c-t and
dj are also real. This follows from the fact that the complex conjugate numbers
have equal moduli. In this case, if, moreover, the function F{x) is real valued, the
unique continuous solution of equation A3.25) must also be real ('). To prove this,
consider an equation of form A3.21) with a real-valued continuous right-hand side
?i(x). If Aj= + 1, then by theorem 2.11 a continuous solution of A3.21) must have
the form
and thus is real provided that the constant t\ is allowed to assume real values only.
Now suppose that A,-# + l and put Ч = ?№)/0-—&{), W*-i(x) = ?i-i(.x)~tl-
by A3.21)
A3.27) ?t ,[/(*)]
If the solution of A3.25) is determined up to an additive constant, it is of the form y/(x)
, where \i/o{x) is real and the constant ц may take real or complex values, according
to whether we consider A3.25) in the real or in the complex field.

2. Equation with constant coefficients 265
and hence, on setting x=?, y/*-i(O—^ results. By induction we obtain from A3.27)
vt-i(x)=- Y тят + p
p-0 I4 Aj
whence, since y/f_l(?)=O and |A||>1, the continuous solution of A3.21) must be
of the form
Now, if Я; is real, function A3.28) must be real. If kt is not real, function A3.28)
need not be real. But then also the conjugate to X-t occurs among the roots of equation
A3.18), and we may assume that it is Xt_!. So At_ i =lj. Then the function ^t_2(x)
satisfies the equation
A3.29) ^i_2[/(x)]-A^i_2(x) = ^_1(x)
and again the continuous solution of A3.29) must be of the form
1E
1—/; 4=0
Inserting A3.28) into A3.30), we obtain
A3.31) yi-iW^ T+ E
A^)(lAJ
A3.31) is a real-valued function. This shows that, though some of the functions
i//t(x) may be not real, the function y/k{x) must be real, of course in the case where a,
are real and F(x) is real.
Finally, we make one more remark. If at least one root of A3.18) is equal to
+ 1, then y/k(x) in A3.23) is determined up to an additive constant (if it exists at
all). Then A3.23) represents a family of functional equations. Solutions of two
equations of this family differ by an additive constant. Namely, if <p(x) is a solution
of equation A3.23), then the function
A3.32) p*(.
is a solution of the equation
A3.33)
In other words, knowing the general solution (p{x) of equation A3.23), we may
obtain the general solution <p*{x) of equation A3.33) by formula A3.32). Thus it
is sufficient to determine the general continuous solution q>{x) of equation A3.23)
with a single function y/k(x) from the family (with an additive parameter) of con-
continuous solutions of equation A3.25), and then the general continuous solution

266 CHAPTER XIII. Linear equations of higher orders
<p*(x) of equation A3.17) is given by formula A3.32), where n is an arbitrary constant.
Writing
</¦=,
we may give A3.32) a simpler form
where n* may again be regarded as an arbitrary constant.
We may gather the conclusions of the above considerations in a theorem.
Let 3? be the field of complex numbers or the field of real numbers and let]* [I\
be the class of functions which are defined in an interval / and take values in !F.
Theorem 13.3. (Kordylewski-Kuczma [4].) Let /еЛ°[/], ?е/, let F(x) be a
continuous function on I belonging to Ф [Г\, and let Oj e fF be constant coefficients,
а„ф§. Further, let X-t be the roots of equation A3.18) and suppose that \Xt\<l for
i=\,...,k, |A,-|>1 for j=k+l, ...,n. Then equation A3.17) is equivalent to the
system of equations A3.23) and A3.25) (i// = i//^), where the coefficients c; and dj are
determined by A3.24) and A3.26).
If none of the roots of A3.18) has modulus equal to one, then equation
A3.25) has a unique continuous solution ц/еФ[Г\ in I. In this case a function
<реФ[1\ is a continuous solution of equation A3.17) in I if and only if it is a con-
continuous solution of equation A3.23) in I.
If there are roots of A3.18) with modulus equal to one, then two cases are pos-
possible.
(I) Equation A3.25) has no continuous solution in I. Then equation( A3.17) has
no continuous solution in I either.
(II) Equation A3.25) has a continuous solution y/(x) in I. Then there are further
two possibilities:
(a) None of the roots of A3. Щ is equal to +1. Then the solution y/(x) is unique
and belongs to *[/]• A function <p еФ[Г\ is a continuous solution of equation A3.17)
in I if and only if it is a continuous solution of equation A3.23) in I.
(b) There are roots of A3.18) equal to +1. Then the general continuous solution
of equation A3.25) has the form y/(x) + n, where ц is an arbitrary constant. We
may choose у/еФ[1]. Then a function <реФ[1] is a continuous solution of equation
A3.17) in I if and only if it differs from a continuous solution of equation A3.23)
by an additive constant
If y/(x) is a continuous function belonging to Ф[Г\, then equation A3.23) has
in I a continuous solution <p e Ф[1\ depending on an arbitrary function. Every so-
solution (p(x) of equation A3.23) in I which is continuous in I— {?,} is also continuous
at x = ?.
A similar case is presented by the equation with automorphic coefficients
(Ghermanescu [20], [22]), i.e. equation A3.1) with the coefficients fulfilling a;[/(x)]

2. Equation with constant coefficients
267
= a;(x), i=0, ..., n. Then we may assume that A[/(x)] = A(x) and equation A3.5)
becomes
i.e. an algebraic equation of degree n. The details are obvious and will not be dealt
with here.
§ 3. Finite groups of substitutions. Quite different is the caseA) where the
function/(x) in A3.1) satisfies
A3.34) fn+\x) = x.
Then, replacing in A3.1) x by/(x),/2(x), ...,/"(x), we obtain л + l equations (to-
(together with A3.1)):
A3.35)
+ *„[/(*)] 9 [/(*)] + aH [/(x)] <p(x) = F [/(x)],
A3.35) forms a system of linear algebraic equations with л + l unknowns (regarded
as independent): (p{x), (p[f(x)],..., (p[f"(x)]. For a fixed x the set {x,/(x), ... ,/"(*)}
is the orbit C(x). System A3.35) allows us to determine B) the values of <p on C(x).
The solution is uniqueC) if det [aB_i_j(/'+-'(x))]#0 D), otherwise it may contain
parameters, or depend on an arbitrary function, or not exist at all.
Example. Consider in /=(—oo, 0)u@, l)u(l, °o) the equation
A3.36)
Here the function f(x)=(x-\)jx satisfies A3.34) with n=2. System A3.35) takes the form
/ 1 \ /x-l\
p -2p ¦
\l-x/ \ x I
I 1 \ fx-\\
-2p(- + p
\l-x/ \ x 1
x-l
1-х
A) Grevy [2], Popovici [6], [12], Ghermanescu [1], [2], [3], [14], [15], [22], Valeiras [1], Aczfl,
Ghermanescu, Hosszii [1], Presic [3], [4], Mitrinovic [1], Hosszii [2], Crstici[l].
B) If system A3.35) has no solution, then A3.1) has no solution either.
C) This is, of course, the general solution of equation A3.1). The problem of the regularity
of solutions is not discussed here.
D) The indices in an_t_j are to be taken modulo л+l.

268
CHAPTER XIII. Linear equations of higher orders
The last equation is the sum of the first two multiplied by — 1. From the first two equations we
find
A3.37)
<p(x)=<)
-l
x2+2x-\
We may define <p(x) arbitrarily on one of the intervals (— oo, 0), @, 1), A, oo), and formulae A3.37)
yield an extension of <p(x) onto the whole of /. Thus equation A3.36) has in / a solution (of arbitrary
regularity) depending on an arbitrary function.
The same procedure is applicable to the more general case of equation A2.10)
or A2.11), where the functions /,(x) generate a finite group. However, we shall
describe here an elegant method given by S. Presic [3], [4] for the homogeneous
linear equation (')
A3.38) an(x) <p [/„ (x)] + а„_ ,(x) <p [/„_ ,(*)] +... + a,{x) <p [/, (*)] +
(fo(x)=x), which gives an effective formula for the general solution of A3.38).
We assume that the functions ft{x) are defined in a common submodulus set E
of an arbitrary nature and that they form a group ^ of order n +1 (with respect
to the operation of substitution). Thus
A3.39) 1 /,[Л(*)]=/р(*),
O^p^n, and the index p=pik is by A3.39) unambiguously determined. Because of
the associativity of the composition of functions, we have for arbitrary indices
df
Let matrices Mk = [a\j\, 0^i,j, k^n,be defined by
0 if ]фрл.
If we multiply Mk by a one-column matrix [Xj]0<i<n, we obtain a one-column
matrix whose first element is xk. (Note that pQk=k.) Matrices Mk form a group
isomorphic to ^: MtMj=MPi .
We shall express the general solution of equation A3.38) in the form
A3.40)
ф)
= B(x)
9
(') The values of mix) and q>(x) may be real or complex. Some coefficients are allowed to
vanish identically, i.e. not all functions fi(x) need appear explicitly in the equation.

3. Finite groups of substitutions
269
where B(x) is a square matrix of order я + l, and g(x) is an arbitrary function.
In general, the expressions obtained from A3.40) for <p[fi(x)] are contradictory. If
they are not, the matrix B{x) is called compatible with the group ^.
Lemma 13.1. If
A3.41)
k = 0,
xeE,
then the matrix B(x) is compatible with the group
Proof. Let В(х) = [Ьц(х)] and
A3.42) <p(x) = boo(x)g(x)+t
Write
9>o(x)'
A3.43) ^l(x)
Multiplying A3.43) from the left by Mk we obtain
whence by A3.41)
i.e. by A3.42) and A3.39) <pk(x) = <p\fk(x)]. Thus B(x) is compatible with <S.m
Setting in A3.38) in tum/0(x),/i(x),... ,fn{x) in place of x, we obtain a system
of n +1 equations, which can be written in the form
A3.44)
A{x)
?[/.(*)]
= 0.
In the rth row and the /th column (') of the matrix A{x) we find aiL/i(*)] where /
is defined by pu =j. Thus in the rth row and the _/th column of A [fk(x)] we have
<*i[/PlJix)]. Hence the (i,j)ih element of A[fk(x)]Mk is
(!) The rows and columns of the matrices occurring in the piesent section are numbered
from 0 to n.

270 CHAPTER XIII. Linear equations of higher orders
equal to
n
1=0
where m is defined by the condition j=pPmlk =pmik • Similarly, the (i,j)th element of
MkA(x)h
4 = 0
equal to a,[fPtk(x)], where / is defined by j=Piptk =pUk. Hence pmik=p,ik, which
implies m = l. Consequently, A[fk(x)] Mk=MkA(x), i.e.
A3.45)
This is valid for arbitrary O^k^n, and so by lemma 13.1 A(x) is compatible with
the group &.
Now let r{x) be the rank of the matrix A{x) and let D(x) be a diagonal matrix
with r{x) units (and n+1 — r(x) zeros) on the main diagonal. Then there exist regular
matrices P(x) and Q(x) such that
A3.46) A(x)=P(x)D{x)Q{x).
(The matrices P(x) and Q{x) can be eflfectively constructed for a given A(x); the
construction can be found in any book on the matrix calculus.) The matrix
A3.47) B0{x)=-Q-\x)D{x)P^\x)
fulfils the condition
A3.48) A(x)B(x)A(x)+A(x) = Q for xeE.
Write
A3.49) B(x) = —t МГ%[/,(*)]Mt.
n + 1 ;=o
On account of A3.45) function A3.49) fulfils condition A3.48). We shall prove that
it fulfils also condition A3.41). We have
, 1
-1
By lemma 13.1 B{x) is thus compatible with the group ^.

3. Finite groups of substitutions
271
Now we are able to prove the following
Theorem 13.4. (Presic [3], [4].) The general solution of equation A3.38) in E is
determined by the formula
A3.50)
<p(x)
?[/.(*)]
9 [/„(*)]_
= (B(x)A(x
<7 [/.(*)]
where B(x) is defined by A3.49), A3.47) and A3.46), U is the unit matrix of order
n+l, and q{x) is an arbitrary function defined in E. In other words, the general solution
of A3.38) is given by formula A3.42), where bOi(x) are the elements of the 0-th row of
the matrix B(x)A(x)+U, and g(x) is an arbitrary function.
Proof. We have by A3.41) and A3.45)
в ш*)] а [/*(*)]+и=мкв (x) m; X a (x) Aft' + мк м; •
= Mk(B(x)A(x)+U)M;1,
and so, in virtue of lemma 13.1, the matrix B(x) A(x) + U is compatible with the
group &. Consequently, formula A3.50) is not contradictory and defines a function
(p{x) unambiguously for arbitrary^ g(x). It follows from A3.48) that A3.44) is
fulfilled, i.e. (p{x) satisfies equation A3.38).
On the other hand, if <p{x) is a solution of A3.38), then A3.44) holds and hence
(B(x)A(x
<p(x)
<p(x)
This means that <p (x) may be obtained from formula A3.50) on choosing g(x)=<p (x).
Consequently, A3.50) gives the general solution of equation A3.38).и
§ 4. Characterization of polynomials. If <p(x) is a polynomial of degreed я, then
A3.51) 4i+V(*) = 0
for every ю>0. A converse theorem is also true to a certain extent: T. Angheluta
proved D) that the only measurable functions satisfying equation A3.51) for all x
and all a» 0 are polynomials of degree ^n. Equation A3.51) contains two variables,
x and со, and thus is not of the type treated in this book. However, if we make stronger
regularity suppositions regarding <p{x), then it is enough to assume that A3.51)
holds for some values of со in order to obtain polynomials as the only solutions B).
Below we prove two theorems to this effect.
(!) Cf. e. g. Ghermanescu [22], pp. 305-318. Cf. also Kuczma [25] concerning further ref-
references. t
B) Montel [3], [5], [81-[12], Popoviciu [2], Ghermanescu [22], Kuczma [25].

272 CHAPTER XIII. Linear equations of higher orders
Theorem 13.5. (Popoviciu [2]; cf. also Montel [3], [14].) If <p(x) is continuous
at n +1 points and satisfies the two equations
A3.52) <
for all real x, where co1/co2 is irrational, then <p(x) is a polynomial of degree<и.
Proof. (*) From A3.52) we get in particular
A3.53) Anlo+1
for every x and any integers p, q. Let us keep x and q fixed. Then it follows from
A3.53) that there exists a polynomial P(t) (depending on x + qco2) of degree<n
such that <p(x+pco1 + qco2) = P(p). We may write
i = 0
whence
л
<p (x + pcOi + qco2) = ? at{x + qco2) p'.
t = 0
Now, A3.52) gives A"^ 1q>(x+pco1 + qco2)=0, whence, since the operators/ is linear,
i=0
This holds for every integer p, whence
+1 0, i = 0, ...,«.
Consequently, afa+qcoj (with x fixed) coincides for integral q with a polynomial
Qt(q) of degree <и. Hence we obtain
л
A3.54) <p(x+poj1+qco2)= X ai}{x)plqJ=Wx(j>,q),
for all integers p, q and every x. For a fixed x the left-hand side of A3.54) depends
only on pa>1+qco2. Therefore Wx must be a polynomial of pco1 + qco2: Wx{p,q)
= Ux(pco1 + qco2), and the degree of Ux cannot exceed n. Setting Vx(t)=Ux(t—x)
we finally obtain from A3.54)
A3.55) <p(x + pco1 + qco2)=Vx(x + pco1 + qco2),
where Vx(t) is a polynomial (depending on x) of degree < n.
Now suppose that <p is continuous at a point x0. Since оз1/оз2 is irrational,
for every x there exist sequences of integers pv, qv such that lim. (pv со ! + qv со2)
V->00
= x0 — x. Vx(t), being a polynomial, is continuous everywhere. Therefore, setting
in A3.55) p =pv, q=qv and passing to a limit, we obtain <p(xo)= Vx(x0). This shows
(') For n=0 this theorem results immediately from theorem 11.1.

