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Fixed Points and
Topological Degree
in Nonlinear Analysis
Jane Cronin
American Mathematical Society
Providence, Rhode Island
1991 Mathematics Subject Classification. Primary 47H15;
Secondary 47H10, 47H11, 47H20, 55M20, 55M25, 35J65, 58C30, 34C25, 45G10.
Library of Congress Catalog Number: 62-21550
International Standard Book Number 0-8218-1511-3
International Standard Serial Number 0076-5376
Copying and reprinting. Individual readers of this publication, and nonprofit libraries acting
for them, are permitted to make fair use of the material, such as to copy a chapter for use
in teaching or research. Permission is granted to quote brief passages frpm this publication in
reviews, provided the customary acknowledgment of the source is given.
Republication, systematic copying, or multiple reproduction of any material in this publication
(including abstracts) is permitted only under license from the American Mathematical Society.
Requests for such permission should be addressed to the Assistant to the Publisher, American
Mathematical Society, P. O. Box 6248, Providence, Rhode Island 02940-6248. Requests can also
be made by e-mail to reprint-permissionQams.org.
© Copyright 1964 by the American Mathematical Society.
Reprinted with corrections, 1972
Printed in the United States of America.
The American Mathematical Society retains all rights
except those granted to the United States Government.
@ The paper used in this book is acid-free and falls within the guidelines
established to ensure permanence and durability.
10 9 8 7 02 01 00 99 98
PREFACE
Since this book is an introduction to the application of certain topological
methods to nonlinear differential and integral equations, it is necessarily an
incomplete account of each of these subjects. Only the topology which is
needed will be introduced, and only those aspects of differential and integral
equations which can be profitably studied by the use of the topological
methods are discussed. The bibliography is intended to be representative,
not complete.
I am greatly indebted to Professor Lamberto Cesari who read the entire
manuscript and made a number of valuable suggestions and corrections
concerning both the mathematics and the exposition in this book.
The preparation of this book was supported by the Air Force Office of
Scientific Research. I wish to express my thanks to the Air Force Office
of Scientific Research and also to the editorial staff of the American
Mathematical Society for their work in preparing the manuscript for
publication.
Jane Cronin.
FOREWORD
Our aim is to give a detailed description of fixed point theory and topo
logical degree theory starting from elementary considerations and to explain
how this theory is used in the study of nonlinear differential equations,
ordinary and partial. The reader is expected to have a fair knowledge of
advanced calculus, especially the point set theory of Euclidean n-space, but
no further knowledge of topology is assumed. (Following this introduction
is a list of definitions and notations from advanced calculus which will be
used in the text.)
Why topological methods are used. It is easy to see that the solution
x(t) of the differential equation
such that x(0) — 0 &nddx(0)/dt = 1, is o; = sint. In finding this solution, we
have obtained complete quantitative information, i.e., for any given value
of t, we can find the corresponding value x(t) by referring to a table of trigono
metric functions. However, we have also obtained other useful information
called qualitative or "in the large" information: we know that the values
x(t) are between — 1 and + 1 and that x(t) has period 2n.
If all differential equations could be solved as easily, it would be unnecessary
to introduce the distinction between quantitative and qualitative informa
tion concerning solutions. But this example is an exceptionally simple one.
Most differential equations, especially nonlinear equations, must be studied
with one technique to obtain quantitative information and by another to
obtain qualitative information. If a method can be derived for finding the
numerical values of the solution corresponding to given values of the in
dependent variable, this method will usually not give us qualitative informa
tion. For example, to determine if the solution is periodic requires a different
approach.
The topological methods we describe will give us qualitative information,
e.g., information about the existence and stability of periodic solutions of
ordinary differential equations and the existence of solutions of certain partial
differential equations. However, they will give no quantitative informa
tion, i.e.. no information about how to compute any of the solution values.
At best, we will get upper and lower bounds for solution values. The
information that the topological methods give us is thus incomplete.
Nevertheless, topological methods are used because at present there are
no other methods that yield as much qualitative information with so little
effort. It is possible to envision a future in which topological methods will
vi
FOREWORD Vll
be supplanted by more efficient methods, but so far there is little encourage
ment for doing so. Such a venerable technique as the Poincare-Bendixson
Theorem has not been essentially improved in over sixty years although it is
widely used. (The Poincare-Bendixson Theorem is not one of the techniques
which we will describe but as will be seen in Chapter II, it is intimately
related to one of these techniques.)
The topological techniques to be developed. The two techniques to be
developed, the fixed point theorem and local topological degree, are closely
connected. The fixed point theorem has the advantage of being a com
paratively elementary theorem (it can be proved without using any "topo
logical machinery") which has many useful applications. The topological
degree theory requires lengthier considerations for its development, but it
has an important advantage over the fixed point theorem: it gives informa
tion about the number of distinct solutions, continuous families of solutions,
and stability of solutions.
When to use topological techniques. We shall mostly be concerned with
the question of how to apply topological techniques. The question of when
to apply them is equally important. There can be no precise answer to the
question, but we can formulate a rough rule. If we use an analytical method
(like successive approximations), we establisli existence and uniqueness of
solution and a method (not necessarily practical) for computing the solution.
