Nonlinear Functional Analysis A First Course 2nd Edition

Description

he purpose of this book is to provide an introduction to the theory of the topological degree and to some variational methods used in the solution of some nonlinear equations formulated in Banach or Hilbert spaces. While the choice of topics and the treatment have been kept sufficiently general so as to interest all students of higher mathematics, the material presented will be particularly useful for students aspiring to work in the applications of mathematics, especially in the area of partial differential equations.

The first chapter gives a brisk introduction to differential calculus in normed linear spaces and some important classical theorems of analysis are proved. The second chapter develops the theory of the topological degree in finite dimensional Euclidean spaces while the third chapter extends this study to cover the theory of the Leray-Schauder degree in infinite dimensional Banach spaces. Applications, especially to fixed point theorems, are presented. The fourth chapter gives an introduction to bifurcation theory. The last chapter studies some methods to find critical points of functionals defined on Banach spaces, with emphasis on min-max methods. The text is punctuated throughout with exercises which prove additional results or indicate applications, especially to nonlinear partial differential equations.

The first edition of this book has been very well received and it is hoped that this (second) edition will prove to be even more user-friendly. The presentation has been completely overhauled, without altering the structure of the earlier edition. Many definitions and statements of results, and their proofs, have been rewritten in the interest of greater clarity of exposition. A section on monotone mappings has been added and a few more important fixed point theorems have been covered.

Table of Contents

1 Differential calculus on normed linear spaces 1
1.1 The Fr´echet derivative . . . . . . . . . . . . . . . . . . . . 1
1.2 Higher order derivatives . . . . . . . . . . . . . . . . . . . 15
1.3 Some important theorems . . . . . . . . . . . . . . . . . . 20
1.4 Extrema of real-valued functions . . . . . . . . . . . . . . 27
2 The Brouwer degree 33
2.1 Definition of the degree . . . . . . . . . . . . . . . . . . . 33
2.2 Properties of the degree . . . . . . . . . . . . . . . . . . . 41
2.3 Brouwer’s theorem and applications . . . . . . . . . . . . 46
2.4 Monotone mappings on Hilbert spaces . . . . . . . . . . . 49
2.5 Borsuk’s theorem . . . . . . . . . . . . . . . . . . . . . . . 57
2.6 The genus . . . . . . . . . . . . . . . . . . . . . . . . . . . 63
3 The Leray-Schauder degree 68
3.1 Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . 68
3.2 Definition of the degree . . . . . . . . . . . . . . . . . . . 71
3.3 Properties of the degree . . . . . . . . . . . . . . . . . . . 73
3.4 Fixed point theorems . . . . . . . . . . . . . . . . . . . . . 76
3.5 The index . . . . . . . . . . . . . . . . . . . . . . . . . . . 82
3.6 An application to differential equations . . . . . . . . . . . 87
4 Bifurcation theory 91
4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . 91
4.2 The Lyapunov-Schmidt method . . . . . . . . . . . . . . . 95
4.3 Morse’s lemma . . . . . . . . . . . . . . . . . . . . . . . . 97
4.4 A perturbation method . . . . . . . . . . . . . . . . . . . 104
4.5 Krasnoselsk’ii’s theorem . . . . . . . . . . . . . . . . . . . 107
4.6 Rabinowitz’ theorem . . . . . . . . . . . . . . . . . . . . . 109
4.7 A variational method . . . . . . . . . . . . . . . . . . . . . 112
5 Critical points of functionals 120
5.1 Minimization of functionals . . . . . . . . . . . . . . . . . 120
5.2 Saddle points . . . . . . . . . . . . . . . . . . . . . . . . . 126
5.3 The Palais-Smale condition . . . . . . . . . . . . . . . . . 130
5.4 The deformation lemma . . . . . . . . . . . . . . . . . . . 136
5.5 The mountain pass theorem . . . . . . . . . . . . . . . . . 142
5.6 Multiplicity of critical points . . . . . . . . . . . . . . . . 146
5.7 Critical points with constraints . . . . . . . . . . . . . . . 150
Bibliography 159
Index 163