4. Characterization of polynomials 273
that all the polynomials Vx(t) must coincide at every point of the continuity of <p.
Since <p is continuous at n +1 points and the degree of any Vx(t) does not exceed n,
all the Vx must be identical, i.e. Vx in A3.55) is independent of the index x.m
If we assume more about <p, then a single equation A3.51) will be sufficient
to draw the same conclusion. Without loss of generality we may assume that со = 1.
Theorem 13.6. (Kuczma [25].) If cpeMn[(a, oo)],a> — oo, and satisfies
A3.56) <
for xe(a, oo), then <p(x) on (a, oo) is a polynomial of degree ^.n.
Proof. Put \i/j{x)=A{q>{x),j=Q,...,n. By lemma 0.12 ^eM""'[(fl, oo)]. We
shall show that y/j{x) is a polynomial of degree<и—/ For j=n we have y/ne
M°[(a, oo)] and by A3.56)
for x e (a, oo). In virtue of theorem 5.3 i^n(x)=const, i.e. it is a polynomial of degree
zero. Now suppose that for a certain j < n the function y/J+1(x) is a polynomial of
degree <n —j— 1. Then the function y//x), belonging to the class M"~J[(a, oo)],
satisfies in (a, oo) the equation
By A3.56) J"-V;+i(x)=^"+V(x) = 0 and y/j+1(x), being a polynomial of degree
<и—7 —1 <и— 7, belongs to M""y[(a, oo)] (cf. lemma 0.11). Thus in virtue of theorem
5.13 y/j(x) is a polynomial of degree < n —j.
It follows by induction that <p(x) = y/0(x) is a polynomial of degree <и on
(а, оо).и
Since the class M"[(a, oo)] may be alternatively defined as the class of those
measurable functions <p(x) which satisfy
or <0 for all x>a and ю>0,
(Popoviciu [1]), theorem 13.6 may be given the following equivalent formulation.
Theorem 13.7. If <p(x) is a measurable function in (a, oo),a>— oo, such that
Л"а>+ 1<р(х) has a constant sign for x>a, co>0, andvanishes (identically in x)for a single
value of со, then <p(x) on (a, oo) is a polynomial of degree<,n.
This is an improvement of the theorem of Angheluta.
18 Functional equations

CHAPTER XIV
INVARIANT CURVES
§ 1. Unique invariant curve. Suppose that we are given a transform on the
real plane
A4.1) x'=f(x,y), y'=g(x,y).
Let y=q>{x) be the equation of a curve ^. By A4.1) *% is transformed into another
curve ^". If the equation of ^" can be written in the form y = y/(x), we shall call
the function y/ the transform of cp and we shall write y/= T[q>].
It may happen that ^ is transformed by A4.1) into itself, i.e. Т[ф\ = <р. Then ^
is called an invariant curve under transform A4.1). Analytically this means that
y = q>(x) implies y' = q>(x'), i.e.
A4.2) (p\_f{x,(p{x))-\=g{x,(p{x)).
Equation A4.2) is therefore called the equation of invariant curves (*).
We shall study invariant curves in a neighbourhood of a fixed point of transform
D.1). Without loss of generality we may assume that that fixed point is placed
at the origin. But we shall also make more restrictive hypotheses regarding the
functions/and g.
Let Kr denote a disc of radius r centred at the origin.
Hypothesis 14.1. Functions f(x,y) and g{x,y) are defined in a region Q con-
confining the origin as. an inner point. Moreover,
A4.3) f(x,y)=Xx + u{x,y), g(x,y)
where the functions и and v fulfil
J'l-^l].
(!) Concerning this and simflar equations cf. Poincare [1], [2], Hadamard [1], Lattes [l]-[8],
[12], Buhl [1], Birkhoff [5], Curtiss [1], Lewis [1], [3], [4], H. Schmidt [1], Urabe [3], McCarthy
[1], J. Moser [1], Siegel [8], Montel [13], Kyner [1], Bass, Lewis [1], Yorinaga [1], Hartman [2], Ni-
shino, Yoshioka [1], Brydak [4].
A4.4)
and
/1 Л C\
A4.5)
for(xt,y
ii(O,O) = »(O,Q) = O
" (^i, >>i)-" (x2, y2)\ <e(r) [\xt -x2
v (xi, yt)-v (x2, y2)\ <е(г)[|хх -х2
+
+
i)> (.x2, y2) e KrcQ, where lim e(r)=O.
r->0

1. Unique invariant curve 275
This hypothesis is certainly fulfilled if/and g are of form A4.3), of class C1
in Q, and the derivatives of u, v vanish at the origin. On the other hand, if/, g are
of class C1 in Q and the characteristic roots X, ц of the matrix
A4 6) P°'0) />'@'°)l
are real and distinct, then transform A4.1) can be given form A4.3) by choosing
the eigenvectors of matrix A4.6) as the axes of the coordinate system.
Theorem 14.1. (Hadamard [1], Lattes [3], Montel [13].) Let hypothesis 14.1
be fulfilled and
A4.7) 0<ц<Л, л>1.
Then there exists a unique solution q>(x) of equation A4.2) defined and fulfilling a
Lipschitz condition in a neighbourhood of x=0. This solution is tangent to the x-axis
at x = 0 and is given by the formula
A4.8) q>(x)=1imq>Jx),
n->oo
where <pn+i = T[<pn] and <p0 is an arbitrary function defined and fulfilling a Lipschitz
condition in a neighbourhood ofx=0 and such that <po@)=0.
Proof. Let us choose positive numbers с and L in such a manner that the set
be contained in Q. The number с will later be subjected to further restrictions.
Let !F be the space of functions cp(x) defined on /=< —c,c>, fulfilling the
Lipschitz condition
A4.9) ^(x^-yix^L^i-x^ for Xl,x2el
and the condition
A4.10) <p@) = 0.
Endowed with the metric
p -._ <Pi(x)-<P2(x)
i
is a complete metric space. (Note that by A4.9) and A4.10) we have p [q>t,
and, moreover, sup \<Pi(x) — q>2(x)\^cp[p1, q>2\) The graphs of functions tpelF lie
inside the set S.
We shall show that if с is sufficiently small, then the transform T is defined
in J*" and maps it into itself. Let <p e 3F, Then y/=T[q>] is defined by the condition
18»

276 CHAPTER XIV. Invariant curves
that the value of y/ at x'=f(x, <p(xj) is y'=g(x, <p(pcj). In order to prove that this
rule actually defines a function on / we must show that for every x' el the equation
x'=f(x,<p(x)), i.e.
A4.11) x'=Xx+u{x,(p(xj),
has a unique solution xel.
Suppose that for a certain x' e 7A4.11) does not hold for any xel. It follows
from A4.4), A4.5), A4.9) and A4.10) that
where r = с \IL2 +1, whence
|m(x,9»(x))|<|x|
if с is chosen small enough. Hence it follows by A4.7) that
A4.12) sgn [Xx + u(x,q> (x))] = sgn x
for xel. In virtue of A4.5) and A4.9) the right-hand side of A4.11) is a continuous
function of x and by A4.4) and A4.10) it equals zero for x=0. Thus our supposition
and condition A4.12) imply that
for all x e I such that sgn x = sgn x'. This gives by A4.4) and A4.5)
A4.13) [x'|>A|x|-e(r)[|x| + |9»(x)|],
(r = c sib2 +1). Dividing A4.13) by c, we obtain in view of A4.10) and A4.9)
Letting |x|-»c, sgn x=sgn x', we obtain
A4.14) l>A-e(r)(l+L),
which is impossible since the right-hand side of A4.14) exceeds 1 provided that с
is small enough. Consequently, our supposition that A4.11) has no solution must
have been false.
Similarly, supposition that A4.11) has two distinct solutions xt, x2 leads to a
contradiction. For then

1. Unique invariant curve
277
i.e.
, (р{х1))-и{х2, q>{x2))\
by A4.5), and hence Л<е(г)A+Г) by A4.9). But e(r)(l+L) approaches zero as
c->0, and thus it is smaller than 1 if с is small enough. This contradicts A4.7).
Thus y/= T[<p] defines a function on /, provided с has been chosen small enough.
y/(Q)=0 results from A4.10) and A4.4). Now let us take arbitrary x[,x'2el, х\Фх'2>
and let x1,x2el be such that x'1=f(x1,<p(x1)), x'2=f(x2, <p(x2)). We have by
A4.3), A4.5) and A4.9)
g{x1,<p(x1))-g(x2,<p(x2))
x\-x2
f(x1,<p(x1))-f(x2,<p(.x2))
v(x1,<pOq))-v(x2, q>(x2))
1,(р (xj) -u(x2,<p (x2))
by A4.7), provided с is sufficiently small.
Next we show that if с is small, then Г is a contraction map. Let q>1, <p2 be in
J5" and let x'el. Define х1гх2 by x' =f(xt, (Piixi)), x'=f(x2, <p2(x2)). Then
A(x1-x2) = u(x2, (p2{x2))-u(xl, ipiixj)), whence
i.e.
A4.15)
Now
A4.16)
U-e(r)
-<P2(x2)\ +s(r) [|xx -x
A-e(r)
by A4.15). Further, we have
A4.17)
But
A4.18)
)-<P2(xi)\ + \<P2{x1)-<p2{x2)\

278 CHAPTER XIV. Invariant curves
Inequalities A4.17), A4.18) and A4.15) give
Ix'l
)()|
A4.19)
and A4.16) and A4.19) yield
A4.20) "lC°-ftM
If с is small enough, the coefficient of p in A4.20), say S, is smaller than 1 in virtue
of A4.7). Taking in A4.20) the supremum over,*' e /, we obtain
i.e. Г is a contraction map. The existence and uniqueness of a solution <p
of equation A4.2) and formula A4.8) now follow directly from Banach's fixed
point theorem.
If 0<L' <L, then the solutions <pL and <pL. of equation A4.2) corresponding
to the constants L and L' in A4.9) must coincide in the smaller of the corresponding
intervals IL and Iv. In other words, the solution of A4.2) furnished by the present
theorem fulfils A4.9) with arbitral ily small L if we confine ourselves to a sufficiently
small neighbourhood of x = 0, i.e. the curve y=<p(x) is tangent at x=0 to the
x-axis.a
§ 2. Lack of uniqueness. Condition A4.7) is essential for the uniqueness of the
solution of A4.2). If this condition is replaced by
A4.21)
then equation A4.2) has a solution depending on an arbitrary function.
Theorem 14.2. (Montel [13].) If hypothesis 14.1 and condition A4.21) are ful-
fulfilled, then equation A4.2) has a continuous solution fulfilling a Lipschitz condition
depending on an arbitrary function. More exactly, to every constant L > 0 positive
constants R, M can be found such that for every (x0, y0) e KR, хоФ0, and for every
function <Po(x) continuous and fulfilling the Lipschitz condition with the constant L
on /0 = <x1, xoy resp. <x0, *!> and such that
A4.22) ?'o(^o) = J'o» PoOO^i, |Ы*)|<Я f°r
A4.23) |^0(x)|<M|x| for xelo,
where
A4-24) *! =/(x0, y0), У! = д (х„, y0),
there exists a unique function <p(x) which satisfies equation A4.2) in /=<0, xo>, resp.
(xo > 0>, and fulfils the condition
A4.25) <p(x)=<po(x) for xelo.

2. Lack of uniqueness
279
This function is continuous in I, fulfils the Lipschitz condition A4.9), is differentiable
at x=0and(p'@) = 0.
Proof. Chose a S such that A<5<1 and R such that
A4.26)
for
R will subsequently be subjected to further restrictions. The constant M will be
specified later. For the moment we regard it as fixed.
Suppose that \x\^r, \y\<r, r^R. Then by A4.5), A4.4), A4.21) and A4.26)
\в(х, у)\<11
This proves that the origin is an attractive fixed point of A4.1) and KR is contained
in the attractive domain. Consequently, the sequence (хп, у„) defined by
A4.27) xn+! =/(*„,yn), у„+1=д(х„, у„)
tends to the origin. (Note that A4.27) agrees with A4.24).) Moreover, if
A4.28)
then
Уп+1
^N+EW[H
provided R has been chosen sufficiently small. By A4.23), A4.28) holds for и = 0,
and thus it is valid for every и>0. Now, if R is such that е(г)<A -Л)/A +М) for
, then
.A4.29)
On the other hand, according to A4.5), A4.4) and A4.28) we have
—X
A4.30)
if R is small enough. A4.30) proves that xn+1/xn>0, i.e. in view of A4.29) the
sequence х„ is strictly monotonic. In the sequel we assume that xo>0, so that xn
is decreasing. The proof in the other case is similar.
Let q>0(x) be a continuous function on /0 fulfilling the Lipschitz condition
with the constant L and conditions A4.22), A4.23). The function f(x, <po(x)) is
continuous in /0 and by A4.22) and A4.27) f(x0, <po(.xo)) = x1, f(xlt <Po(.x1)) = x2.
As in the proof of theorem 14.1, it is shown that x'=f(x, <po(x)) may have at most
one solution 0<x<R for any 0<x'<R provided R has been chosen sufficiently
small. Consequently, f(x, <Po(x)) maps /0 onto (х2,х^ in a one-to-one manner

280
CHAPTER XIV. Invariant curves
and the formula <Pi[f(x, <po(x))]=g(x,<po(x)) defines a continuous function ^(x)
for x e <x2, X!>. Again, in quite the same way as in the proof of theorem 14.1,
we check that the function <Pi(x) fulfils in <x2,X!> the Lipschitz condition with
the constant L provided R has been chosen sufficiently small.
Now we put
A4.31) 9n+iU(x><Pn(.x))l = g(x,q>n(xJ) for хе(х„+1,х„} .
It can easily be shown by induction that if R is chosen sufficiently small, then
for every n the function <р„(х) is defined and continuous in <хп+1, х„>, fulfils
the Lipschitz condition with a constant L and fulfils the conditions
A4.32) q>?xJ = yH, <pn(xn+1) = yn+l, \(pn(x)\^8"R for хе<х„+1,х„>,
A4.33) |9>„(х)|<М|х| for хб<х„+1,хп>.
Consequently, the formula
pn(x) for хе<х„+1,х„>,
) for x=0,
defines a function <p(x) on /. In view of A4.31) <p(x) is a solution of equation A4.2)
fulfilling A4.25), and evidently it is the only solution with this property. According
to A4.32) <p(x) is continuous in / and fulfils A4.9) since every function ^„(x) does.
In order to prove the differentiability of <p(x) at x=0 we put a(x) = <p(x)/x for хфО,
<x@) = 0, and h(x)=f(x, <p(x)). The function h(x) belongs to S%[I], and the function
a(x) is by A4.33) bounded: |a(x)|<M, and satisfies in /the equation
where
for x2+y2>0, G@,0) = 0.
G{x,y)^\
f(x,xy)
In view of A4.4) and A4.5) the function G(x, y) fulfils the inequality
which shows that G(x, y) is continuous at @, 0) (the continuity of G in a vicinity
of the origin follows from A4.3) and A4.5)), and
\G{x,y)-G{x,y)\
g(x,
[цМ
xy)[f(x,
+ e(l+Af
[X
ху)-/(х,хуУ]+/(х
f(x,xy)f(x,
)]е + [Я+еA+М)][у
-e(l+M)]2
,xy)[g(x,xy)-g(x,
xy)
\y-y .
*y)]

2. Lack of uniqueness 281
where e=e(\/x2(l + M2)). When x tends to zero, the coefficient of \y—y\ in the
above inequality approaches ц/Л<1. By theorem 3.6 (J2=/x< — M, M}) we have
lima(x) = 0 provided that M is sufficiently small ('). Consequently, <р'Щ exists
jc-»O
and equals zero.B
Hypothesis 14.1 implies that transform A4.1) is invertible in a neighbourhood
of the origin. In fact, suppose that in every disc Kr there are distinct couples (x,, yx),
(х2,У2) such that
A4.34) f(x1,y1)=f(x2,y2), g(x1,y1) = g(x2,y2).
The second equality of A4.34) implies in view of A4.5)
A4.35) У1-У2 <—^; *i"*2 ,
and the first equality of A4.34) yields by A4.5) and A4.35)
which shows that xx = x2, and hence by A4.35) also yx=y2, provided r is small
enough.
The inverse transform to A4.1)
x'=f*(x,y), y'=g*(x,y),
has the form
f*(x, у)=Г1х + и*(х, y), g*(x, y)=n~1y + v*(x, y),
where the functions u* and v* fulfil conditions A4.4) and A4.5) (with another func-
function ?(/¦)). The equation
is equivalent to A4.2) in a neighbourhood of the origin; if a curve is invariant under
transform A4.1), it is also invariant under the inverse transform. Thus theorems
14.1 and 14.2 imply the following
Theorem 14.3. (Montel [13].) Let hypothesis 14.1 be fulfilled. If 0<ц<1<Л,
then there are exactly two invariant curves (?) under A4.1): one tangent to the x-axis
and the other tangent to the y-axis. If \<ц<Х, then there is exactly one invariant
curve tangent to the x-axis and infinitely many (depending on an arbitrary curve)
tangent to the y-axis. IfQ<n<X<\, then there is exactly one invariant curve tangent
A) This is the only restriction on M, but even this can be avoided. A positive M may be
fixed quite arbitrarily, but then one more condition on R appears: it must be chosen so that the
inequality \G(x,y)-G(x,y)\<e\y-y\, в<1, hold for \x\<R, \y\<M,\y\<M.
B) By a curve we mean here a continuous curve y—q>(x) or x—<p(y) with <p fulfilling a Lip-
schitz condition in a neighbourhood of the origin and passing through the origin.

