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Softcover ISBN:  9780821890004 
Product Code:  MAWRLD/2 
List Price:  $45.00 
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Product Code:  MAWRLD/2.E 
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AMS Member Price:  $31.20 
Softcover ISBN:  9780821890004 
eBook ISBN:  9781470424701 
Product Code:  MAWRLD/2.B 
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Book DetailsMathematical WorldVolume: 2; 1991; 77 ppMSC: Primary 01; 54
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

Chapters

Chapter 1. Continuous Mappings of a Closed Interval and a Square

Chapter 2. First Combinatorial Lemma

Chapter 3. Second Combinatorial Lemma, or Walks through the Rooms in a House

Chapter 4. Sperner’s Lemma

Chapter 5. Continuous Mappings, Homeomorphisms and the Fixed Point Property

Chapter 6. Compactness

Chapter 7. Proof of Brouwer’s Theorem for a Closed Interval, the Intermediate Value Theorem, and Applications

Chapter 8. Proof of Brouwer’s Theorem for a Square

Chapter 9. The Iteration Method

Chapter 10. Retraction

Chapter 11. Continuous Mappings of a Circle, Homotopy, and Degree of a Mapping

Chapter 12. Second Definition of the Degree of a Mapping

Chapter 13. Continuous Mappings of a Sphere

Chapter 14. Theorem on Equality of Degrees

Solutions and Answers


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


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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.

Chapters

Chapter 1. Continuous Mappings of a Closed Interval and a Square

Chapter 2. First Combinatorial Lemma

Chapter 3. Second Combinatorial Lemma, or Walks through the Rooms in a House

Chapter 4. Sperner’s Lemma

Chapter 5. Continuous Mappings, Homeomorphisms and the Fixed Point Property

Chapter 6. Compactness

Chapter 7. Proof of Brouwer’s Theorem for a Closed Interval, the Intermediate Value Theorem, and Applications

Chapter 8. Proof of Brouwer’s Theorem for a Square

Chapter 9. The Iteration Method

Chapter 10. Retraction

Chapter 11. Continuous Mappings of a Circle, Homotopy, and Degree of a Mapping

Chapter 12. Second Definition of the Degree of a Mapping

Chapter 13. Continuous Mappings of a Sphere

Chapter 14. Theorem on Equality of Degrees

Solutions and Answers

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