Nonlinear Functional Analysis: A First Course

188 Pages·2004·10.482 MB·English

by  S. Kesavan (auth.)

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Nonlinear Functional Analysis
A First Course
TEXTS AND READINGS 28
IN MATHEMATICS
Nonlinear Functional Analysis
A First Course
Texts and Readings in Mathematics
Advisory Editor
C. S. Seshadri, Chennai Mathematical Institute, Chennai.
Managing Editor
Rajendra Bhatia, Indian Statistical Institute, New Delhi.
Editors
R. B. Bapat, Indian Statistical Institute, New Delhi.
V. S. Borkar, Tata Inst. of Fundamental Research, Mumbai.
Probai Chaudhuri, Indian Statistical Institute, Kolkata.
V. S. Sunder, Irrst. of Mathematical Sciences, Chennai.
M. Vanninathan, TIFR Centre, Bangalore.
Nonlinear Functional Analysis
A First Course
S. Kesavan
Institute of Mathematical Sciences
Chennai
[l.dgl@ HINDUSTAN
U llLJ UB OOK AGENCY
Published by
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ISBN 978-81-85931-46-3 ISBN 978-93-86279-21-7 (eBook)
DOI 10.1007/978-93-86279-21-7
Copyright © 2004, Hindustan Book Agency (India)
Digitally reprinted paper cover edition 2011
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Preface
Nonlinear Functional Analysis studies the properties of (con
tinuous) mappings between normed linear spaces and evolves meth
ods to solve nonlinear equations involving such mappings. Two
major approaches to the solution of nonlinear equations could be
described as topological and variational. Topological methods are
derived from fixed point theorems and are usually based on the
notion of the topological degree. Variational methods describe the
solutions as critical points of a suitable functional and study ways
of locating them.
The aim of this book is to present the basic theory of these
methods. It is meant to be a primer of nonlinear analysis and is
designed to be used as a text or reference book by students at
the masters or doctoral level in Indian universities. The prereq
uisite for following this book is knowledge of functional analysis
and topology, usually part of the curriculum at the masters level
in most universities in India.
The first chapter covers the preliminaries needed from the dif
ferential calculus in normed linear spaces. It intro duces the no
tion of the Frechet derivative, which generalizes the notion of the
derivative of areal valued function of a single real variable. Some
classical theorems which are repeatedly used in the sequel, like the
implicit function theorem and Sard's theorem, are proved here.
The second chapter develops the theory of the topological de
gree in finite dimensions. The Brouwer fixed point theorem and
Borsuk's theorem are proved and some of their applications are
presented.
The next chapter extends the notion of the topological de
gree to infinite dimensional spaces for a special class of mappings
known as compact perturbations of the identity. Again, fixed
vi
point theorems (in particular, Schauder's theorem) are proved and
applications are given.
The fourth chapter deals with bifurcation theory. This studies
the nature of the set of solutions to equations dependent on a pa
rameter, in the neighbourhood of a 'trivial solution'. Science and
engineering are full of instances of such problems. A variety of
methods for the identification of bifurcation points - topological
and variational - are presented.
The concluding chapter deals with the existence and multi
plicity of critical points of functionals defined on Banach spaces.
While minimization is one method, other critical points, like sad
dIe points are found by using results like the mountain pass theo
rem, or, more generally, what are known as min - max theorems.
Nonlinear Analysis, today, has a bewildering array of tools. In
selecting the above topics, a conscious choice has been made with
the following objectives in mind:
• to provide a text book which can be used for an introductory
one - semester course covering classical material;
• to be of interest to a general student of higher mathematics.
The examples and exercises that are found throughout the text
have been chosen to be in tune with these objectives (though, from
time to time, my own bias towards differential equations does show
up.)
It is for this reason that some of the tools developed more re
cently have been (regrettably) omitted. Two examples spring to
ones mind. The first is the method 01 concentration compactness
(which won the Fields Medal for P. 1. Lions). It deals with the
convergence of sequences in Sobolev spaces. Its main application
is in the study of minimizing sequences for functionals associated
to so me semilinear elliptic partial differential equations into which
vii
a certain 'lack of compactness' has been built. Another instance
is the theory of r - convergence. This theory studies the conver
gence of the minima and minimizers of a family of functionals.
Again, while the theory can be developed in the very general con
text of a topological space, alot of technical results in Sobolev
spaces are needed in order to present reasonably interesting re
sults. The applications of this theory are myriad, ranging from
nonlinear elasticity to homogenization theory. Such topics, in my
opinion, would be ideal for a sequel to this volume, meant for an
advanced course on nonlinear analysis, specifically aimed at stu
dents working in applications of mathematics.
The material presented here is classical and no claim is made
towards originality of presentation (except for some of my own
work included in Chapter 4). My treatment of the subject has
been greatly influenced by the works of Cartan [4], Deimling [7],
Kavian [11], Nirenberg [19] and Rabinowitz [20].
This book grew out of the notes prepared for courses that I
gave on various occasions to doctoral students at the TIFR Cen
tre, Bangalore, India (where I worked earlier), the Dipartimento
di Maternatica G. Castelnuovo, Universiat degli Studi di Roma
"La Sapienza", Rome, Italy and the Laboratoire MMAS, Univer
site de Metz, Metz, France. I would like to take this oppurtunity
to thank these institutions for their facilities and hospitality.
I would like to thank the Institute of Mathematical Sciences,
Chennai, India, for its excellent facilities and research environ
ment which permitted me to bring out this book. I also thank
the publishers, Hindustan Book Agency and the Managing Editor
of their TRIM Series, Prof. Rajendra Bhatia, for their coopera
tion and support. I also thank the two anonymous referees who
read through the entire manuscript and made several helpful sug
gestions which led to the improvement of this text. At a personal
level, I would like to thank my friend and erstwhile colleague, Prof.
Vlll
v. S. Borkar, who egged me on to give the lectures at Bangalore
in the first place and kept insisting that I publish the notes. I
also wish to thank one of my summer students, Mr. Shivanand
Dwivedi, who, while learning the material from the manuscript,
also did valuable proof reading. Finally, for numerous personal
reasons, I thank the members of my family and fondly dedicate
this book to them.
Chennai. S. Kesavan
October, 2003.
Contents
1 Differential Calculus on Normed Linear Spaces 1
1.1 The Frechet Derivative . . . 1
1.2 Higher Order Derivatives 16
1.3 Some Important Theorems 22
1.4 Extrema of Real Valued Functions 29
2 The Brouwer Degree 36
2.1 Definition of the Degree · ....... 36
2.2 Properties of the Degree · ....... 45
2.3 Brouwer's Theorem and Applications . 50
2.4 Borsuk's Theorem 57
2.5 The Genus 63
3 The Leray - Schauder Degree 69
3.1 Preliminaries ...... 69
3.2 Definition of the Degree 72
3.3 Properties of the Degree 74
3.4 Fixed Point Theorems 78
3.5 The Index ........ 82
3.6 An Application to Differential Equations . 87
4 Bifurcation Theory 92
4.1 Introduction ... ......... 92
4.2 The Lyapunov - Schmidt Method 97
4.3 Morse's Lemma ..... 99
4.4 APerturbation Method · .... 107