Mathematical World 1991; 77 pp; softcover Volume: 2 Reprint/Revision History: reprinted with corrections 1995; third printing 1997 ISBN10: 082189000X ISBN13: 9780821890004 List Price: US$20 Member Price: US$16 Order Code: MAWRLD/2
 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 conceptssuch as continuity, compactness, degree of a map, and so onare 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 fixedpoint 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
