Nonlinear Functional Analysis - A First Course

Description

The book discusses the basic theory of topological and variational methods used in solving nonlinear equations involving mappings between normed linear spaces. It is meant to be a primer of nonlinear analysis and is designed to be used as a text or reference book by graduate students. Frechet derivative, Brouwer fixed point theorem, Borsuk's theorem, and bifurcation theory along with their applications have been discussed. Several solved examples and exercises have been carefully selected and included in the present edition. The prerequisite for following this book is the basic knowledge of functional analysis and topology. In the present second edition, the presentation has been completely overhauled without changing the basic structure of the book. The statements of results, definitions and remarks have been modified wherever necessary, and many proofs have been rewritten, in view of greater clarity of the exposition. Some examples and exercises have been added. A completely new section on monotone mappings has been added, and the proofs of a few more important fixed point theorems have been included. Preface to the Second Edition Preface to the First Edition Contents About the Author 1 Differential Calculus on Normed Linear Spaces 1.1 The Fréchet Derivative 1.2 Higher-Order Derivatives 1.3 Some Important Theorems 1.4 Extrema of Real-Valued Functions References 2 The Brouwer Degree 2.1 Definition of the Degree 2.2 Properties of the Degree 2.3 Brouwer's Theorem and Applications 2.4 Monotone Mappings on Hilbert Spaces 2.5 Borsuk's Theorem 2.6 The Genus References 3 The Leray–Schauder Degree 3.1 Preliminaries 3.2 Definition of the Degree 3.3 Properties of the Degree 3.4 Fixed Point Theorems 3.5 The Index 3.6 An Application to Differential Equations References 4 Bifurcation Theory 4.1 Introduction 4.2 The Lyapunov–Schmidt Method 4.3 Morse's Lemma 4.4 A Perturbation Method 4.5 Krasnoselsk'ii's Theorem 4.6 Rabinowitz' Theorem 4.7 A Variational Method References 5 Critical Points of Functionals 5.1 Minimization of Functionals 5.2 Saddle Points 5.3 The Palais–Smale Condition 5.4 The Deformation Lemma 5.5 The Mountain Pass Theorem 5.6 Multiplicity of Critical Points 5.7 Critical Points with Constraints References Index