A Topological Introduction to Nonlinear Analysis

by Robert F. Brown

Overview

This third edition is addressed 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. Included in this new edition are several new chapters that present the fixed point index and its applications. The exposition and mathematical content is improved throughout. This book is ideal for self-study for mathematicians and students interested in such areas of geometric and algebraic topology, functional analysis, differential equations, and applied mathematics. It is a sharply focused and highly readable view of nonlinear analysis by a practicing topologist who has seen a clear path to understanding. "For the topology-minded reader, the book indeed has a lot to offer: written in a very personal, eloquent and instructive style it makes one of the highlights of nonlinear analysis accessible to a wide audience."-Monatshefte fur Mathematik (2006)

Product Details

ISBN-13:	9783319117935
Publisher:	Springer International Publishing
Publication date:	11/28/2014
Edition description:	3rd ed. 2014
Pages:	240
Product dimensions:	6.10(w) x 9.25(h) x 0.02(d)

About the Author

Robert F. Brown is a Professor of Mathematics at UCLA. His research area includes algebraic topology that is included within topological fixed point theory. Professor Brown's most recent research concerns the fixed point theory of fiber maps of fiberings with singularities.

Table of Contents

I Fixed Point Existence Theory.- 1 The Topological Point of View.- 2 Ascoli-Arzela Theory.- 3 Brouwer Fixed Point Theory.- 4 Schauder Fixed Point Theory.- 5 The Forced Pendulum.- 6 Equilibrium Heat Distribution.- 7 Generalized Bernstein Theory.- II Degree Theory.- 8 Brouwer Degree.- 9 Properties of the Brouwer Degree.- 10 Leray-Schauder Degree.- 11 Properties of the Leray-Schauder Degree.- 12 The Mawhin Operator.- 13 The Pendulum Swings Back.- III Bifurcation Theory.- 14 A Separation Theorem.- 15 Compact Linear Operators.- 16 The Degree Calculation.- 17 The Krasnoselskii-Rabinowitz Bifurcation Theorem.- 18 Nonlinear Sturm-Liouville Theory.- 19 More Sturm-Liouville Theory.- 20 Euler Buckling.- IV Appendices.- A Singular Homology.- B Additivity and Product Properties.- C Bounded Linear Transformations.- C Bounded Linear Transformations.- References.

Editorial Reviews

Based on ideas from several branches of topology, and illustrated by numerous figures that attest to the geometric nature of the exposition, provides mathematically sophisticated readers with an understanding of the mathematics of the nonlinear phenomena that characterize the real world. A classical example is given in the differential equation problem that models the maximum weight that a column can support without buckling. Annotation c. Book News, Inc., Portland, OR (booknews.com)

Booknews

"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)

"For the topology-minded reader, the book indeed has a lot to offer: written in a very personal, eloquent and instructive style it makes one of the highlights of nonlinear analysis accessible to a wide audience."

- Monatshefte für Mathematik

"Written by an expert in fixed point theory who is well aware of the important applications of this area to nonlinear analysis and differential equations, the first edition of this book has been very well received, and has helped both topologists in learning nonlinear analysis and analysts in appreciating topological fixed point theory. The second edition has kept the freshness and clarity of style of the first one. The new version remains more than even an excellent introduction to the sue of topological techniques in dealing with nonlinear problems." —-Mathematical Society

From the Publisher