If analytical means fail and especially if there seems to be no way to establisli
uniqueness, then we turn to the weaker question of establishing mere
existence. For answering this weaker question, the topological methods
(cruder and yielding less information than analytical methods) sometimes
suffice. Thus topological methods should be regarded as a last resort or at
least a later resort than analytical methods.
Summary of contents. In Chapter I, a definition of the local topological
degree in Euclidean rc-space is given, the basic properties of topological
degree are derived, and some methods for computing the degree are described.
Also the Brouwer Theorem (the fixed point theorem in Euclidean n-space) is
obtained. In Chapter II, the techniques described in Chapter 1 are applied
to some problems in ordinary differential equations: existence and stability
of periodic and almost periodic solutions. In Chapter III, the Euclidean
n-space techniques developed in Chapter I are extended to spaces of arbitrary
dimension. We obtain the Leray-Schauder degree and the Schauder and
Banach fixed point theorems for mappings in Banach space. We obtain
also a combination of analytical and topological techniques which can be
used to study local problems in Banach space. Finally in Chapter IV, the
theory developed in Chapter III is applied to integral equations, partial
differential equations and to some further problems on periodic solutions of
ordinary differential equations.
Vlll FOREWORD
The applications in Chapters II and IV are treated in varying detail.
Existence of periodic solutions of nonautonomous ordinary differential
equations is treated in complete detail. Stability of periodic solutions is
treated in full detail except for the basic stability theorem of Lyapunov
which is stated without proof. The other topics in Chapter II are similarly
treated: certain theorems, particularly those from other disciplines, are
stated without proof. In Chapter IV, an elaborate apparatus of theorems
from analysis must be used in applying the Leray-Schauder theory and the
Schauder Fixed Point Theorem. Some of these theorems have lengthy and
complicated proofs and we restrict ourselves to giving references for these
proofs.
The numbering of definitions, theorems, etc., is done independently in
each chapter. Unless otherwise stated, references to a numbered item
means that item in the chapter in which the reference is made. E.g., if in
Chapter II, reference is made to Theorem (3.8), that means Theorem (3.8)
in Chapter II.
Some Terminology and Notation Used in This Text
| denotes end of proof.
nasc is abbreviation for necessary and sufficient condition.
Set notation
e: element of
£: not an element of
(J: union
P): intersection
—: difference
0: null set
C : contained in
Ac: complement of set A
A x B: Cartesian product of sets A and B, i.e., the set of all
ordered pairs (a, b) where a e A, b e B.
The set of elements having property P is denoted by:
[x J x has property P].
If a, b are real numbers such that a < b, the following notation is used to
indicate the various intervals:
[a, b] = [x real / a g x g b]
y
[a, b) = [x real / a ^ x < b],
(a, b] = [x real / a < x ^ 6],
(a, b) = [x real / a < x < b].
[a, 6] is called a closed interval and (a, b) is called an open interval. Rn
denotes real Euclidean n-space, i.e., the collection of n-tuples of real numbers
FOREWORD IX
(x • • •, x ). The elements of Rn will be denoted by single letters, p, q, • • •
lf n
when this is possible. If p = {p • • •, p ) and A is a real number, then
1} n
Xp = {Xp^- • ., Xp ).
n
In particular,
(~l)P = (-^i>- ' •> -Pn)-
If p = {p ---,p ) and ? = (q • • •, g ), then
lf n lt n
p + q = (Pi + qi,--,p + ffn).
n
The distance between points p and q, denoted by \p — g|, is:
b - ?l = [(^i - ?i)2 + • • • + (Pn - g )2]1/2.
n
If p e i2n, the neighborhoods of p are the sets:
N(P) = [?/ |P " q\ < «]
e
where £ is an arbitrary positive number. A set 0 in i?n is open if for each
point p e 0, there is a neighborhood N (p) of p such that N (p) C 0. A set
E €
JP in ^n is closed if i^c is open. A point p is a K??n7 ^OZ'TZ/ (cluster point,
accumulation point) of a set E in i?n if each neighborhood of p contains at
least one point of E distinct from p. (Point p may or may not be in E.) If
D C Rn> & boundary point of D is a point p such that each neighborhood of
p contains a point of D and a point of Dc. (A boundary point of D may or
may not be in D.) The collection of boundary points of D is denoted by D'.
The set D \J D' (also denoted by D) is called the closure of D. The set
D U D' is a closed set. If there is a neighborhood N (p) of a point p such
£
that N (p) is contained in a set E, then ^) is an interior point of E.
E
A metric space is a collection ilf of points p, q, • • • for which a function p
from M x M into the non-negative real numbers is defined such that:
(1) p(p, q) > 0 if and only if p ^ q\
(2) ^p, q) = (q,p);
P
(3) p(y, r) <: p(p q) + p{q, r).
9
All the concepts defined for i?n, i.e., neighborhood, open set, closed set, etc.,
may be defined for a metric space by using the same definitions given for
Rn only with \p — q\ replaced by p(p, q).
A separable metric space is a metric space M such that there is a denumer-
able subset [x ] of M such that the closure of [x ] is M.
n n
A connected set in a metric space M is a set $ such that S is not the union
of two disjoint nonempty sets A and B which are contained in disjoint open
sets.
A component of an open set 0 in a metric space is a maximal connected
subset of 0.
A locally connected metric space is a metric space M such that if p e M
and iV (^) is a neighborhood of _p then there is a connected neighborhood
£
of p which is contained in N (p).
E
