Synopses & Reviews
"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.
Review
Aus den Rezensionen zur 2. Auflage: "... Um auf nicht einmal 200 Seiten ein sinnvolles Bild des umfangreichen Gegenstandes zu zeichnen, muss nämlich eine sehr selektive Stoffauswahl getroffen werden. Sehr wohl werden ... drei Ansprüche erhoben, welchen das Buch auf sicherlich vorbildliche Weise gerecht wird ... Dies wird durch eine klare Gliederung in drei Teile und einen Anhang erreicht. ... Mit seinem ... verbalen und doch sehr präzisen Stil versteht es der Autor hervorragend, die entscheidenden Ideen zu vermitteln. ... Deshalb kann das Buch als Einführung in den Gegenstand uneingeschränkt empfohlen werden ..." (R. Winkler, in: IMN - Internationale Mathematische Nachrichten, 2005, Vol. 198, S. 33)
Synopsis
"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." -review of the first edition. New to this edition: additional applications of the theory and techniques, as well as several new proofs. This book is ideal for self-study for mathematicians and students interested in geometric and algebraic topology, functional analysis, differential equations, and applied mathematics.
Table of Contents
Preface.- Part I: Fixed Point Existence Theory.- The Topological Point of View.- Ascoli-Arzela Theory.- Brouwer Fixed Point Theory.- Schauder Fixed Point Theory.- The Forced Pendulum.- Equilibrium Heat Distribution.- Generalized Bernstein Theory.- Part II: Degree Theory.- Brouwer Degree.- Leray-Schauder Degree.- Properties of the Leray-Schauder Degree.- The Mawhin Operator.- The Pendulum Swings Back.- Part III: Bifurcation Theory.- A Separation Theorem.- Compact Linear Operators.- The Degree Calculation.- The Krasnoselskii-Rabinowitz Bifurcation Theorem.- Nonlinear Sturm-Liouville Theory.- Euler Buckling.- Part IV: Appendices.- Singular Homology.- Additivity and Product Properties.- Bounded Linear Transformations.- References.- Index