A Topological Introduction to Nonlinear Analysis

by Brown, Robert F.

• ISBN13: 9780817632588
• ISBN10: 0817632581
• Edition: 2nd
• Format: Paperback
• Copyright: 2003-12-01
• Publisher: Birkhauser

Summary

"The book is highly recommended as a text for an introductory course in nonlinear analysis and bifurcation theory . . . reading is fluid and very pleasant . . . style is informal but far from being imprecise."-MATHEMATICAL REVIEWS (Review of the First Edition)Here is a book that will be a joy to the mathematician or graduate student of mathematics---or even the well-prepared undergraduate---who would like, with a minimum of background and preparation, to understand some of the beautiful results at the heart of nonlinear analysis. Based on carefully expounded ideas from several branches of topology, and illustrated by a wealth of figures that attest to the geometric nature of the exposition, the book will be of immense help in providing its readers with an understanding of the mathematics of the nonlinear phenomena that characterize our real world.New to the second edition: New chapters will supply additional applications of the theory and techniques presented in the book. * Several new proofs, making the second edition more self-contained.

Table of Contents

Preface	ix
Part I Fixed Point Existence Theory
1 The Topological Point of View	3	(6)
2 Ascoli-Arzela Theory	9	(10)
3 Brouwer Fixed Point Theory	19	(4)
4 Schauder Fixed Point Theory	23	(6)
5 The Forced Pendulum	29	(10)
6 Equilibrium Heat Distribution	39	(6)
7 Generalized Bernstein Theory	45	(6)
Part II Degree Theory
8 Brouwer Degree	51	(4)
9 Properties of the Brouwer Degree	55	(8)
10 Leray-Schauder Degree	63	(6)
11 Properties of the Leray-Schauder Degree	69	(10)
12 The Mawhin Operator	79	(6)
13 The Pendulum Swings Back	85	(10)
Part III Bifurcation Theory
14 A Separation Theorem	95	(4)
15 Compact Linear Operators	99	(12)
16 The Degree Calculation	111	(8)
17 The Krasnoselskii-Rabinowitz Bifurcation Theorem	119	(14)
18 Nonlinear Sturm-Liouville Theory	133	(10)
19 More Sturm-Liouville Theory	143	(12)
20 Euler Buckling	155	(8)
Part IV Appendices
A Singular Homology	163	(6)
B Additivity and Product Properties	169	(6)
C Bounded Linear Transformations	175	(6)
References	181	(2)
Index	183