282 CHAPTER XIV. Invariant curves
to the y-axis and infinitely many {depending on an arbitrary curve) tangent to the
x-axis.
The cases where matrix A4.6) has equal roots {X=fi), or one of the roots equals
1 or 0, or transform A4.1) cannot be given form A4.3) by a real change of variables,
are more involved and will not be dealt with here. In the case where the functions
f{x,y) and д{х, у) are strictly monotonic with respect to each variable equation
A4.2) has recently been investigated by D. Brydak [4].
§ 3. A problem of continuation. The solutions of equation A4.2) obtained in the
preceding two sections are local. They are denned only in a small neighbourhood
of x=0. However, the problem of extending these solutions onto a larger x-interval
is much more difficult than in the case of equation A.1). Here we shall treat one
example (which, however, does not enter under the case previously discussed).
We shall deal with the equation (Kuczma [10], M. K. Fort [2], Lush [1])
A4.36) q>{q>{x)) = g{x,(p{x)),
which is a particular case/(x, y)=y of A4.2). The function g{x, y) will be subjected
to the following conditions:
Hypothesis 14.2. The function g{x,y) is defined in the set
is continuous in Q and strictly increasing with respect to each variable; moreover,
A4.37) <?@,0)=0, g{a,a)=a,
A4.38) g{x,x)<x, g{x,y)<y for 0<x<a, fi{x)<y<x,
A4.39) g(x,p{x)) = p{x) for
It follows from the above conditions that the function /?(x) is continuous and
strictly increasing in <0, a>, /?@)=0, fi{a)=a, p{x)<x in @, a).
Lemma 14.1. Let hypothesis 14.2 be fulfilled and let <p{x) be a continuous solution
of equation A4.36) defined in an interval /<=@, a) and passing through an inner point
of Q. Then for every xel we have (x, q>{xj) e int Q and cp{x) is strictly increasing.
Proof. If (x, q>{xj) eint Q does not hold for every xel, then on account
of the continuity of <p there must exist an x0 e I such that either cp{x0)=x0 or cp{x0)
=j3{x0). The first possibility yields, by A4.36), g{x0, xo) = xo, which is impossible
according to A4.38). The other possibility gives <p [/?(*o)] =/?(xo)> i-e- with x1=p{xQ),
<p{xi)=x1. This again is possible only if x±=0 or x1=a, whereas in / we have
0</?(x)<a. Consequently, (x, g>{x)) e int Q for x e /.
Now we shall prove that the function <p{x) is single valued. Let <p{x1) = <p{x2)
= <p0 for х1г х2 е /. From A4.36) we get д{х1г <ро)=д{х2, <p0) and, since g is strictly
increasing with respect to x, Xi = x2. Consequently, <p{x), being continuous, is
strictly monotonic. Let us take an xt e /and put x2 = <p(x1). Since (xl3 ^(*i)) e int Q,

3. A problem of continuation 283
we have x2<xy. Hence by A4.36) and A4.38) <p(x2)=g(x1, ?'(x1))<^(x1). Thus
<p(x) must be increasing. ¦
Lemma 14.2. Let hypothesis 14.2 be fulfilled. For an arbitrary point (xo,yo)
e int Q and an arbitrary function (po(x) which is defined, continuous and strictly in-
increasing on <j0, *o) and fulfils the conditions
A4.40)
there exists exactly one solution <p(x) of equation A3.36) in <0, xo> such that
A4.41) <p(x) = <po(x) for xe(yo,xo}.
This solution is continuous and strictly increasing in <0, xo> .
Proof. We put Xi=j0 and
A4.42) х„+2=д(х„,хп+1).
It follows from conditions A4.37)-A4.39) that the sequence xn is strictly decreasing
and tends to zero. We define functions <pjx) for x e <xn+1, xn> by
<Pn(x)=9(<Pn-i(x),x), хе(хя+1,хпУ, n = l,2,...,
and put
>„(*) for xe(xn+1,xny,
<1443) '«^0 for x=0.
It can easily be verified that function A4.43) is the unique solution of A4.36) ful-
fulfilling condition A4.41) and that it has the required properties.B
Now we are going to handle the delicate question of the possibility of a con-
continuation of the solution obtained to the right of x0 (Kuczma [10], M. K. Fort
12]). Let R{x, y) be the homeomorphism of Q into itself defined by
R(.x,y) = (y,g(x,y)).
Solutions of A4.36) are invariant curves with respect to the transform (*', y')=R{x,y).
In order to continue <p(x) to the right of x0 we must form a curve consisting of the
points J?^, (p{x)). This is possible if (x, q>o(.x)) e R(Q) for x e <x1;- xo>. We can
then define the function <p(x) for xe<xo,x_1>, where x_j is determined by
^(x_!, xo)=x1. If we want to continue the 'solution obtained further to the right,
we must have (x, <p(x)) e R(Q) for xe<xo,x_1>, i.e. (x, <po(xf) e R2(Q) for xe
(xy,xoy. Generally, in order to continue q>(x) to the right up to the point x_n
(where xn is defined by A4.42) also for negative ri) we must have (x, (po{xjj e R"(Q)
for xe<x1; xo>.
Now, it follows from hypothesis 14.2 that R(Q)c=Q, but R(Q)^Q. More exactly,
R(Q) = {(x,y): 0^x^a,g(x,x)^y^x}. For, if (x,y)eQ and (x',y')=R(x,y),
then 0<y?(xKj = x'sSx<a and g(x',x')=g(y,y)^g(x,y)=y'^y = x'. On the
other hand, let O^x'^a, g(x', x')^y'^x', and put u=fi~\x')>x'. Then, as t
increases from x to u, git, x) increases fromg(x', x) to g(u, x')=g(u, P(u))=fS(u)

284 CHAPTER XIV. Invariant curves
= x' and there is exactly one x e <x', u) such that g(x, x')=y'. Hence (x', y') =
R(x,x')eR(Q).
We see that in general it will not be possible to continue cp(x) to the right as
far as x=a. For this purpose we must have
A4.44) (x,(po{xj)eC\R\Q)=Q* for xe(Xl,xoy,
n=0
which is certainly quite an exceptional case. Anyhow, following M. K. Fort [2],
we shall show that Q* is not empty and that it is possible to find a continuous and
strictly increasing function <po(x) on <x1; xo> fulfilling A4.40) and A4.44).
Let f be the space of all continuous and strictly increasing functions y/(x)
on <0, a> such that (x, y/(x)) e Q for x e <0, a) with the usual metric
i, У2З = sup W^x) - y/2(x)\ .
<o,a>
Я* is partially ordered by the relation ^i<^2 if and only if y/y{x)^y/2{.x) for all
x e <0, a>. We define the map T[y/] of 4> into Ф by
T[?-\(x)=g(?-\x),x).
(Note that this map is identical with that described in § 1.) It is easily seen that
Г is one-to-one, continuous, and order reversing, i.e. ^i<^2 implies Г[^2]^Л^1]-
When p=(x, y) runs over a curve y=y/(x) with у/еФ, then R(p) runs over the
curve y=y/{x), where y/ = T[y/], and conversely.
Lemma 14.3. Let hypothesis 14.2 be fulfilled. Then for any e>0 there exists a
S>0 such that for arbitrary u, v, 0^u<v^a, v — u^-e, andfor an arbitrary у/ е T(V),
we have {ц/(р) — ц/ (u))/(v—u)^6.
Proof. Let us assume that lemma is false. Then there would exist e>0, ц/„
в Т(Х), un, vn such that 0^un<vn^a, vn—un^s and
,...„ Vn(vn)-4/n(un) 1
A4.45) <—, и = 1,2,...
Since Q is compact, we may assume the sequences pn = (un, у/п(и„)) and qn = (vn, y/n(vn))
to be convergent, and it follows from A4.45) that the limit points have equal ordi-
nates. Let the limit points be (и, у) and (v, y) respectively, and put p*=R~1(u, y)
and q*=R~1(v,y). Then p*=limR~1(pn) and q*=limR~1(qn), since R'1 is con-
continuous. The function i? is defined at the points in question, since by hypothesis
у/„ в T(T) we have pn,qne R(Q) and, since R(Q) is closed, also lirapn = (u, y) and
lim qn=(v, y) belong to R(Q).
Now set p* = (t,u), q*=(s,v); then g(t, u)=g(s, v)=y, whence t>s, since
u<v and g is increasing with respect to either variable. But, on the other hand,
R'1^) an<i R^) lie on the graphs of strictly increasing functions T~y[y/n],
and consequently, t>s implies u>v. Thus we have obtained a contradiction, which
proves the lemma. ¦

3. A problem of cotinuation 285
Lemma 14.4. Under hypothesis 14.2 the closure T{V) of T{?) is contained in
Proof. Let y/eTDr). This means that ^(x)=lim^n(x) uniformly in <0, a),
П-> GO
where y/n e Г(Я*). Evidently y/(x) is continuous in <0, a> and (x, y/(xj) e Q for
xe <0,a>, since Г(!Р)с!Р and Q is closed. We must show that y/(x) is strictly in-
increasing.
Take 0 «S и < v < a. By lemma 14.3 there exists a 3 > 0 such that (y/n(v) - y/n(u))l(v - u)
~^8 (as e one simply takes v—u). Hence (y/(v) — y/{uj)j{v—u)^S>0, which proves
that y/(x) is strictly increasing. ¦
Lemma 14.5. Under hypothesis 14.2 Т2(Ч*) forms an equicontinuous set of
functions.
Proof. Let us suppose the contrary. Then there exist an e>0 and sequences
0^x'n<x'n'^a, y/neT2(T) such that x'n'-x'n<l/n and y/n{x';)-y/n{x'n)^?. We write
pn=(x'n, ^„(O), qn=(x'n', y/n(x'n')), and we may assume that the sequences converge:
lim pn=p, limtfn=4.
n-*GO n-*GO
The limit points evidently have the same abscissa but different ordinates. Hence
R^) and R~\q) have the same ordinate but different abscissae. But this is not
possible by lemma 14.3, since J?~1(/3)=limi?~1(^n), R~1(q)=limR~1(qn), and
n-*GO n->GO
R~Kpn), R~y{qn) lie on the graph of Г^] belonging to TD>).m
Theorem 14.4. (M. K. Fort [2].) Under hypothesis 14.2 equation A4.36) has
continuous and strictly increasing solutions <p(x) in @, a)>.
Proof. Put y/0(x)=P(x) for xe@, a} and let y/n = T"[t//o], n = l,2, ... Thus
y/iix) is the identity function on <0, a) and we have
A4.46) , Vo<V2^V
It is easily shown by induction that
R2n(Q) = {(x,y):
R2n+\Q)={(x,y):
It follows from A4.46) that
A4.47) T(x)=lim^2n(x) and a(x)=limy/2n+1(x)
exist in <0, a> and r(x)^a(x) for 0<x<a; moreover,
00
Q*= p| R\Q)={(x, y): O^x^a, r(x)^y^a(x)} с intfl.
n=0
Now, for n ^2, y/n e T2CP) and by lemma 14.5 convergence A4.47) is uniform
in <0, a>. Since y/n e TD>) for и> 1, т and <т belong to Г(!Р) and, in virtue of lemma
14.4, т and a are in Я*. In other words, т(х) and <т(х) are continuous and strictly
increasing functions on <0,a>, x{x)^a{x). Furthermore, if (x0, y0) eQ*, then

286 CHAPTER XIV. Invariant curves
(yo>g(xo,yo))=R(xo, Jo)ef2*. Therefore it is possible to find a continuous and
strictly increasing function q>0(x) on <x1; xo> = <jo. *o> fulfilling conditions A4.40)
and A4.44). By lemma 14.2 this function can be uniquely extended to a (continuous
and strictly increasing) solution of equation A4.36) in <0, x0}, and in view of the
subsequent remarks this solution can be continued to the right up to the point
x=a.m
If we fix a point x0 e@, a), then, taking all possible y0 e <т(х0), <т(хо)> ап^
all possible continuous and strictly increasing functions cpo{x) on <jo> *o> fulfilling
A4.40) and A4.44), we obtain all continuous and strictly increasing solutions g>(x)
of equation A4.36) in <0, a}. (In virtue of lemma 14.1 these are in fact all the con-
continuous solution of A4.36) in <0, a>.) If r(x)=a(x), the solution is unique: cp{x)
= r(x)=a(x). If т{х)фа{х), the solution depends on an arbitrary function. M.
K. Fort [2] has shown that the latter case can actually happen.
§ 4. Euler's equation. This name has been given to the equation (*)
A4.48) <p(x + (p(x))=<p(x),
which is a particular case of A4.2). We shall prove the following
Theorem 14.5. (Kuratowski [1], Wagner [1].) The only solutions of equation
A4.48) in (—oo, oo) possessing the Darboux property are constant functions.
Proof. Let <p(x) be a function with the Darboux property satisfying A4.48)
for all x e (— oo, oo). If ^(x)=0, then it is constant in accordance with the theorem.
So let us assume that there exists an a such that cp{a)^Q, e.g.
A4.49) q>(a)>0.
(If <p (a) < 0, the proof is analogous.)
We write
A4.50) y/(x)=x
and we prove that
A4.51) V\x)=
for n= 1,2, ... For n = \, A4.51) holds by definition. Assuming it true for an
n^ 1 we have by A4.50) and A4.48)
y,n+ \x) = y/"{V (x)) = y,(x) + n<p(y, (x))
Thus A4.51) is generally valid. Equation A4.48) can now be written as д>(у/(хУ)
= <p{x) and by induction we obtain hence
A4.52) <p[y,Xx)-] = <p(x)
for n = \, 2, ...
(!) Euler [4], Kuratowski [1], Lussy [1], Wagner [1], Brydak [1], [2]. This equation has no
connection with Euler's functional equation A1.18) of the Gamma function.

4. Euler's equation 287
In virtue of A4.49) and A4.51) the sequence ^"(a), n = 0,1, 2, ..., is strictly
increasing and tends to infinity. We shall show that for x e <a, у/(а)У and n = 1, 2,
3, ..., we have
A4.53) V\a)^yf(x)^?n+ \a) .
If A4.53) were not true, then there would exist a b e (a, y/(a)) such that either
A4.54) ?\b)<?n(a)<?n+1(a),
or
A4.55) y/\a)<?n+1(a)<y/\b) .
The function y/^x) has the Darboux property, like <p(x). Thus in case A4.54) there
must be a ce(b,y/{dj) such that ц/п{с) = ц/\а). By A4.52) we obtain hence <p(c)
= (p{d) and by A4.51) c=a results. But this is impossible, since a<b<c. Similarly
in case A4.55) there must be а с е (a, b) such that y/"(c) = уУ+ 1(а), this in turn implies
<p(c) = <p(a) and c=y/(a), which contradicts ce{a,b). Thus A4.53) has been proved.
From A4.53) and A4.51) we obtain
which is possible for all n only if g>(x) = g>(a) for xe <a, y/(a)y. Replacing point
a by y/\d) we obtain (since <p[y/k(a)] = (p(a)) <p(x) = <p(a) for x e <M/\d), y/k+1(a)y.
Consequently, (p{x)=cp{a) for all x^a.
The point a might have been an arbitrary point from (-00,00) at which <p
is positive. Thus what has been proved so far shows that <p{x) cannot assume
two different positive values. Since <p(x) has the Darboux property (and was assumed
to be positive at a point), it must be constants
The assumption of the Darboux property is essential for the validity of the
above theorem. The function
(h±>0 for x>0,
for x=0,
U2<0 for
satisfies A4.48) in (— 00, 00) but is not constant.
Continuous solutions of the more general equation
have been determined by D. Brydak [2] under the condition that the function/(x, y)
is continuous and strictly monotonic with respect to each variable.

CHAPTER XV
FRACTIONAL ITERATES
§ 1. The Babbage equation. In Chapter VI, § 7, we encountered the condition
that an iterate of f(x) should be the identity function. Functions fulfilling similar
conditions also played a rdle in the considerations of Chapter XIII, § 3. Now we
are going to general investigation of functions <p(x) such that their nth iterate is
the identity function:
A5.1) (p\x)=x, n>\.
A5.1) belongs to the oldest functional equations (J). In the mathematical litera-
literature it is known as the Babbage equation, since Charles Babbage was the first to
investigate it.
In the case of equation @.49) the set of x's in which a solution was sought
was to a certain degree determined — or at least restricted — by the given func-
functions f(x) and F(x, y, z). In the case of equation A5.1) the situation is different;
here the set in which a solution is to be constructed can be prescribed a priori. For
convenience we shall adopt the following definition, already for the case of more
general equations.
Definition. We say that a function <p(x) satisfies in a set E the functional equa-
equation
A5.2) F[x,<pn\x), ...,^(x)]=0,
where п1г ..., nm are positive integers, if ? is a submodulus set for <p(x) and A5.2)
holds for every xeE.
As pointed out by P. I. Haidukov [1], the condition that E should be a sub-
modulus set for q> causes no loss of generality.
We start with the following lemma, which we prove already for the more
general equation
A5.3) <p\x) = g(x).
(!) Babbage [l]-[4], Serret [1], Johnson [1], Livencov [1], Cayley [4], Tanner [3], [4], Rausen-
berger [1], Pellet [1], [2], Laisant [1], Leau [3], Lemeray [13], Jaggi [1], Spiess [1], Pincherle [3],
[4^ Ritt [1], Bennett [4], Julia [2], Chajot [2], Hornich [1], Ghermanescu [8], Lojasiewicz [1], Heins
[2], Haidukov [1], Vincze [1], McShane [1], Kister [1], Bogdanov [1], Wallace [1].

