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Fixed Points
Yu. A. Shashkin, Institute for Mathematics and Mechanics, Ekaterinburg, Russia
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Mathematical World
1991; 77 pp; softcover
Volume: 2
Reprint/Revision History:
reprinted with corrections 1995; third printing 1997
ISBN-10: 0-8218-9000-X
ISBN-13: 978-0-8218-9000-4
List Price: US$19
Member Price: US$15.20
Order Code: MAWRLD/2
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The theory of fixed points finds its roots in the work of Poincaré, Brouwer, and Sperner and makes extensive use of such topological notions as continuity, compactness, homotopy, and the degree of a mapping. Fixed point theorems have numerous applications in mathematics; most of the theorems ensuring the existence of solutions for differential, integral, operator, or other equations can be reduced to fixed point theorems. In addition, these theorems are used in such areas as mathematical economics and game theory.

This book presents a readable exposition of fixed point theory. The author focuses on the problem of whether a closed interval, square, disk, or sphere has the fixed point property. Another aim of the book is to show how fixed point theory uses combinatorial ideas related to decomposition (triangulation) of figures into distinct parts called faces (simplexes), which adjoin each other in a regular fashion. All necessary background concepts--such as continuity, compactness, degree of a map, and so on--are explained, making the book accessible even to students at the high school level. In addition, the book contains exercises and descriptions of applications. Readers will appreciate this book for its lucid presentation of this fundamental mathematical topic.

Reviews

"This pleasant little book tries to stimulate the mathematical appetite of bright senior high school and beginning university students by introducing them to some concepts from topology, with emphasis on Brouwer's fixed point theorem, and it should succeed in this very well ... All of the material hangs together very nicely, and makes enjoyable reading. There are 54 helpful and sometimes amusing exercises with answers ... The English of the translation is fluent."

-- Mathematical Reviews

"Makes available to the young anglophone undergraduate math major a lovely, thoughtful, and thorough exposition of elementary fixed-point theory in Euclidian space, along with some of its simpler topological applications, whose more customary presentations rely on the sort of easy familiarity with homology theory that only the most unusual undergraduate is likely to have ... No detail is omitted, even to the inclusion of solutions to all the pleasant problems rounding out each chapter ... Highly recommended. Undergraduate through faculty."

-- CHOICE

Table of Contents

  • Continuous mappings of a closed interval and a square
  • First combinatorial lemma
  • Second combinatorial lemma, or walks through the rooms in a house
  • Sperner's lemma
  • Continuous mappings, homeomorphisms, and the fixed point property
  • Compactness
  • Proof of Brouwer's Theorem for a closed interval, the intermediate value theorem, and applications
  • Proof of Brouwer's Theorem for a square
  • The iteration method
  • Retraction
  • Continuous mappings of a circle, homotopy, and degree of a mapping
  • Second definition of the degree of a mapping
  • Continuous mappings of a sphere
  • Lemma on equality of degrees
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