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.
Author(s): S. Kesavan
Series: Texts and Readings in Mathematics 28
Edition: 2
Publisher: Springer Nature Singapore
Year: 2022
Language: English
Pages: 150
City: Singapore
Tags: Functional Analysis, Frechet Derivative, Brower Degree, Leray-Schauder Degree, Bifurcation, Critical Points
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