1. The Babbage equation 289
Lemma 15.1. If the function g(x) is invertible in E and <p(x) satisfies equation
A5.3) in E, then also <p(x) is invertible in E. If g(x) mapsE onto E, then so does <p(x).
Proof. Assume that (p(x1) = (p(x2), х1г x2 e E. Then by induction <pi(x1) =
<p\x2), /=1,2, ..., whence for i=n we get g(x1)=g(x2). If g is invertible, this
implies x±=x2, and consequently q> is also invertible.
By induction we also get <р\Е)^<р(Е)<=Е, 7=1, 2, ..., whence for j= n
g{E)^q>(E). Thus g(E)=E implies <p(E)=E.m
Now we are going to describe the general solution of equation A5.1) in a set E.
By lemma 15.1 <p(x) must be invertible. It follows from A5.1) that there exist only
closed orbits under cp in E and that Er\(&k[g>]^0 implies that к divides n. Thus
let l=no<...<nr=n be the complete set of divisors of и and let
A5.4) ?=U U U)
j=o y=i
be an arbitrary decomposition of E into disjoint sets such that for every / the sets
U\, ..., U\t have the same cardinality. Further, for 1^/^r and 1<_/<иг — 1, let
<Pifx) be an arbitrary one-to-one map of U) onto U]+1.
Theorem 15.1.0 The formula
A5.5) <p(x)=
for xeUl,
for xeUlj, j = l
dD") f°r
defines the general solution of equation A5.1) in E.
Proof. It can easily be seen that (p{x) is by A5.5) denned in the whole of E
and maps E onto E. Moreover, the reader will verify that we have
A5.6) <pni(x)=x for xeyjlfj.
A5.1) results directly from A5.6), i.e. A5.5) actually defines a solution of A5.1)
in?.
On the other hand, let cp{x) be a solution of equation A5.1) in E. For every
0</<r, let UI be an arbitrary set containing exactly one point of every orbit con-
contained in Щ[<р\, and put и)=ср{и)^^) for j=2, ...,щ. Then, according to lemma
"<
15.1, the sets U),j=\, ...,щ, have the same cardinality, are disjoint, and \J Uj —
j=i
%[ф\- Hence decomposition A5.4) follows in view of the previous remarks. Now,
if we put (Pij{x) = q>{x) for xe U), l^i^r, 1<7<и, —1, then formula A5.5) results
from A5.1), and by lemma 15.1 the function <ри maps U) onto U}+1 in a one-to-one
manner. Thus formula A5.5) contains all solutions of equation A5.1) in E.m
@ Lojasiewicz [1]. Cf. also Haidukov [1], Isaacs [1].
19 Functional equations

290 CHAPTER XV. Fractional iterates
Functions satisfying A5.1) with и = 2:
A5.7) (p\x) = x,
are called involutory (*). By a slight modification of the above argument the fol-
following result may be proved.
Lemma 15.2. Equation A5.7) has in a real interval I={a,b) [or /=<a,Z>>] a
continuous and strictly decreasing solution depending on an arbitrary function. More
exactly, for an arbitrary ce(a,b) and an arbitrary continuous and strictly decreasing
function (po(x) on (a, c> [or <a, c>] such that
lim <po{x) = b, <po(c) = c,
there exists exactly one continuous and strictly decreasing solution <p(x) of equation
A5.7) in I fulfilling <p(x) = (po(x) for x e (a, c). This solution is given by the formula
( q>o (x) for x e (a, c> [or <a,c>],
A5.8) <p(x)=< _!,,.. / 1л г t u\i
{ <p0 \x) for xe(c,b) [or (c,b}J.
Thus formula A5.8) gives the general continuous and strictly decreasing solution
of equation A5.7) in I.
It follows from theorems 15.2 and 15.3 below that A5.8) does in fact give the
general strictly decreasing solution of A5.7) in / and, except for g>(x)=x, the
general continuous solution.
As a corollary to theorem 15.1 we find that, unless the set E is finite, equa-
equation A5.1) has infinitely many solutions in E. This need not be true any longer if
we impose some further conditions on q> B).
Lemma 15.3. The only increasing solution of equation A5.1) in a real set E is
<p{x) = x.
Proof. The statement of the lemma is equivalent to E=<&1[g>]. Thus for an
indirect proof, let us assume that <p(x) is an increasing solution of A5.1) in E and
that there is an x0 in Er\(?k[q>], k>\. Let xm be the minimal element in Cv(x0).
The function q> maps СДхо) onto itself; consequently, min cp(x)=xm. But (p(x),
since it is increasing, assumes its minimum on СДхо) at xm. Hence q>(xm)=xm and
xme^1[<p], whereas by theorem 0.1 xmeC<l,{xQ)<^^&k[(p\, k>\. The contradiction
obtained proves the lemma. ¦
Theorem 15.2. (Vincze [1], McShane [1].) If <p(x) is a monotonic solution of
equation A5.1) in a real set E, then either g>(x)=x or cp{x) is a decreasing involutory
function.
Proof. If <p{x) is increasing, then q>{x) = x by lemma 15.3. If <p(x) is decreasing,
(!) Laisant [1], Lattes [4], van Uven [2], Sierpinski [4], Aczel [1], Hornich [1], Revuz [1],
Heins [2].
B) Revuz [1], Haidukov [1], Vincze [1], McShane [1], Bogdanov [1].

1. The Babbage equation 291
then (p\x) is increasing for even / and decreasing for odd /. Consequently, n must
be even and y/(x) = cp2(x) is an increasing solution of y/"l2{x) = x in E. By lemma
15.3 y/(x)=x, i.e. q>{x) is an involutory function.B
In the case where E is an interval the general construction of a decreasing
involutory function is described in lemma 15.2. In fact, we have the following
Theorem 15.3. Let cp{x) be a solution of equation A5.1) in a real interval I. Then
(p{x) is continuous if and only if it is monotonic.
Proof. If <p(x) is continuous, then it is strictly monotonic by lemma 15.1. On
the other hand, suppose that <p is monotonic. If q> were not continuous, it would
have to make a jump; in other words, there would be points in / that are not con-
contained in <p{t). But since <p(x) satisfies A5.1) in /, we have <р(Г)=/in virtue of lemma
15.1. Consequently, q> must be continuous.¦
The above theorem will no longer be true if we replace the interval / by an
arbitrary subset E of the real axis. E.g. the function g>(x)=(x— l)/x satisfies A5.1)
with и=3 in ?=(—00, 0) и @, 1) и A, oo) and is continuous in E but is not mo-
monotonic.
For the complex variable we have the following result.
Lemma 15.4. If q>{x) is a meromorphic solution of equation A5.1), then
A5.9) ф)^^
ух + д '
Proof. This follows from lemma 15.1 since other meromorphic functions are
not invertible.B
Finally we shall determine all solutions of equation A5.1) of form A5.9)A). At
first let us note that if L(x)=Ax+B is an invertible linear transform, then cp(x)
satisfies A5.1) if and only if L~1(<p[L(x)]) does so. If y#0, then with L(x) = y~1x—
-y~18 we have L~1(q>[L(x)]) = ((x + d)-((xd-fiy)x~1. If y=0, but афЬ, then with
L(x) = x + P(e-a)~1 we have L~1{q>[L(x)])=«d~1x. If y = 0 and a = S, then <p(x)
= x+/3S~1. Thus it is enough if we confine ourselves to the three cases:
A5.10) <p(x) =
A5.11) <p(x) = ax,
A5.12) <p(x)=x + b.
Let (p{x) have form A5.10) and define the sequence Kj by the recurrence
A5.13) Х_1 = 0, X0=l,
It is easily shown by induction that
A5.14) VJM=K
@ Serret [1], Johnson [1], Cayley [1], Tanner [3]', [4], Julia [8], Chajot [2], Ghermanescu
[8], Haidukov [1J.
19*

292 CHAPTER XV. Fractional iterates
Inserting A5.14) into A5.1) and making use of A5.13), we obtain АГп-^х2— ах—b) = 0,
whence Kn-1 = 0. Thus function A5.10) satisfies A5.1) if and only if Кп^г=0.
Now, A5.13) implies (by induction)
where \Ja2+4b denotes either of the two possible values. Hence Kn^1=0 is equiv-
equivalent to (a+si a2 + 4b)n = (a- sj a2+Щ", a2+46 #0 O, i.e.
A5.15) a+\/a2+4b=(a - \/a2+4b) (cos 9 + i sin 9),
where &=2kn/n, k=\,...,n-\ (fc=0 leads to a2+4b=0). A5.15) yields
a2 n
Ь=~л 2/, , ч» fc=l, ...,и-1, fc#—,
4 cos (kn/n) 2
or (forS=n/2)
a=0, Ь arbitrary.
Taking into account that together with <p(x) also L~1[q>[L(x)]) satisfies A5.1) we
come in this case to the solutions
A5.16)
a X
cos (knIn)/
X k=l,...,nl,k^,
\ 4L (x) cos (knIn)/ 2
and if n is even (writing a instead of b)
A5.17) "W = L
If k divides n, say n=mk, then function A5.16) satisfies also the equation g>m(x)=x.
Function A5.17) is involutory (i.e. satisfies A5.7)).
Now let <p(x) have form A5.11). Then (p!{x)=a'x, which, inserted into A5.1)
gives a" = l, i.e. a=cosBkn/ri) + isinBkn/ri), fc=0, ...,n — 1. Thus this case leads
to the solution
A5.18) <p(x) = LT1(( cos — + /sin— JL(jc)J, k=0,...,n-l.
Lastly, if <p{x) has form A5.12), then <p'{x) = x+jb and A5.1) is not possible
unless b=0, which reduces A5.12) to the case A5.11) already considered.
Putting together the results obtained, we may formulate the following
Theorem 15.4. The only meromorphic solutions of equation A5.1) are functions
A5.16), A5.18) and, if n is even, A5.17), where L(x)=Ax+B is an arbitrary linear
function anda^O is an arbitrary constant.
(i) If O2+46=0, then K)=(J+i)(iay>?=0 for/=1,2,...

2. Fractional iterates 293
§ 2. Fractional iterates. The solutions of equation A5.3) may be regarded as
iterates of order l/n of the function g(x) (*). The member glln(x) of every iteration
group gu(x) of g(x) evidently satisfies A5.3). But iteration groups were studied in
Chapter IX under rather restrictive conditions. Now we are faced with a more
general situation: we shall describe the general solution of equation A5.3) for an
arbitrary invertible map g(x) of E onto EB). Also, the problem of solving A5.3)
for a single value of n is more general than that of constructing an iteration group
of g (x). And in fact, with a given g (x), equation A5.3) may be solvable for one value
of n and not solvable for another; and the same is true of regular solutions C).
We start with the following
Lemma 15.5. Let g(x) be a one-to-one map of a set E onto E and let <p{x) satisfy
equation A5.3) in E. Then, for every fc^O, the set &k[g] is a modulus set for (p.
Proof. By lemma 15.1 <p(x) maps E onto E in a one-to-one manner. Thus we
must prove that for every xoeE and x1 = <p(x0) the relation gJ(x0)=x0 is equiv-
equivalent to ^•/(jc1) = jci, or, by A5.3), <pni{x0) = x0 is equivalent to <pnj+1{xo)=<p{x<y).
But this is obvious in view of the invertibility of q>{x).n
According to the above lemma we may construct a solution of equation A5.3)
independently in each of the sets <&k[g], fc=0, 1, 2, ... In the sequel we shall assume
that
E=^k[g~\ for a certain integer fc^O.
Let Q=E/i be the set of all orbits under g in E and let Ф(х) be an arbitrary
solution of the equation
A5.19) Ф\Х)=Х, XeQ,
obtained from a decomposition of Q
where, however, the sets ?/j, j=l, ...,nt, are empty whenever и/и, has a
divisor (>1) in common with k. (This is certainly true if /=0.) If fe=0, we as-
assume U) = 0 for /=0, ..., r— 1. Thus, modifying our notation, we may write
@ Gersevanov [2], Bodewadt [1], Massera, Petracca [1], Isaacs [1], Lojasiewicz [1], Collatz
[1], Fine, Schweigert [1], Schobe [1], Haidukov [1], Kuczma [12], Lillo [1], [2], Bajraktarevic [20],
A. Smajdor [1], Michel [2]; cf. also Brauer [1], [2].
B) For non-invertible g mapping E into E and n=1 the general solution of equation A5.3)
has been given by R. Isaacs [1]. For non-invertible continuous maps of a real interval / into itself
the general continuous solution of A5.3) is not known. (For a partial result, cf. Lillo [2]).
C) E. g. the equation q>4x)=e1(x) obviously has an analytic solution q>{x)=e{x), and this
is the only regular solution in a neighbourhood of x=0. Therefore an analytic solution of q>4(x)
= e2ix) would have to satisfy <p2(x)=e(x); but by theorem 15.13 the latter has no analytic solution
in a neighbourhood of x=0.

294 CHAPTER XV. Fractional iterates
decomposition A5.20) as
A5.21) Q=
UUj
i=0j=l
(J Vf for k=0,
where m—n/nf, and 1=hJ<...<«* is the complete set of divisors of n which are
relatively prime to k.
If k^\, then there exist unique integers p-t, qt such that
A5.22) ptnf-qtk=l.
If k=0, we take po = l. Let Л be an arbitrary set containing exactly one element
of every orbit under g contained in E (*).
Theorem 15.5.B) Let g{x) be a one-to-one map of E onto E, E=<&k[g]. The
general solution of equation A5.3) in E is given by the formula
A5 23) (x)=\ 9(A П Ф[С*(Х)]) whenever ВД e VJ ' ¦/ =
Ьд+"DпФ[ОД]) whenever Сд{х)еУ1т1,
where the integer ц=ц(х) is chosen so that
A5.24) д\АпСв(х))=х,
and all the remaining symbols have the meaning described above.
Proof. If k=0, ц fulfilling A5.24) is unique. If k^ 1, and A5.24) holds for
fix and^j then by lemma 0.2 (take xl=x2,p = q, п=цх, т=ц2) И\— Рг is a multiple
of к and, since g\x)=x in E, the right-hand side of A5.23) is independent of the
choice of /г.
In order to prove that function A5.23) satisfies equation A5.3) let us fix an
arbitrary jc0 e E, and put Cv = (Pv[Ce(*<>)], v=0,... ,n. It follows from A5.23) by in-
induction that ф"{хо)еС„ for v=0, ...,n. We writeyv = A n Cv, xv = <pv(x0), v=0, ...,n,
and let fiv be such that g^{yv)=xv. Formula A5.23) may now be written as
/v(jv+i) whenever Cv e VJ, j = l, ..., m,-l,
+"CK,+,) whenever Cv e ^,.
Hence, if Cv e F}, 1 ^/<mi -1, then /iv+l=ftvC); if Cv e F^, then mv + 1=mv +P-X3)-
Thus when v increases by тг, fiv increases by p,. And when v increases from 0 to
n, fiv increases by pt multiplied by njmh i.e. by n*Pi=qik+ 1, in virtue of A5.22).
On account of A5.19) we have Cn=C0, whence yn=y0. Consequently, xo=g'lo(yo)
and <p\xo)=xn=g»»(jn) = g>'»(yo) = g»°+<'<k+\yo) = g»°+1(jo) = g[g*°(yo)] = g(xo), i.e.
A5.3) holds for x=x0- (The argument in the case k=0 is similar.) Since x0 has
(•) Here we make use of ths axiom of choice.
B) Lojasiewicz [1]. Cf. also Haidukov [2], Michel [2], Bajraktarevic [20].
C) mod к if k>\.

