A Fixed-Point Farrago

1. Front Matter

   Pages i-xiv

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2. Introduction to Fixed Points

   1. Front Matter

      Pages 1-2

      PDF

   2. From Newton to Google

      • Joel H. Shapiro

      Pages 3-17

   3. Brouwer in Dimension Two

      • Joel H. Shapiro

      Pages 19-26

   4. Contraction Mappings

      • Joel H. Shapiro

      Pages 27-37

3. From Brouwer to Nash

   1. Front Matter

      Pages 39-40

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   2. Brouwer in Higher Dimensions

      • Joel H. Shapiro

      Pages 41-50

   3. Nash Equilibrium

      • Joel H. Shapiro

      Pages 51-64

   4. Nash’s “One-Page Proof”

      • Joel H. Shapiro

      Pages 65-71

4. Beyond Brouwer: Dimension = ∞

   1. Front Matter

      Pages 73-74

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   2. The Schauder Fixed-Point Theorem

      • Joel H. Shapiro

      Pages 75-81

   3. The Invariant Subspace Problem

      • Joel H. Shapiro

      Pages 83-97

5. Fixed Points for Families of Maps

   1. Front Matter

      Pages 99-100

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   2. The Markov–Kakutani Theorem

      • Joel H. Shapiro

      Pages 101-119

   3. The Meaning of Means

      • Joel H. Shapiro

      Pages 121-129

   4. Paradoxical Decompositions

      • Joel H. Shapiro

      Pages 131-144

   5. Fixed Points for Non-commuting Map Families

      • Joel H. Shapiro

      Pages 145-162

   6. Beyond Markov–Kakutani

      • Joel H. Shapiro

      Pages 163-180

6. Back Matter

   Pages 181-221

   PDF

This text provides an introduction to some of the best-known fixed-point theorems, with an emphasis on their interactions with topics in analysis.  The level of exposition increases gradually throughout the book, building from a basic requirement of undergraduate proficiency to graduate-level sophistication. Appendices provide an introduction to (or refresher on) some of the prerequisite material and exercises are integrated into the text, contributing to the volume’s ability to be used as a self-contained text. Readers will find the presentation especially useful for independent study or as a supplement to a graduate course in fixed-point theory.

The material is split into four parts: the first introduces the Banach Contraction-Mapping Principle and the Brouwer Fixed-Point Theorem, along with a selection of interesting applications; the second focuses on Brouwer’s theorem and its application to John Nash’s work; the third applies Brouwer’s theorem to spaces of infinite dimension; and the fourth rests on the work of Markov, Kakutani, and Ryll–Nardzewski surrounding fixed points for families of affine maps.

• Analysis
• Banach Spaces
• Fixed-Point Theory
• Hilbert Spaces
• Set-Value Analysis

“This is a well-written and essentially self-contained survey on several applications of fixed-point theory. … The book makes easy reading because the proofs are remarkably well structured. They are broken up into easily digestible parts and the claims are always clearly stated. There are five appendices which help to make the book almost self-contained.” (Christian Fenske, zbMATH 1352.47002, 2017)

“The chapters provide models for students to appreciate the art of a well-thought out seminar talk … on a particular topic or theme. There are also a variety of exercises integrated into the lectures, ‘to encourage active participation’, which allow the text to be used for a course or an independent study. … The book appears to be written to engage a young audience, yet there is more in it to provide pleasure to mathematicians further on in their career.” (Tushar Das, Maa Reviews, maa.org, October, 2016)

• Portland State University, Portland, USA

  Joel H. Shapiro

Joel H. Shapiro is an adjunct professor of Mathematics and Statistics at Portland State University. He received his PhD from the University of Michigan. 

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