A Topological Introduction to Nonlinear Analysis

1. Front Matter

   Pages i-ix

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2. Fixed Point Existence Theory

   1. Front Matter

      Pages 1-1

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   2. The Topological Point of View

      • Robert F. Brown

      Pages 3-7

   3. Ascoli-Arzela Theory

      • Robert F. Brown

      Pages 8-17

   4. Brouwer Fixed Point Theory

      • Robert F. Brown

      Pages 18-21

   5. Schauder Fixed Point Theory

      • Robert F. Brown

      Pages 22-28

   6. Equilibrium Heat Distribution

      • Robert F. Brown

      Pages 29-33

   7. Generalized Bernstein Theory

      • Robert F. Brown

      Pages 34-40

3. Degree and Bifurcation

   1. Front Matter

      Pages 41-41

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   2. Some Topological Background

      • Robert F. Brown

      Pages 43-47

   3. Brouwer Degree

      • Robert F. Brown

      Pages 48-56

   4. Leray-Schauder Degree

      • Robert F. Brown

      Pages 57-62

   5. Properties of the Leray-Schauder Degree

      • Robert F. Brown

      Pages 63-71

   6. A Separation Theorem

      • Robert F. Brown

      Pages 72-74

   7. Compact Linear Operators

      • Robert F. Brown

      Pages 75-86

   8. The Degree Calculation

      • Robert F. Brown

      Pages 87-94

   9. The Krasnoselskii-Rabinowitz Bifurcation Theorem

      • Robert F. Brown

      Pages 95-107

   10. Nonlinear Sturm-Liouville Theory

       • Robert F. Brown

       Pages 108-117

   11. Euler Buckling

       • Robert F. Brown

       Pages 118-131

4. Back Matter

   Pages 133-146

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Nonlinear analysis is a remarkable mixture of topology, analysis and applied mathematics. Mathematicians have good reason to become acquainted with this important, rapidly developing subject. But it is a BIG subject. You can feel it: just hold Eberhard Zeidler's Nonlinear Functional Analysis and Its Applications I: Fixed Point Theorems [Z} in your hand. It's heavy, as a 900 page book must be. Yet this is no encyclopedia; the preface accurately describes the " ... very careful selection of material ... " it contains. And what you are holding is only Part I of a five-part work. So how do you get started learning nonlinear analysis? Zeidler's book has a first page, and some people are quite comfortable beginning right there. For an alternative, the bibliography in [Z], which is 42 pages long, contains exposition as well as research results: monographs that explain portions of the subject to a variety of audiences. In particular, [D} covers much of the material of Zeidler's book. What makes this book different? The answer is in three parts: this book is (i) topological (ii) goal-oriented and (iii) a model of its subject.

• Mathematica
• addition
• applied mathematics
• boundary element method
• distribution
• form
• function
• functional analysis
• learning
• minimum
• model
• online
• review
• techniques
• topology
• ordinary differential equations
• partial differential equations

"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

• Department of Mathematics, University of California, Los Angeles, Los Angeles, USA

  Robert F. Brown

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