2. Fractional iterates 295
been chosen arbitrarily in E, this proves that function A5.23) satisfies equation
A5.3) in E.
Now we shall prove that every solution <p(x) of equation A5.3) in E can be
given from A5.23). For Xe Q we put Ф(Х) = <р{Х). This definition is correct, since <p
maps orbits under g onto orbits under g. Since orbits under g are modulus sets for g,
the function Ф{Х) so defined satisfies equation A5.19) in Q. In virtue of theorem
15.1 Ф(Х) must be obtained from a decomposition A5.20) and satisfies Ф"'(Х)=Х
for Xe \JU) (cf. formula A5.6)). Suppose that U)=?& for a certain /, and let,
xoeXoeU'j. Then (р"'(хо)е <рщ{Х0)=Фщ{Х0)=Х0, i.e. хокрщ{хо). Consequently,
there . must exist an integer p such that <p"'(xo)=gp(xo). Hence д(.хо)=ф\хо)
=gnplnii.x0), i.e.
A5.25) gnp""~\x0)=x0.
This means that, if k~^ 1, k must divide рп]щ — \ and и/и,- must be relatively prime
to k. If k=0, relation A5.25) is not possible unless пр]щ—\ =0, which in turn implies
п\щ=\ and и=и,-, i=r. Thus decomposition A5.20) can be written in form A5.21).
Now let A\ be an arbitrary set which has exactly one element in common with
every orbit contained in V[ and put Aij = ipJ~1(Ai1), A= (J \JA). The set A con-
•=o j=i
tains exactly one element of every orbit contained in E. Define pt by A5.22), and
let us fix an x0 eE. Suppose that C0 = Cg(x0) e V), l^y^m,--1, and put CV = ^V(CO),
yv=Ar\Cv. We have in particular уо=АпСо=А)пСо, y1 = An C1=Aii+1n Ct
= <р(А))с\<р{С0). For//fulfillingA5.24) withx=x0 we have g>l{y0)=x0 and <p\g\y0)]
= (p(x0). But q>, as a solution of A5.3), commutes with every iterate of g. Hence
i.e. formula A5.23) holds for x=x0. Similarly, if CoeVlmi, then yo=AnCo =
Alt n Co and yi =A n Cl=A\ nC^y1 ~mi{Aim)n4>{C0). But we have Co = <pm'(C0),
whencey1 = <p1~mi(AimnC0)=fp1-m%y0) and
A5.26) <р(*о) = 911ЫУо)\=911Ъ>тШ ¦
Since (pm'(C1) = C1, we have jt i^""(j!), i.e. there exists an integer p such that <pm'( t)
=ff''(j;i)- Iterating nf times we obtain (рпчп*(у1)=дрп*(у1), i.e. ^(yi)=^'*(ji). If
A: = 0, this implies pnf = 1 and/?=po=l. If k>l, pnf — l must be a multiple of A:,
say pnf — 1 =<jrfc. Thus, up to a multiple of/:, /? coincides with/?,- (cf. formula A5.22))
and fmi(yi)=9Pi(ji)- Hence, according to A5.26),
i.e. formula A5.23) holds for x=xo.a
The existence of a solution of equation A5.3) depends on the existence of de-
decomposition A5.21). The latter surely does exist if Q is infinite. If Q is finite and

296
CHAPTER XV. Fractional iterates
contains, say, L elements, then decomposition A5.21) is possible if there exist non-
negative integers Kt ks such that
A5.27) L = m0K0 + .
Note that for /=0, ...,s, nf divides n* and hence ms divides mt. Thus condition
A5.27) is equivalent to the divisibility of L by ms. If k = 0, then A5.27) becomes
L = m0K0, where m0 = n. Hence we obtain the following
Theorem 15.6. (Lojasiewicz [1]. Cf. also Michel [2].) Let g be a one-to-one map
of a set E onto itself and let Lk be the number of orbits under g in (?*[#]¦ In order that
equation A5.3) have a solution in E it is necessary and sufficient that, for every k,
Lk is either infinite, or divisible by dk, where do = n and, for k~^\, dk = n/n*k, n% being
the largest divisor ofn that is prime to k.
Example. (*) Let E be the unit circle
x=cos2nt, y=sia2nt, te@,1),
so that the points in Emay be represented by the parameter t, t=0 and t=\ being identified. De-
Define on E the function git) as follows:
|,+i for ,6<0,i>,
I t-i+d(t) for /e(*,l),
where 3(t) is continuous on 0,1), positive on (|, 1), <5(?)=<5(l)=O, and such that /—i+3(t) is
strictly increasing in ($, 1]> (cf. fig. 10, which represents the graph of g(t))- F°r f e @, J) we have
g2(t)=t+S(t+i)jtt. Similarly, for /e(i, 1) we have g4t)=t+8(t)j=t. On the other hand, the
points f=0 and t=i form the unique two-point orbit under g.
It follows from theorem 15.6 that the equation 92(t)=g(t) has no solution in E. For here
Z,2=l> whereas the largest divisor of n=l prime to k—1 is n*—\ and thus dz=2. Consequently,
d2 does not divide ?2 •
I owe this example to W. Holsztynski.

2. Fractional iterates 297
This example is particularly interesting, because g(t) is an order preserving
homeomorphism of E onto itself. In the next section we shall show that on the
real line such a situation is impossible. For a continuous and strictly increasing
map g(x) of a closed real interval onto itself there always exist (continuous and
strictly increasing) solutions of equation A5.3) (cf. the corollary to theorem 15.7).
§ 3. Continuous increasing solutions. Now we are going to describe the general
continuous solution of equation A5.3) under the assumption that the function g(x)
is continuous and strictly monotonic in a real interval / (open or closed) and maps
/ onto itself (*). (One or both ends of / may also be infinite.) By lemma 15.1a con-
continuous solution <p(x) of equation A5.3) must then be strictly monotonic and maps
/ onto /.
We may assume that the function g(x) is defined on the closed interval /, for
otherwise g (x), being monotonic, could be extended onto the closure I of I. In the
present section we are going to investigate the continuous increasing solutions of
equation A5.3) in the case where the function g is increasing. It can easily be seen
that the ends of / must belong to G^ [g]. By lemma 15.5 G^ [g] is a modulus set for
<p, and consequently <p(x) satisfies on <&t [g] the Babbage equation A5.1). By lemma
15.3 q>(x)=x on ©! [g]; in other words, we have
The set /—/ is open and consists of at most denumerably many disjoint open inter-
intervals. Each of those intervals is evidently a modulus set for q> as well as for g. There-
Therefore, it is enough to describe the construction of q>(x) independently in each of
those intervals.
Let (a, b) be one of the intervals of the set /-/. The expression g(x)—x has a
constant sign in (a, b). We may assume that^(x)<x in (a, b); in the other case the
considerations run similarly.
Lemma 15.6. Let g eR°[(a, b)], where a, be<&t[g], and let x0, ..., xn_t be
arbitrary points from (a, b) such that
A5.28) g(xo)<xn_1<...<xo.
Further, let <p\(x), ..., cpn-i[x) be arbitrary, continuous and strictly increasing func-
functions on the intervals <xl5 xo>, ..., <xn_1, xn_2>, respectively, fulfilling the conditions
A5.29)
Put
A5.30) xv+n = g(xv), v=0, ±1, ±2,...,
(i) Kuczma [12]. a. also Sibiriani [1], Collatz [1], Fine, Schweigert [1], Haidukov [2],
Lillo [1]. The case where the interval / is closed on one side and/or g maps /into itself, presents
no particular difficulties; cf. Kuczma [12].

298 CHAPTER XV. Fractional iterates
and define the function q>v(x) for x e <xv, xv_ t> by
A5.31) fv+,W=ff(ft+1i(".fc1»-iD)...)). v=0, ±1,±2,....
Then the formula
A5.32) tp(x) = (pv(x) for xe<xv,xv_!>
defines a continuous and strictly increasing solution of equation A5.3) in (a, b), which
fulfils the conditions
A5.33) lira <p(x)=a, lira <p(x) = b.
x-»a + O x->6-0
Taking all possible systems of points x{ fulfilling A5.28) and all continuous and
strictly increasing functions q>{(x) fulfilling A5.29), we obtain from A5.32) all con-
continuous and strictly increasing solutions of equation A5.3) in (a, b).
Proof. It is easily seen (cf. in particular theorem 0.4) that the sequence xv
defined by A5.30) is strictly decreasing and fulfils
A5.34) limxv=a, lim xv = b.
V~* 00 V~* — 00
+ CO
Moreover, (a, b)= (J <xv, xv-i>. Making use of A5.31X1) and of the properties of
— со
the functions <Pi{x), ..., ^„.Дх) one proves by induction that for every v the func-
function (pv(x) is defined, continuous and strictly increasing in the interval <jcv, jcv_i>
and fulfils the conditions
A5.35) ?>v(*v) = *v+l> ?>v(*v-l) = *v
Hence it follows that formula A5.32) defines a function cp(x) on (a, b) and that
this function is continuous and strictly increasing. Relations A5.33) result from
A5.35) and A5.34) in view of the fact that q>(x) is monotonic.
Relation A5.31) implies the formula
A5.36) ?,+,(?,+,-i(...(p,+iD»))=9W for *e<*v+i>*v>-
In virtue of A5.32) and A5.35), A5.36) means cp\x)=g[x) for xe<xv+1, xv>.
Since v is arbitrary, this proves that <p(x) satisfies equation A5.3) in (a, b).
Now we shall show that every continuous and strictly increasing solution <p{x)
of equation A5.3) in (a, b) can be obtained in this manner.
The sign of <p(x)—x in (a, b) agrees with that of g(x) — x. (<p(x)<x implies
cp\x)<x.) Thus we have q>(x)<x in (a, b). We choose arbitrarily an x0 e (a, b) and
put xv = q>\x0), v=0, ±1, ±2,... The points Jto."->*n-i then fulfil condition
A5.28). We define functions q>v(x) by A5.32). Then the functions g>i(x), ..., ^„-Дх)
In order to determine (/>,{x) for v<0 we must write A5.31) in the equivalent form
.(р~1п(д(х)))...\

3. Continuous increasing solutions 299
are continuous and strictly increasing on <xl5 xo>, ..., <xn-i, xn_2>, respectively,
and fulfil conditions A5.29). Relation A5.31) holds in virtue of A5.3) and A5.32).и
Thus we have the following
Theorem 15.7. Let g(x) be a continuous and strictly increasing function on a
modulus interval I {open or closed), д{х)фх on I. Then equation A5.3) has on I a con-
continuous and strictly increasing solution q>(x) depending on an arbitrary function. This
solution is obtained by putting <p{x) = x for x e J=d1 [g] and by constructing a con-
continuous and strictly increasing solution of A5.3) independently in every interval of the
set I—J by the method described in lemma 15.6.
Corollary. If g(x) is a continuous and strictly increasing function on a modulus
interval (open or closed) J, then equation A5.3) has a continuous and strictly increas-
increasing solution in I.
If д(х)фх, this results from the above theorem; if g(x) = x, then q>(x)=x is a
solution.
§ 4. Continuous decreasing solutions for decreasing g. If g(x) is decreasing in a
modulus interval /, then equation A5.3) may have a continuous (and then necessarily
strictly decreasing) solution only if n is odd:
The set &i[g] = &i[(p] consists now of a single point: (Si[#] = {?}. By lemma 15.5
<?2[g] is a modulus set for q>. Thus q> satisfies on (?2Ы equation A5.3) and hence
(p2n(x) = x for x e (?2 [#]• The function q>2{x) is increasing, consequently, q>2(x) = x for
x e &2[g]- Writing equation A5.3) in the form <p(<p2m(x'j)=g(x) we obtain hence
A5.37) 9(x) = g(x) for хеадиЩ.
(Note that
The set /— (Ci[^]uC2[ff]) is a union of at most denumerably many disjoint
open intervals. If (a, b) is one of those intervals, then so is also (g(b), g(a)). The
set
A5.38) (a,b)u(g(b),g(a))
is a modulus set for q> (x). We shall construct a continuous and strictly decreasing
solution of equation A5.3) independently in every set of form A5.38).
Let us note that the expression g\x)-x has opposite signs in {a, b) and in
(g(b), g(aj). For, if e.g. g2(x)<x in (a, b), then g3(x)>g(x), and setting y=g{x) e
(g(P), g(a)) we obtain hence g2(j)>y in (g(b), g{a)). As in the preceding section,
we may assume that g2(x)<x in (a, b).
Lemma 15.7. Let g be a continuous and strictly decreasing map of an interval
{a, b) onto {дф), д(а)) and let g2 e R°a[(a, b)], a, be C^] и &2\д]. Let x0, ..., jcn_t
(!) Since &i[g] и ^2^] is a modulus set for g and g^(x)=x in it, we have by A5.37) (p(a)=
=g(a), <p(b)=g(b), <p[g(b)]=g2(b)=b, <p[g(a)]=g2(a)=a, and, since q> is monotonic, <p(x) e
(g(.b),g(a)) for x e (a, b), q>{x) e (a, b) for x s (дф), д(а)).

300 CHAPTER XV. Fractional iterates
be arbitrary points of set A5.38) such that xoe{a, b), xt e(g~1{x0), g{xoj) (*) and
xo>x2>...>xn_1>g{x1),
A5.39)
X1<x3<...<Xn_2<g{x0).
Define the sequence xv by A5.30) and let (px{x), ..., (pn-x{x) be arbitrary functions
which are defined, continuous and strictly decreasing on <x2,x0>, <хь x3>,...,
<xn_3, xn_!>, <xn_2, х„У, respectively, fulfilling the conditions
A5.40) Vi{xi-1) = xi, (Pi{xi+1) = xi+2, i=l,...,n-l.
Then the formula
A5.41) <p{x)=<pv{x) for xe(xv+11j:v.i> or <xv_t, xv+1>,
where the functions q>v{x) are defined by A5.31), defines a continuous and strictly
decreasing solution of equation A5.3) in set A5.38), which fulfils the conditions
lim <p{x) = g{a), lim y{x)=g{b),
x-»a + 0 x->b — 0
lim <p{x)—b, lim cp{x)=a.
x->g(a)-0
Taking all possible systems of points xt fulfilling A5.39) and all continuous and
strictly decreasing functions срг{х) fulfilling A5.40), we obtain from A5.41) all the con-
continuous B) solutions of equation A5.3) in {a, b) и (g{b), g{a)).
The proof of this lemma is analogous to that of lemma 15.6. From lemma 15.7
we obtain the following
Theorem 15.8. Let g{x) be a continuous and strictly decreasing function on a
modulus interval I {open or closed) and let n be odd. If д2{х)фх, then equation A5.3)
has in la continuous {andstrictly decreasing) solution cp{x) depending on an arbitrary
function. This solution is obtained by putting (p{x)=g{x) for x e G^ \g] и <?2[д] and by
constructing a continuous and strictly decreasing solution o/A5.3) independently in
every set A5.38) {where a and b belong to (SJg] и(?2Ы) by the method described in
lemma 15.7.
If g\x) = x on I, then cp{x) = g{x) is the only continuous solution of equation
A5.3) in I.
§ 5. Continuous decreasing solutions for increasing g. If g{x) is strictly increasing
and n is even, say n = 2m, then equation A5.3) may also have strictly decreasing
solutions. If (p{x) is such a solution, then y/{x)=(p2{x) is an increasing solution of
y/m{x)=g{x), which may be found according to theorem 15.7. cp{x) is then to be
determined from the relation cp2{x)=y/{x), in which y/{x) is now regarded as given.
@ Setting y=g-i(x) in the inequality g2(y)>y, ye{g(b), g(a)), we obtain g(x)>g~i(x)
for xe(a, b). Thus the interval {g~l(xo), g(xQ)) is not empty. Moreover, xie{g-i(xa), g(x0))
implies xi>g'l(xo) and g(xl)<xo. Consequently, conditions A5.39) are not contradictory.
B) Note that in this case every continuous solution of A5.3) must be strictly decreasing.

5. Continuous decreasing solutions for increasing g 301
Therefore we shall confine ourselves to discussing decreasing solutions of equation
A5.3) in the case и = 2, i.e. we shall consider the equation
A5.42) <p\x) = g(x).
Definition. A point ? e &i[g] is called regular if there exists an order reversing
map/(x) of &i[g] onto itself, leaving the point <j; fixed and such that in the intervals
(a, b) and (f(b), Да)) with no points in common with (SJg] the expression g(x) — x
has opposite signs. The function /(x) will then be said to map &i\gf regularly onto
itself.
We have the following
Lemma 15.8. Let g{x) be a continuous and strictly increasing function on a modulus
interval I. If q>(x) is a continuous and strictly decreasing solution of equation A5.42)
in I, then the point ? such that <р(?,)=Е, is regular and cp(x) maps G^ [g] regularly onto
itself.
Proof. The function q>(x) is strictly decreasing (i.e. order reversing) on (&х[д],
and, of course, leaves ? unmoved. If xeG^fg], then q>(x)e (SJg], since g[<p(x)] =
*p\x)=<p\g{x)\ = <p{x). Lastly, if a<b are two consecutive points of (^[gr] and e.g.
#(x)<x in (a, b), then <p\g{x)\><p{x), i.e. g [<p{x)]xp(x) in (a, b). Setting y=<p(x) we
obtain hence g(y)>y in (y(b), ?>(a)), i-e. g[x) — x has opposite signs in [a, b) and
O(Z>), <p(a)).u
Let ? be a regular point of <&x\g] and /(x) a corresponding regular map of
&t[g] onto itself. The set I-*&i\g] consists of at most denumerably many disjoint
open intervals. Let (a, b) be such an interval; then (f(b),f(a)) is another such inter-
interval. By theorem 3.1, (Q = (a, b) x (f(b), /(a))) the equation
<15.43) a>|>(x)] = 0[a>(x)].
has in (a, b) a continuous solution assuming values in (/F), /(a)) depending on an
arbitrary function. It may easily be proved (cf. also the proof of theorem 10.1) that
if ш(х) is prescribed as strictly decreasing on <x0, ^(xo)> resp. <^(x0), xo>, then it
is strictly decreasing on (a, b) and fulfils
<15.44) lim ш(х)=/(а), lim co(x)=/(Z>).
x-»a+0 x->b~ 0
Consequently, equation A5.43) has in (a, b) a continuous and strictly decreasing
solution ш(х) fulfilling A5.44) and depending on an arbitrary function.
Lemma 15.9. Let g{x) be a continuous and strictly increasing function on a modulus
interval I (open or closed) and let ? be a regular point of G^ \g] and /(x) a correspond-
corresponding regular map of<&x \g] onto itself. Let {a, b) be an interval of the set I—^lg]. Then
the formula
М f°r xe(a,b),
<15.45, fW{,-,M for

302 CHAPTER XV. Fractional iterates
where co(x) is an arbitrary continuous and strictly decreasing solution of equation
A5.43) in (a, b) taking values in (f(b),f(d)), defines the general continuous and strictly
decreasing solution of equation A5.42) in (a, b) и (f(b), f(a)\.
Proof. Let co{x) be a continuous and strictly decreasing solution of equation
A5.43) in (a,b) taking values in (f(b),f(a)). According to A5.44) co((a,b)) =
{f(b), f{d)) and thus со ~1 (y) is defined for у e (f(b), f(a)). Since each of the intervals
(a,b), (f(b),f(d)) is a modulus interval for g{x), the function tp(x) is by A5.45)
defined in (a, b)v(f(b),f(a)). It is evidently continuous and strictly decreasing there..
Now, if x e (a, b), then cp(x)=co(x) e (f(b),f(a)) and by A5.43)
If x e (f(b),f(a)), then <p(x) = co-1[g(x)] e (a, b) and
<p2(x) = <p(a>-1[g(x)-]) = co(co-1[g(x)]) = g(x).
Consequently, function A5.45) satisfies equation A5.42) in (a, b)<o[f(b),f(a)).
Now suppose that q> (x) is a continuous and strictly decreasing solution of equa-
equation A5.42) in (a, b)vj(f(b),f(a)). Since <p(x) is continuous, it maps connected sets
onto connected sets. Therefore either q>((a, b))c(a, b) or <p((a, b))<=(f(b),f(a)). la
the former case there would have to exist a point с e (a, b) such that (p(c) = c. But,,
since f{x) satisfies A5.42), this would imply с е<&1 [g], which contradicts the defini-
definition of (a,b). Consequently, we must have <p((a,b))<=(f(b),f(a)) and similarly
(p((f(b),f(a)))<=(a, b). (Actually in both relations the equality must hold.) By A5.42)
the function q>(x) commutes with g(x). Thus the function co{x)=f{x) for x e (a, by
is a continuous and strictly decreasing solution of equation A5.43) in (a, b) taking
values in (f(b),f(a)). Formula A5.45) results from A5.42). Thus A5.45) actually
gives the general continuous and strictly decreasing solution of equation A5.42) in
(a,b)v(f(b),f(a)).m
Now we can prove the following
Theorem 15.9. Let g(x) be a continuous and strictly increasing function on a
modulus interval I (open or closed) and let ? be a regular point off&^g] and f^(x) a
corresponding regular map o/GJgr] onto itself. Then equation A5.42) has in la con-
continuous and strictly decreasing solution depending on an arbitrary function. This solu-
solution is obtained by putting on ©i[gr]
^x) for xee^gr], x<i,
A5.46) cp(x) =
for x =
\x) for хе^О], x>?,
and by constructing a continuous and strictly decreasing solution of A5.42) in the
intervals of the set I— ®i[gr] by the method described in lemma 15.9.
Taking all possible regular points ^ в ©Jgr], all regular maps f$(x), and all solu-
solutions in the intervals of the set I— ©Jgr], we obtain all the continuous and strictly de-
decreasing solutions f(x) of equation A5.42) in I.

