**Mathematical World**

Volume: 2;
1991;
77 pp;
Softcover

MSC: Primary 01; 54;

Print ISBN: 978-0-8218-9000-4

Product Code: MAWRLD/2

List Price: $20.00

Individual Member Price: $16.00

**Electronic ISBN: 978-1-4704-2470-1
Product Code: MAWRLD/2.E**

List Price: $20.00

Individual Member Price: $16.00

# Fixed Points

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*Yu. A. Shashkin*

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.

#### Table of Contents

# Table of Contents

## Fixed Points

- Cover Cover11 free
- Title iii4 free
- Copyright iv5 free
- Table of Contents v6 free
- Preface vii8 free
- Chapter 1 Continuous Mappings of a Closed Interval and a Square 110 free
- Chapter 2 First Combinatorial Lemma 514
- Chapter 3 Second Combinatorial Lemma, or Walks through the Rooms in a House 716
- Chapter 4 Sperner's Lemma 918
- Chapter 5 Continuous Mappings, Homeomorphisms, and the Fixed Point Property 1524
- Chapter 6 Compactness 2130
- Chapter 7 Proof of Brouwer's Theorem for a Closed Interval, the Intermediate Value Theorem, and Applications 2534
- Chapter 8 Proof of Brouwer's Theorem for a Square 3342
- Chapter 9 The Iteration Method 3948
- Chapter 10 Retraction 4352
- Chapter 11 Continuous Mappings of a Circle, Homotopy, and Degree of a Mapping 4756
- Chapter 12 Second Definition of the Degree of a Mapping 5362
- Chapter 13 Continuous Mappings of a Sphere 5564
- Chapter 14 Theorem on Equality of Degrees 6170
- Solutions and Answers 6574
- References 7786
- Back Cover Back Cover187

#### 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