A Fixed-Point Farrago

Description

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. Front Matter....Pages i-xiv Front Matter....Pages 1-2 From Newton to Google....Pages 3-17 Brouwer in Dimension Two....Pages 19-26 Contraction Mappings....Pages 27-37 Front Matter....Pages 39-40 Brouwer in Higher Dimensions....Pages 41-50 Nash Equilibrium....Pages 51-64 Nash’s “One-Page Proof”....Pages 65-71 Front Matter....Pages 73-74 The Schauder Fixed-Point Theorem....Pages 75-81 The Invariant Subspace Problem....Pages 83-97 Front Matter....Pages 99-100 The Markov–Kakutani Theorem....Pages 101-119 The Meaning of Means....Pages 121-129 Paradoxical Decompositions....Pages 131-144 Fixed Points for Non-commuting Map Families....Pages 145-162 Beyond Markov–Kakutani....Pages 163-180 Back Matter....Pages 181-221