5. Continuous decreasing solutions for increasing g 303
Proof. Only the last statement requires a proof. Let q>{x) be a continuous and
strictly decreasing solution of equation A5.42) in /. Then by lemma 15.8 the point ?
such that <p(?) = ? is regular and the function
ft(x) = (p(x) for xeGiCsr]
maps ejgr] regularly onto itself/, (x) is involutory on ©Jgr], and thus fi(x)=ff1(xy
Consequently, relation A5.46) holds.
Let (a,b) be an interval of the set /-(^Ы- Then <p{a)=f(a), f{b)=f{b\
<p[f(b)]=b, <p[f(a)]=a, and, since <p(x) is monotonic, q>((a,b)) = (f(b), f(a)),
<p((f(b),f(d))) = (a, b). Consequently (a, b)v(f(b),f(a)) is a modulus set for tp and q>
satisfies equation A5.42) in it. Thus <p(x) must be obtained by the construction
described in lemma 15.9.И
Let us note that if g (x) = x, then every inner point of / is regular and every
strictly decreasing function from / onto / maps / regularly onto /. Formula A5.46)
is then identical with A5.8).
As a corollary to lemma 15.8 and theorem 15.9 we obtain the following
Theorem 15.10. Let g(x) be a continuous and strictly increasing function on a
modulus interval I {open or closed). Then equation A5.42) has in I a continuous and
strictly decreasing solution if and only if the set ©i[gr] contains at least one regular
point.
The function g(x)=2~S8axx, /=(-oo, oo), yields an example where the above
condition is not fulfilled. In fact, we have g(x)—x<0 in (— oo, 0) u @, oo), which
shows that 0 (the only point of ©Jgr]) cannot be a regular point.
§ 6. Regular solutions. The present section contains a few remarks concerning
analytic solutions of equation A5.42)A) in a neighbourhood of a fixed point of
g (x) B). Without loss of generality we may assume that the fixed point is placed at
the origin. Thus g (x) has an expansion
A5.47) g(x)=fianx'1
convergent in a neighbourhood of the origin. We shall restrict ourselves to a treat-
treatment of the case where at фО.
Let
A5.48) ?>(*)= ?*„*".
n=l
Inserting A5.48) and A5.47) into A5.42) we obtain
A5.49) ax = b\,
A5.50) a^b^bl +
(') In the more general case A5.3) the discussion is similar.
B) Koenigs [5], [6], Pfeiffer [2], [4], Crum [1], Kneser [1], Baker [1], [2], Thron [1],
serman [1], Schobe [1], Myrberg [10].

304 CHAPTER XV. Fractional interates
where Pn is a polynomial. Equation A5.49) has exactly two solutions. For each
fixed bi fulfilling A5.49) further coeficients hn can be uniquely determined from equa-
equations A5.50) provided Ь\+Ь1ф0. This will certainly be the case if at is not a root
of unity. Thus we have the following
Lemma 15.10. If g(x) is analytic (with expansion A5.47)) in a neighbourhood of
the origin anda± ^0 is not a root of unity, then equation A5.42) has exactly two formal
solutions of form A5.48).
If, moreover, \а^\Ф 1, the two solutions actually exist:
Theorem 15.11. If g(x) is analytic {with expansion A5.47)) in a neighbourhood of
the origin and |«i|^0 or 1, then equation A5.42) has exactly two analytic solutions
{15.48) in a neighbourhood of the origin.
Proof. Let 0< |ai| < 1. Then the Schroder equation
has an analytic solution y/(x) = x+terms of higher degrees (theorem 6.4). Thus
i/f~1(x) exists and is analytic in a neighbourhood of the origin, and
where bt may be either of the two solutions of A5.49), represents the desired solu-
solutions. If |«i|> 1, we solve the equationx2(x)=g~1(x) as above, and then <p(x)=x~1(x)
yields the solutions of the original problem. It follows from lemma 15.10 that there
are no other solutions. ¦
If 1^1 = 1, but at is not a root of unity, then it may happen that both formal
solutions of A5.42) have a positive radius of convergence, exactly one solution has
such a radius or no formal solution has one (Pfeiffer [2], [4]). However, the above
argument may be repeated whenever there exists an invertible analytic solution of
equation A5.51). Thus we have by theorem 6.13 the following
Theorem 15.12. If g(x) is analytic (with expansion A5.47)) in a neighbourhood of
the origin and at=s fulfils condition F.43), then equation A5.42) has exactly two
analytic solutions A5.48) in a neighbourhood of the origin.
The case where at is a root of unity is more complicated. Equation A5.42) may
have no formal solution, one such solution or infinitely many of them. The first
case is exemplified by the function
g(x)=-x+x2+x3.
Equation A5.49) gives bt=±i, A5.50) for n = 2 gives b2 = (-l±i)~1, and for n = 3
{15.50) becomes 1 = —1, which is impossible. An example of the second case is
offered by the function
{15.52) g(x)=e(x)=x+ix2 + ix3 + ...
Here the two possible values of bt are bt = l and bl = — \. For the first we have
О, and thus the determination of the formal series A5.48) is possible.

6. Regular solutions 305
For bt = — 1 equation A5.50) for и=2 becomes i=0; consequently, there is no
second solution. Lastly, we meet the third case with involutory functions. If y/(x) is
an arbitrary analytic function in a neighbourhood of x=0, y/@)=0, \1/'ф)ф0, then
<p(x) = y/~1(— y/(x))is an analytic solution of equation A5.7) in a neighbourhood of
the origin (*).
The convergence of these formal solutions is a difficult problem and there exist
no general results in this respect. An important partial result known is that in the
case of function A5.52) the solution has the zero radius of convergence:
Theorem 15.13. (Baker [2].) The equation
A5.53) <p\x) = e(x)
has no solution that would be analytic at x=0.
This follows from theorem 10.11 in view of the fact that every solution of equa-
equation A5.53) commutes with e(x).
Equation A5.42) is of particular interest and has been widely studied in the case
where g(x) is an exponentially growing function. Its analytic solutions have also
been sought in the real case. For equation A5.53) we may easily indicate a real
analytic solution in @, со) B): <p(x) = a~ 1(a(x) +J). For the equation
A5.54) cp2{x)=ex
a real analytic solution has been found by H. Kneser [1]. This solution, however, is
not single-valued (Baker [1]) and, as pointed out by G. Szekeres [3], there is no
uniqueness attached to the solution. It seems reasonable to admit <p(x)=fll2(x),
where f"(x) is the regular iteration group of f(x) = ex, as the "best" solution of
equation A5.54) (best behaved at infinity). However, we do not know whether this
solution is analytic for x>0.
It is also an open question under what conditions equation A5.3) has a (real)
solution f e M1 and whether this solution is unique (Kuczma [24], Aczel [6]). As
A. Smajdor [1] has shown, the condition g e М1[Г\ о R^ [Ц alone is not sufficient for
the existence of a <p bM1[I] satisfying A5.3). U. T. Bodewadt [1] has conjectured
that for completely monotonic g equation A5.3) has a unique completely monotonic
solution.
§ 7. A generalization. The equation C)
A5.55) <p"+1(x)=g[<p(x)-]
is a generalization of A5.3). If we assume that cp is a one-to-one map of a set E onto
itself, A5.55) immediately reduces to A5.3). We shall show that it is possible to
reduce the solution of A5.55) to that of A5.3) also in the general case D).
(*) These are not only formal, but even actual solutions.
B) This solution is of course not analytic at x=0\
C) Sierpinski [1], Ritt [3], Gol^b [1], Ewing, Utz [1], Kuczma [11].
D) Cf. Kuczma [11], also for the more general equation <pn(x)=g[<pm(x)\; cf. also Ritt [3],
Chajot [2], Ewing, Utz [1].
20 Functional equations

306 CHAPTER XV. Fractional iterates
Suppose that the function g (x) is defined in a set F and takes values in a set E.
We denote by V the class of sets V<=F such that
A5.56) g(V)czVczEr\F.
Further, for an arbitrary set Fe V we denote by Tv the class of functions y/{x)
which are defined in E— Fand take values in V. The following lemma is an immediate
consequence of the definition of a solution of equation A5.55) (cf. § 1).
Lemma 15.11. If f{x) satisfies equation A5.55) in E, then <p(E) e V.
Now we shall prove
Theorem 15.14. (Kuczma [11].) Let g(x) be a map from a set F into E. The
general solution of equation A5.55) in E is given by the formula
A5.57) f{x) = \^X) f°r X?V>
K ' VK \w(x) for xeE-V,
where V is an arbitrary set from the class V, <po(x) is an arbitrary solution of equation
A5.3) in V, and y/{x) is an arbitrary function from the class Tv.
Proof. Let f be given by A5.57) and let us take an arbitrary x e E. If x e V,
then we obtain by induction <p'(x)=<pl0(x) for i=l, 2, ..., and f"+1{x) = fn0+l{x) =
<Po[<Po(x)]=g[<Po(x)]=0[<p(x)]> i-e- A5.55) holds. Similarly, if xeE— V, then <p(x) =
y/(x)e Fand by induction (pt(x) = qfr1[?(x)], i=2, 3, ... Hence (р"+1(х)=<рп0[ч/(х)]
=g [w(x)]=g [<p(x)], i.e. A5.55) holds. Thus function A5.57) satisfies equation A5.55)
inE.
Now suppose that a function q>(x) satisfies equation A5.55) in E. Put V=q>{E).
By lemma 15.11 Fe V. If xeE-V, then q>{x)eq>(E)=V, which means that the
function f{x) restricted to the set E— V belongs to the class фу. If x e V, then by
A5.56) <p(V)<=<p(E)= V. Moreover, there is a teEsuch that x=<p(t). Hence f"(x) =
(pn[(p(t)\ = fn+\t)=g[(p(t)\=g(x), which means that <p(x) restricted to the set V
satisfies equation A5.3). Hence it follows that (p{x) has form A5.57).и
The general continuous solution of equation A5.55) may be achieved in a similar
manner. Suppose that F and E are real intervals (x) and g(x) is a continuous func-
function defined in F and taking values in E. Let F*c F be the class of real intervals (x)
F fulfilling A5.56). For an arbitrary interval V e V* and an abritrary function/(x),
defined and continuous in F, we denote by ФУу{ the class of functions y/{x) which
are defined and continuous in E— V, take values in F and fulfil the condition
A5.58) lim y/(x)= lim f(x)
x->u,x?E-V i->«,iEK
for every endpoint и of interval F belonging to E. (If for an endpoint ueE the
limit on the right-hand side of A5.58) does not exist, the class TVff is empty.) By
an argument analogous to that employed in the proof of theorem 15.14 one can
prove the following
Open or closed; or closed on one side; possibly infinite or degenerated to a single point.

7. A generalization 307
Theorem 15.15. (Kuczma [11].) Let g(x) be a continuous map froma real inter-
interval F into a real interval E. The general continuous solution of equation A5.55) in E
is given by the formula A5.57), where V is an arbitrary interval from the class V*,
<p0 (x) is an arbitrary continuous solution of equation A5.3) in Vandy/(x) is an arbitrary
function from the class ?"*,w.
A number of other functional equations containing iterates of the unknown
function have been dealt with by various authors (x).
(!) Lemeray [6], [10], [14], Spiess [2], Touchard [1], Azevedo do Amaral [1], [2], Kreveras
[1], Targonski [1], Sierpinski [6], Heinhold [1], Srivastava [1], Bajraktarevid [15], [17]-[19], [21],
Pelczar [2], Adamczyk [2], [3].

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[5] Badanie nielinwwego zagadnienia Hilberta-Hasemana metodq Schaudera, Roczniki P.T.M.,
Prace Mat. 9 A965), pp. 35^18.
Zarankiewicz, K.
See Sikorski, R., Zarankiewicz, K.
Zolotarev, M., Koroliuk, V. S. (Золотарев, М., Короток, В. С.)
[1] Об одной гипотезе Б. В. Гнеденко, Теория Вероятностей 6 A961), pp. 469^74.
Исправление: ibid. 7 A962), р. 120.
Zverovic, E. I. (Зверович, Э. И.)
[1] Краевая задача типа задачи Карлемана для многосвязной области, Докл. Акад. Наук
СССР 156 A964), pp. 1270-1272.
[2] Краевая задача типа задачи Карлемана для многосвязной области, Мат. Сб. (Н. С.)
64 A06) A964), pp. 618-627.
[3] Краевые задачи со сдвигом на абстрактных римановых поверхностях, Докл. Акад.
Наук СССР 157 A964), pp. 26-29.
Zverovic, E. I., Litvincuk, G. S. (Зверович, Э. И, Литвинчук, Г. С.)
[1] Об одностронных краевых задачах теории аналитических функций, Докл. Акад.
Наук СССР 145 A962), pp. 266-269.
[2] Одностронное краевые задачи теории аналитических функций, Изв. Акад. Наук СССР,
Сер. Мат. 28 A964), pp. 1003-1036.
24*

INDEX OF SYIVIBOLS
(Numbers refer to the page where the symbol in question in defined)
¦ 6 ^24 Нк{х,у,У1,...,ук) 84
fW 13 Ъ\1\,Ъ 24 Gk(x,y,yi,...,yk) 84-85
i 14 Af/r) 24 Л" 121
C,(x), C(x) 14 Ф 29, 156 Г(х) 128
©*[/],©* 14 Qx 29,67,245 5(x, .y) 130
Jk(x) 14 Г, 29,68,245 Г 156
©oo[/], ©oo 15 gn(x, y) 29 Л 156
©oi[/],©oi 15 Q% 30 L/x) 157,210
Af(x0), A(xo) 19 F[x] 30-31, 36 b(J) 157
Cn[I],Cn 20 9ЧЯ,] 32,36 O(x) 174
^m.-S? 20 FnlodM 35 f(*) 193
*SW.*? 20 !Ршоа№] 35 r"W 216
M"[/],M" 22 ^-i^n 36 /„(x) 216
M+"[/],M» 22 36 /.0 217
M«[I],M! 22 /\ 217
^ 22 ^(X'^} 36 г 218
Ф [/] 46, 68, 245, 246, 266
{/} 23 >|x 244
23 G"« 47'137 r[f] 274,284
23 Q* 68 - *(*,,) 283
^"/} 23 й^ 68 у 306
Я 23 Лх 68,245 К* 306
«W 23 ?х 78 <f^ . 306

SUBJECT INDEX
Abel's equation 43, 111, 145, 163-179, 198,
204, 208
analytic iterate 209, 226
attractive domain 19
attractive fixed point 18
attractive fixed point of order к 19
automorphic coefficients 266-267
automorphic functions 41-43
Babbage's equation 288-292, 293, 297
Banach's fixed point theorem 6, 74, 98, 191,
249, 278
Beta function 130
biperiodic functions 227
branching processes 131
Cauchy's equation 26
Cebysev polynomials 216, 218
Cesaro method of summation 65
characteristic equation 262
closed orbit 15, 289
commuting functions 213-226, 301
compatible matrix (with a group) 269
completely monotonic functions 23, 118, 128,
131, 172, 174, 305
conjugacy 156-159
cosine 104, 231
continuous iteration group 197, 198, 199
convex functions 22, 45, 104, 105, 114, 115,
116, 131, 133, 143, 144, 159, 171, 202,
206, 305
convex functions of order n 22, 118, 119,
126, 131, 273
difference equations 5, 6, 114-121, 185
domain of attraction 19
doubly periodic functions 227, 229
elliptic functions 229
equivalent P-chains, SP-chains 215
equivalent points 14
Euler's equation (for the Gamma function) 235
Euler's equation (invarient curves) 286
Euler's function see Gamma function, Beta
function
exponential functions 162, 222-226
exponentially growing functions 174-179, 207-
209
Fatou's exceptional value 24, 218, 223
fixed point 14, 24, 222
fixed point of order к 14
formal conjugacy 157
formal solution 188, 304
formal solution of the Schroder equation 147
fractional iterates 293
functional equation 25, 26, 288
functional equation of associativity 26
functional equation of invariant curves 274-287
functional equation of order n 27
functional equation of order zero 27-28
functional equation xy=yx 28
functions, {/}-convex 23
functions, {/}-convex of order n 23, 121, 126
functions, {/}-monotonic 23, 61, 127, 130
Gamma function 26, 66, 105, 127-131, 191-193,
232-236
Gauss' multiplication formula 234
Gaussian law of errors 254
geometric difference equations 259
Goursat problem 101, 102
Hardy's scale 23, 129
hut-function 15
honest probability distribution 131, 133
implicit functions 28
indeterminate case 57, 104
involutory functions 26, 290, 291, 292, 303, 305
irrational case 148, 183
Isbell problem 214
iterates 13

376
Subject index
iteration group 197-212, 293
iteration group with respect to <J 197
Jensen's equation 144
Koenigs's equation 135
^-function 23, 24, 129, 178, 209
i-function of type (s, r, ft) 24
Legendre's formula 234
Levy algorithm 167, 171, 173, 206
logarithm with the base a 160
logarithmic functions 159-162
logarithmically convex functions 128, 162, 232
logarithmico-exponential scale see Hardy's
scale
modulus set 13, 24, 293
monotonic functions 22, 45, 66, 105, 107, 109,
110, 111, 113, 127, 129, 130, 133, 199, 241,
242, 290, 297-308
multiplier 18, 24
natural logarithm 161
nowhere differentiable functions 82
orbit 14
order of a functional equation 27
P-chain 215-218
permutable functions 213-226
Picard's exceptional value 24
Picard's transcendents 227
Poincare's equation 141, 148, 185
polynomials 22, 24, 118, 185, 186, 215-218,
220, 222, 271-273
principal iteration group 200, 202, 203, 204,
206, 208
principal solution 45
principal solution of Abel's equation 165, 167,
204
principal solution of difference equations 128
principal solution of Schroder's equation
143-145, 200, 214
Pythagorean theorem 28
^-difference equations 259
recurrent sequences 5, 121-126
reduction of order 259-260
regular iteration group 200, 202, 203, 204,
206, 208, 209, 124
regular map of ©ll^] 301
regular point in d^] 301, 303
repulsive fixed point 18
repulsive fixed point of order к 19
Riemann's boundary value problem 180
Riemann's equation 194
Riemann's function see Zeta function
Schauder's fixed point theorem 94
Schroder's equation 43, 44, 113, 135-162, 163,
180, 185, 198, 199, 214, 304
semipermutable functions 215
semipermutable polynomials 215-218
Sierpinski's carpet 241
sine 231
solution depending on an arbitrary function 45
SP-chain 215-218
Stirling's formula 129, 236
strictly convex function (of order n) 22
strongly attractive fixed point 18, 24, 52, 60
strongly attractive fixed point of order к 19
strongly repulsive fixed point 18, 24
strongly repulsive fixed point of order к 19, 222
submodulus interval 21
submodulus set 13, 15, 16, 31, 35, 39, 288
summability method 62
term 26
uniformization 28
Zeta function 6, 193-196

INDEX OF NAMES
(Pages in italics refer to the bibliography)
Abdi, W. H. 259, 308
Abel, N. H. 5, 163, 308
Acz61, J. 5, 6, 25, 26, 27, 30, 144, 198, 199,
201, 214, 227, 244, 259, 262, 267, 290,
305, 308, 328, 333, 337
Adachi, R. 227, 309
Adamczyk, H. 224, 307, 308
Adams, С R. 259, 309
d'Adhemar, R. 102, 309
Alblas, J. B. 180, 192, 309
Aleksandrija, G. N. 180, 309
Alekseevskil, V. P. 192, 309
Amaldi, U. 259, 309, 354
Ananda-Rau, K. 227, 309
Anastassiadis, J. 23, 113, 128, 129, 130, 131,
232, 310
Anczyk, L. 84, 310
Andreoli, G. 17, 21, 30, 67, 180, 236, 310
Anfertieva, E. A. 194, 310
Angheluta, T. 271, 273
Apostol, Т. М. 194, 310
Appell, P. 41, 43, 135, 180, 194, 227, 244, 310
Archimedes 5
Aristov, I. I. 199,311
Arnold, V. I. 230, 311, 361
Artiaga, L. 244, 311
Artin, E. 128, 232, 233, 234, 311
Ayoub, R. 194, 311
Azevedo do Amaral, I. M. 307, 311
Azpeitia, A. G. 244, 311
Baayen, P. С 15, 17, 311, 340, 34/
Babbage, Ch. 288, 311
Badescu, R. 43, 180, 259, 312
das Baghi, Hari 30, 312
Bajraktarevic, M. 17, 21, 46, 58, 59, 60, 61,
62, 63, 67, 74, 106, 130, 131, 163, 198,
199, 244, 248, 259, 262, 293, 294, 307,
312
Baker, I. N. 17, 24, 28, 159, 180, 199, 214,
218, 220, 222, 224, 226, 303, 305, 313,
327
Barba, G. 135, 199, 313
Barna, B, 15, 17, 18, 19, 313
Barnes, E. W. 192, 313
Barrow, D. F. 314
Barvinek, E. 163, 314
Bass, R. W. 274, 314, 343
Bateman, P. T. 194, 314
Batty, J. S. 199, 214, 314, 368
Baxter, G. 214, 314, 335
Beardon.A. F. 2%, 314
Beckenbach, E. F. 244, 259, 262, 314
Bedoeva, M. G. 180, 314
Beeger, N. G. W. H. 194, 314
Belardinelli, G. 58, 314
Bell, E. T. 194, 314
Bellman, R. 26, 131, 194, 259, 315
Bendersky, L. 192, 315
Bennett, A. A. 163, 199, 288, 315
van der Berg, J. 30, 46, 67, 84, 315
Berg, L. 194, 199, 214, 315
Berkovic, F. D. 180, 315
Bernays, P. 227, 315
Bernstein, V. 180,575
Bertrand, J. 180, 244, 315
Bielecki, A. 46, 58, 60, 84, 102, 103, 315,
337
Biermann, K.-R. 198, 316
Birkhoff, G. D. 41, 180, 230, 244, 259, 274,
316, 329
Block, H. D. 214, 215, 218, 316, 365
Bochner, S. 194, 244, 316, 320, 346
Bodewadt, U. T. 84, 163, 199, 293, 305, 316
Bogdanov, Ju. S. 288, 290, 316
Bohr, H. 128, 2X9,316,348
Boole, G. 5, 316
Boros,J3. 259, 317,368

378
Index of names
Bottcher, L. E. 17, 163, 180, 192, 199, 244,
259, 317
Bourbaki, N. 128, 317
Bourlet, С 135, 199, 259, 262, 317
Bouton, С L. 199,577
Bouzitat, J. 180, 317
Boyce, W. M. 214, 317
Bradley, F. W. 135, 199, 214, 317, 368
Brauer, G. 293, 318
Braun, S. 28, 318
Brolin, H. 17, 140, 318
Bronstein, B. S. 194, 318
Browder, A. 41, 318, 369
Browne, P. 180, 318
deBruijn, N. G. 17, 26, 163, 193, 199, 318
deBrun, F. 141, 244, 318
Brun, V. 318
Brydak, D. 23, 121, 124, 125, 126, 127,
274, 282, 286, 287, 318, 338
Buharinov, G. N. 180, 318
Buhl, A. 274, 318
Burek, J. 46, 106, 107, 111, 113, 114, 319,
340
Burkhardt, H. 192, 319
Le Caine, J. 259, 319
Cahen, E. 194, 319
Cakvetadze, S. S. 180, 319
Campagne, С 17, 319
Caratheodory, C. 128, 319
Caresse, P. 28, 319
Carleman, T. 180, 319
Carlitz, L. 194, 259, 319
Carman, M. G. 259, 319
Carmichael, R. D. 180, 192, 259, 319
Cartan, H. 17, 135, 319
Catalan, E. 46, 320
Cauchy, A. 5
Cayley, A. 58, 180, 227, 288, 291, 320
Chajot, Z. 41, 288, 291, 305, 320
Chandrasekharan, K. 194, 316, 320, 346, 351,
Charzynski, Z. 28, 320
Chayoth, W. 41,67, 320
Chen, Kuo-Tsai 214, 244, 320
Cherry, Т. М. 17, 321
Choczewski, B. 46, 48, 50, 52, 57, 59,
«4, 87, 90, 95, 99, 100, 101, 244, 248, 250,
252, 254, 321, 340
Ciesielski, Z. 230, 321
Cimmino, G. 102, 321
Cioranescu, N. 102, 321
Clunie, J. 25, 321
Cohen, H. 214, 321
Coifman, R. R. 46, 58, 135, 143, 163, 171,
197, 198, 199, 200, 204, 214, 321, 340
Collatz, L. 293, 297, 522
Colombo B. 102, 244, 322
Cooke, K. L. 67, 322
Cooper, R. 103, 104, 106, 322
van der Corput, J. G. 103, 244, 322
Cosnita, С 322
Courant, R. 128, 232, 322
Cremer, H. 135, 139, 147, 322
Crstici, B. 259, 267, 322
Crum, M. 135, 138, 303, 322
Curtis, M. F. 274, 322
Czerwik, S. 121, 259, 262, 322
Darling, H. В. С 323
Da vies, H. С 192, 323
DeMarr, R. 214, 323
Demeczky 214, 323
Dencev, R. 180, 323
Denjoy, A. 17, 193, 323
Deuring, M. F. 194, 323
Dinghas, A. 114, 128, 323
Dokovic, D. 1. 244, 323, 348
Domb, С 323, 325
Dracinskii, A. E. 180, 323
Drazin, M. P. 214, 323
Dubikajtis, L. 231, 323
Dufresnoy, J. 119, 323, 354
Dwork, B. 194, 324
Eagle, A. 128, 324
Ehrenpreis, L. 194, 324, 347
Eichler, M. 194,524
Ellis, A. J. 135, 324
Epstein, P. 194, 324
Erdos, P. 199, 209, 210, 212, 324, 334
v. Escherich, G. 227, 324
Esclangon, E. 227, 324
Etherington, I. M. H. 82, 83, 324
Euler, L. 5, 28, 193, 194, 235, 286, 324
Ewing, G. M. 305, 324, 366
Farkas, J. 135, 199,524
Fatou, P. 17, 19, 24, 41, 135, 140, 141, 147,
163, 180, 199, 214, 215, 222, 227, 324
Ferrand, J. 17,525
Ferrar, W. L. 41, 194, 325
Feyer, W. 180, 325

Index of names
379
Fine, N. J. 157, 199, 293, 297, 325, 338, 360
Fischer, W. 194, 325
Fisher, M. E. 323, 325
Flechsenhaar, A. 28, 325
Forder, H. G. 103, 104, 325
Formenti, С 163, 198, 326
Fort, Jr., M. K. 199, 202, 214, 226, 282, 283,
284, 285, 286, 326
Fort, T. 6, 326
Fortert, R. 17, 230, 326
Foures, L. 41, 326
Froberg, C.-E. 194, 326
Fueter, R. 326
Fujii, S. 326
Fulco, P. 259, 326
Fuller, F. B. 67, 163, 198, 199, 326, 368
FUrle, H. 180, 326
Ganapathy Iyer, V. 28, 180, 182, 183, 185,
186, 187, 214, 220, 326
Ganguillet.V. 41,526
Gauss, F. 234
Gelfond, A. O. 6, 326
Geppert, H. 180, 527
Gersevanov, N. M. 30, 58, 293, 327
Gerst, I. 141, 227, 327
Ghabbour, M. N. 17, 327, 369
Ghermanescu, M. 5, 27, 30, 41, 43 58, 102,
118, 135, 141, 163, 192, 194, 227, 244,
259, 262, 266, 267, 271, 288, 291, 308,
327, 328, 333
Gilman, R. E. 58, 328
Giraud, G. 141, 214, 244, 328
Godefroy, M. 192, 232, 328
Goi^b, S. 5, 198, 305, 308, 328
Goldberg, S. 6, 328
Goralcik, P. 214, 328
GStz, O. 135, 141,525
Goursat, E. 102, 328
Graf, J. H. 232, 328
Gnsvy, A. 244, 259, 267, 328
Gross, F. 28, 180, 313, 329
Guenther, P. E. 259, 316, 329
Guichard, С 185, 227, 259, 262, 52P
Guinand, A. P. 194, 52P
Gumowski, I. 17, 135, 52P, 346
von Gunten, P. 257, 52P
Hadamard, J. 25, 199, 214, 274, 275, 52P
Hadwiger, H. 30, 52P
Hahn, W. 259, 52P
Haidukov, P. I. 288, 289, 290, 291, 293, 294,
297, 330
Halanay, A. 227, 330
af Hallstrom, G. 41, 215, 216, 330
Hamburger, H. 194, 330
Hamilton, H. J. 17, 21, 107, 114, 259, 262,
330
Handt, Th. 199, 330, 337
Hansen, С 194, 330
Hardy, G. H. 23, 24, 41, 58, 62, 66, 129,
180, 193, 194, 209, 330, 331, 366
Harris, Th. E. 131, 135, 199, 331
Harris Jr., W. A. 244, 259, 331, 361
Hartman, Ph. 244, 274, 331
Hasabov, E. G. 180, 331, 344
van Haselen, A. 331
Haseman, Ch. 180, 331
Hasse, H. 194, 331
Heathcote, С R. 131, 132, 331, 360, 368
Hecke, E. 194, 331
Hedrlin, Z. 214, 332
Heilbronn, H. 194, 332
Heinhold J. 307, 332
Heins, M. H. 17, 288, 290, 332
Herve, M. 17, 180,552
Hilb, E. 58, 259, 332
Hille, E. 332
Hirche, J. 259, 262, 332
Holder, O. 180, 192, 194, 244, 332
Holsztynski, W. 296
Homma T. 28, 333
Hornich, H. 288, 290, 333
Hosszu, M. 198, 214, 254, 259, 267, 308.
328, 333, 368
Hou, Tsung-i 180, 333
Howroyd, T. D. 227, 333
Hua, L. K. 227, 333
Hukuhara, M. 135, 244, 333
Huneke, J. P. 214, 333
Hurwitz, Ad. 41, 185, 194, 333
Hutchinson, J. I. 194, 333
Hvedelidze, B. V. 180, 334, 346
Hvialkovskii, S. A. 180, 334
Hyllengren, A. 199, 334
Ingham, A. E. 193, 334
Isaacs, R. 14, 289, 293, 334
Isaacson, E. de St. Q. 30, 334
Isbell, J. R. 214, 334
Iseki, Sho 194, 334
Isenkrahe, С 17, 334

380
Index of names
Jabotinsky, E. 135, 199, 209, 210, 212, 324,
334
Jackson, F. H. 192, 259, 334
Jacobi, С G. J. 198
Jacobsthal, E. 214, 215, 216, 218, 222, 335
Jaggi, E. 41,288, 335
Jaglom, A. M. 131, 132,555
Jensen, J. B. W. V. 192, 232, 335
John, F. 107, 114, 259, 262, 335
Johnson, W. W. 288, 291, 335
Joichi, J. T. 214, 314, 335
Julia, G. 17, 82, 135, 140 141, 147, 163, 180,
214,215,288,291,555
Kaba, S. 141, 336
Kabaila, V. 227, 336
Kac, M. 230, 336
Kahane, J. P. 194, 336, 346
Kalman, R. 244, 336, 363
Karamata, J. 17, 336
Karteszi, F. 199, 214, 308, 337
Kasner, E. 135, 147, 337
Kenzegulov, H. K. 17, 337, 359
Kister, J. M. 288, 337
Kisynski, J. 46, 58, 60, 84, 102, 103, 315,
337
Kitamura, T. 30, 67, 337
Klingen, B. 135, 141, 337
Kneser, H. 135, 138, 139, 199, 303, 305,
330, 337
Knobloch, H.-W. 192, 194, 337
Knopp, K. 62, 65, 337
Kober, H. 194, 337
Koecher, M. 194, 337
Koenigs, G. 135, 139, 163, 180, 199, 214,
303, 338
Koliagin, Ju. M. 41, 338
Kolmogorov, A. N. 230, 338
Kordylewski, J. 30, 46, 53, 56, 57, 58, 59, 60,
67, 70, 72, 73, 77, 78, 79, 80, 84, 121, 125,
126, 127, 244, 246, 248, 259, 262, 266,
318, 338, 340
Korkine, A. 163, 198, 338
Korobeinik, Ju. F. 180, 338
Koroliuk, V. S. 131, 338, 371
Kostant, B. 157, 199, 325, 338
Kratzel, E. 194, 339, 345
Kreveras, G. 307, 339
Krull, W. 114, 119,559
Krzeszowiak, Z. 244, 339
Kucharzewski, M. 259, 262, 339, 340
Kuczma Marek. 5, 14, 17, 21, 23, 25, 27, 30, 35,
41, 43, 44, 46, 48, 50, 52, 53, 56, 57, 58, 59,
60, 61, 67, 70, 73, 75, 84, 95, 106, 107, 111,
113, 114, 119, 121, 122, 126, 128, 130, 135,
137, 141, 143, 156, 157, 159, 162, 163, 165,
171, 174, 180, 199, 202, 206, 214, 227, 230,
231, 244, 248, 259, 260, 262, 266, 271, 273,
282, 283, 293, 297, 305, 306, 307, 319, 321,
338, 339, 340, 362, 364, 368
vanKuik, J. 199, 340
Kuratowski, K. 14, 286, 340
Kuyk, W. 15, 17, 311, 340, 347
Kuylenstierna, N. 227, 341
Kveselava, D. A. 180, 341
Kwapisz, M. 46, 244, 341
Kyner, W. T. 274, 341
Laha, R. G. 254, 341, 345, 357
Laisant, С. А. 288, 290, 341
Lambek, J. 194, 341, 349
Landau, E. 193, 194, 341
Lasota, A. 14, 30, 341, 353
Lattes, S. 17, 135, 140, 141, 244, 259, 274, 275,
290, 341
Leau, L. 135, 148, 288, 342
Lecornu, L. 30, 342
Legendre, 234
Lemeray, E. M. 17, 180, 199, 214, 288, 307»
342
Leonov, V. V. 180, 244, 342
Leptin, H. 194, 343
Lerch M. 180, 194, 232, 343
Lessman, F. 6, 343
Levi, E. E. 141, 244, 343
Levin, B. Ja. 180, 192, 343
Levy, H. 6, 343
Levy, P. 163, 167, 174, 199, 343
Lewin, M. 199, 343
Lewis, D. С 199, 226, 274, 314, 343
Lichtenbaum. P. 194, 344
Lillo, J. С 293, 297, 344
Lindner, E. 17, 344
Lipinski, J. S. 28, 213, 214, 344
Litvincuk, G. S. 180, 331, 344, 371
Livencov, A. I. 288, 344
Lojasiewicz, S. 199, 288, 289, 293, 294, 296,
344
London, Fr. 17, 344
Lowig, H. 227, 244, 259, 345
Lukacs, E. 254, 341, 345, 357
Lundberg, A. 135, 141, 142, 143, 199, 200,
202, 203, 204, 345

Index of names
381
Lundmark, K. 163, 345
Lush, P. E. 282, 345
LUssy, W. 286, 345
Luxenberg, M. 28, 345
MaaB, H. 194, 345
Macintyre, A. J. 17, 345
Maharam, D. 17, 345
Mahler, K. 194, 345
Maier, W. 194, 227, 339, 345
Majcher, G. 102, 244, 248, 259, 346
Malchair, H. 28, 346
Mandelbrojt, S. 194, 320, 335, 346
Mandzavidze, F. G. 180, 334, 346
Marchay, R. 346
Marke, P. W. 194, 346
Martin, W. T. 244, Jl 6, 346
Marty, F. 41, 346
Mason, Th. E. 180, 192, 259, 346
Massera, J. L. 293, 346, 353
Matkowski, J. 180, 346
Matsumoto, T. 17, 347
Mattson, R. 199, 347
Maurice, M. A. 15, 17, 311, 340, 347
Mautner, F. I. 194, 324, 347
Mayer, A. 128, 130, 131, 347
McCarthy, J. 274, 347
McKiernan, M. A. 58, 135, 163, 347
McShane, N. 288, 290, 347
Mehrotra, В. М. 347
Mellin, Hj. 180, 192, 195, 347
Melnik, I. M. 180, 347
Mertnes, F. 227, 347
Me?chkowski, H. 6, 192, 259, 347
Michel, H. 197, 199, 200, 203, 204, 293, 294,
296, 347
Mikusinski, J. 103, 348
Min, Szu-hoa 194, 348
Mioduszewski, J. 28, 214, 348
Mira, Ch. 17, 135, 329, 348
Mitrinovic, D. S. 30, 244, 259, 267, 323, 348,
367
Mizumoto, H. 135, 348
Mohr, E. 348
Moiseev, N. N. 230, 348
Mollerup, J. 128, 316, 348
Montel, P. 6, 17, 23, 41, 135, 140, 163,
199, 228, 229, 271, 272, 274, 275, 278,
281, 348
Moore, E. H. 180, 192, 227, 349
Mordell, L. J. 193, 194, 349
Moret-Blanc, 180, 349
Morris, K. W. 174, 349, 363
Moser, J. 135, 180, 230, 274, 349
Moser, L. 194, 341, 349
Muckenhoupt, B. 148, 158, 159, 349
von der Muhll, K. 180, 349
Myller, A. 102, 350, 367
Myrberg, P. J. 17, 28, 41, 135, 141, 163, 180,
184, 259, 303, 350
Naftalevic, A. G. 185, 227, 244, 259, 350
Nalli, P. 180, 351
Narasimhan, R. 194, 320, 351
Nehari, Z. 180, 194, 351
Netto, E. 351
Neuman, F. 163, 351
Nicoletti, O. 17, 351
Nielsen, N. 180, 232, 351
Nikolaus, J. 215, 351
Nishino, T. 17, 274, 351, 370
Norlund, N. E. 6, 22, 118, 128, 351
Novotny, M. 30, 351
Obrechkoff, N. 17, 352
Oeconomou, A. C. 135, 159, 352
Oltramare, G. 244, 352
Onicescu, O. 244, 352
Osserman, R. 303, 352
Ostrowski, A. 17, 46, 192, 352
Ozawa, M. 352
Panov, A. M. 244, 352
Pastides, N. 41, 135, 141, 148, 180, 214, 352
Pelczar, A. 14, 30, 67, 84, 106, 307, 341, 353
Pellegrino, F. 30, 353
Pellet, A. F. 288, 353
Peres, J. 41, 43, 67, 353, 368
Peterson, H. 194, 353
Petracca, A. 293, 346, 353
Pfeiffer, A. G. 135, 147, 303, 304, 353
Picard, E. 5, 58, 102, 135, 139, 141, 163, 180,
185, 192, 193, 227, 244, 353
Pincherle, S. 5, 17, 41, 135, 163, 180, 199, 259,
288, 309, 354
Pisot, Ch. 119,525, 354
Podetti, F. 17, 354
Poincare, H. 141, 244, 274, 354
Poinsot 65, 354
Polya, G. 25, 28, 354
Pompeiu, D. 30, 46, 65, 354
Popovici, С 30, 41, 43, 46, 58, 67, 84, 102, 180,
244, 259, 267, 355

382
Index of names
Popoviciu, T. 22, 228, 271, 272, 273, 355
Post, E. L. 192, 355.
Pranger, W. 180, 555
Presic, S. 41, 244, 259, 267, 268, 271, 355
Prym, F. F. 192, 355
Pultr, A. 214, 355
Pyhteev, G. N. 180, 356
Raclis, R. 58, 180, 356
Rademacher, H. 41, 193, 194, 356
Radstrom, H. 17, 356
Rajagopal, С. Т. 185, 356
Ran, A. 199, 356
Rankin, R. A. 194, 356
Rasch, G. 192, 356
Rausenberger, O. 41, 135, 148, 180, 227, 288,
356
Rawson, R. 41, 135, 356
Read, A. H. 180, 356
Reddy, A. R. 185, 356
Reghis, M. 14, 30, 35, 244, 357, 368
Reichenbacher 163, 174, 199, 209, 357
Remoundos, G. J. 357
Renyi, A. 180, 254, 341, 343, 357
Renyi, С 180, 357
Revuz, A. 290, 357
de Rham, G. 67, 82, 84, 227, 357
Rieger, G. J. 194, 357
Riemann, B. 5, 193, 194, 357
Ritt, J. F. 17, 140, 141, 180, 192, 214, 215, 227,
288, 305, 357
Robbins, H. E. 227, 231, 358
Robinson, R. M. 358
Rosenbloom, P. Ch. 17, 214, 555
Rowe, Ch. H. 192, 358
Rozmus-Chmura, M. 106, 114, 358
Ryde, F. 259, 358
Sabitov, I. H. 180, 358
Samoloff, J. 244, 358
Sarkovski, A. N. 17, 18, 30, 67, 337,358, 359
Sarsanov, A. A. 199, 359
Schafke, F. W. 128, 359
Schapira, H. 359
Schauffler, R. 17, 21, 359
Scheeffer, L. 192, 359
Schimmack, R. 28, 359
Schmid, H. L. 194, 359, 365
Schmidt, F. K. 194, 359
Schmidt, Hermann, 192, 193, 259, 262, 274,359
Schnee, W. 194, 359
Schneider, A. 180, 360
Schobe, W. 293, 303, 360
Schottky, F. 41, 360
Schroder, E. 135, 198, 360
Schubert, C. F. 135, 199, 214, 360
Schwartz, A. J. 214, 360
Schweigert, E. G. 293, 297, 325, 360
Schweizer, B. 58, 180, 360
Schwering, K. 28, 360
Scott, S. 185, 360
Seneta, E. 131, 132, 163, 331, 360, 368
Sergeev, N. S. 360
Serret, J. A. 288, 291, 360
Sheffer, I. M. 180, 214, 215, 259, 262, 360
Shields, A. L. 214, 361
Shimizu, T. 41, 361
Sibiriani, F. 297, 361
Sibuya, Y. 244, 259, 331, 361
Siegel, С L. 135, 139, 141, 147, 148, 149, 194,
274, 361
Sierpinski, W. 28, 236, 239, 241, 290, 305, 307,
361
Sikorski, R. 28, 361, 371
Silberstein, L. 199, 214, 226, 361
Sinai, Ja. G. 311, 361
Sincov, D. M. 244, 362
Sjostrand, O. 102, 362
Sklar, A. 5, 14, 362
Smajdor, A. 17, 107, 109, 121, 124, 125, 126,
127, 163, 165, 180, 183, 197, 199, 204, 293,
305, 340, 362
Smajdor, W. 180, 183, 188, 244, 362
Smith, E. R. 362
Soni, K. L. 194, 362
Speiser, A. 194, 362
Spiess, O. 180, 199, 259, 262, 288, 307, 362
Srivastava, K. N. 307, 362
Starcher, G. W. 259, 363
Starke, E. P. 41, 363
Steckel, S. 17, 363
Steinberg, R. 46, 106, 363
Steinhaus, H. 46, 58, 66, 363
Sternberg, S. 135, 156, 199, 244, 363
Stridsberg, E. 180, 192, 363
Strodt, W. 244, 259, 363
Szego, G. 244, 336, 363
Szekeres, G. 129, 135, 138, 139, 143, 145, 159,
163, 165, 167, 170, 172, 173, 174, 176, 178,
199, 200, 202, 203, 204, 206, 208, 209, 305,
349, 363
Szmuszkowicz, H. 46, 58, 66, 363
Szymiczek, K. 227, 230, 340, 364

Index of names
383
Talanov, D. I. 17, 364
Tamagawa, T. 194, 364
Tambs Lyche, R. 14, 30, 35, 43, 44, 135, 163,
364
Tanaka, Sen-ichiro 180, 364
Tanner, L. 135, 288, 291, 364
Targonski, Gy. I. 5, 46, 135, 307, 364
Tarski, A. 214, 365
Tauber, S. 244, 246, 259, 365
Taylor, P. R. 194, 365
Teichmuller, O. 194, 359, 365
Thielman, H. P. 130, 214, 215, 218, 316, 365
Thijsen, W. P. 17, 21, 365
Thorn, R. 365
Thomae, J. 259, 365
Thron, W. J. 17, 303, 365
Tietze, H. 180, 192, 365
Titchmarsh, E. C. 41, 193, 331, 366
Topfer, H. 17, 135, 163, 180, 199, 366
Tornquist, M. G. 259, 366
Touchard, J. 141, 180, 192, 244, 307, 366
Tricomi, F. 17, 366
Trjitzinsky, W. J. 244, 259, 366
Tschebotarev, N. 135, 148, 199, 366
Urabe, M. 135, 244, 274, 366
Urysohn, P. 259, 262, 366
Utz, W. T. 305, 324, 364
van Uven, M. J. 135, 163, 199, 290, 366
Vaida, D. 180, 183, 367
Vajzovic, F. 106, 130, 244, 367
Valcovici, V. 102, 180, 350, 367
Valeiras, A. 30, 259, 267, 367
Valiron, G. 17, 28, 135, 141, 163, 180, 227, 244,
367
Vasic, P. M. 244, 348, 367
Vekua, N. P. 180, 367
Vere-Jones, D. 131, 132, 331, 360, 368
Vincze, E. 254, 257, 288, 290, 333, 368
Volterra, V. 41, 43, 67, 353, 368
Vopenka, P. 67, 260, 340, 368
Vuc, L. 14, 30, 35, 244, 259, 317, 356, 368
van der Waerden, B. L. 82, 128, 368
Wagner, R. 286, 368
Walker, A. G. 199, 214, 314, 317, 368
Wallace, A. D. 288, 368
Walsh, С. Е. 259, 262, 368
Ward, M. 67, 163, 198, 199, 326, 368
Watson, G. N. 193, 369
Wavre, R. 28, 369
Weaver, M. W. 30, 369
Weddeburn, J. H. M. 82, 180, 369
Weissinger, J. 194, 369
Werner, J. 41, 318, 369
Whitney, H. 89, 369
Whittaker E. T. 193, 369
Whittaker, J. M. 84, 180, 185, 369
Widder, D. V. 23, 369
Williams, K. P. 192, 369
Williams, W. E. 180, 369
Winn, С. Е. 17, 327, 369
Wintner, A. 227, 230, 369
Wirtinger, W. 180, 193, 227, 370
Wittich, H. 141, 180, 192, 370
Wolff, J. 17, 135, 141, 163, 180, 370
Wright, E. M. 135, 199, 370
Wunderlich, W. 82, 370
Yorinaga, M. 274, 370
Yoshioka, T. 17, 274, 351, 370
Zaharcfik, E. Ju. 41, 370
Zakowski, W. 180, 370
Zarankiewicz, K. 28, 361, 371
Zolotarev, M. 131, 338, 371
Zverovic, E. I. 180, 344, 371

M. Kuczma, Functional equations in a single variable
Page,
line
199
2514
42t7
445
4711
57
83*
123*
134!_3
147!
15416
15911
177! 0
19413
194ц
20414
2094
2453
2472
252! 5
25517
269s
2897;5
295*
2959
3068
ЗО64
3097
309i
320s
331ю
ERR
For
f(V)
puth (x)
<p[f(x)]
hypotheses 1.4 and 1.5
Supossing
Remark in the footnote on
Лх=-(оэ,-^1-2х)
00
>=o
ATA
Read
/-V(F)
put h(x)
<p[fl(x)]
hypotheses 1.4 and 1.6
Supposing
p. 50 applies also to Theorem 2.10.
^ = (-00,-^-2^)
00
X(-1)"^6B+,
v-0
Delete the condition po^O.
log sn -11 < К log n
^, = stjpJ
A-con-
?
ol(x)= J a(t) dt
F. K. Schmid
Schmidt
fulfilling (9.19)
F(x, u + v)s
h(x, y) by g(x,y), h(x,y)
J?-(JC V1 • V ) — ? Ax У л
co@) = Я"@)
¦ ¦ • + bOng[fn(x)]
from
?""( 1)
*V/
A912)
ibid.
ax+ 6
cx + 6
pp. 197-201.
log |5n — 1] ' < К log И
4tpl =¦**,, 1-1
A-con-
X
tx(x) = j a'{t) dt
F. K. Schmidt
Schmid
fulfilling (9.18)
F(x, u+v),
h(x,y) by g(x,y) and А(х,^)
Л> = <*0,й
ft(x.j'i.....j'm)-ft(^1.....5j
co@) = tA"(O)
¦ ¦¦ + w*[/;w]
form
r к,/
A902)
Indian J. Math.
ax + b
cx + d
pp. 341-346.

Errata (cont.)
Page,
line J
338,
3462O
3572O
375 8
376s
For
Korobeniik
Reaekkers
A959)
hut-function
376"
3776 (Adachi)
3777 (Adamczyk)
37910
3795 (Gumowski)
380,2 (Kordylewski)
38112(Mandclbrojt)
381i9
38223 (A. Renyi)
383ц (Utz)
124
297-308
309
224
Fortert
346
Delete 77 and insert
335
Mertnes
343
364
Read
Korobeinik
Raekkers
A859)
hat-function
214
297-303
308
244
Fortet
348
124
336
Mertens
345
366
In Theorems 3.6, 4.8 and 12.9, if ? is an inner point of interval /, con-
condition C.26) or A2.37) must be postulated in (x'0,f(x0))Kj(f(x0) ,xo~) or
<x'0,fm(x'0))Kj(fm(x0),x0), where ? -c < x'o < ? < x0 < ?+c, x'o, xoel.