Fixed PointsThe theory of fixed points finds its roots in the work of Poincare, 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. |
Contents
Continuous Mappings of a Closed Interval and a Square | 1 |
First Combinatorial Lemma | 5 |
Second Combinatorial Lemma or Walks through the Rooms in a House | 7 |
Sperners Lemma | 9 |
Continuous Mappings Homeomorphisms and the Fixed Point Property | 15 |
Compactness | 21 |
Proof of Brouwers Theorem for a Closed Interval the Intermediate Value Theorem and Applications | 25 |
Proof of Brouwers Theorem for a Square | 33 |
The Iteration Method | 39 |
Retraction | 43 |
Continuous Mappings of a Circle Homotopy and Degree of a Mapping | 47 |
Second Definition of the Degree of a Mapping | 53 |
Continuous Mappings of a Sphere | 55 |
Theorem on Equality of Degrees | 61 |
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Common terms and phrases
A₁ A₂ angle angular coordinate antipodal points array assume b₁ Borsuk-Ulam theorem boundary circle Chapter common point compact Consider continuous function continuous mapping contraction mapping converges corresponding curve dead end defined deg f denote displacement vector door edges of type equality of degrees example f₁ Figure fixed point property fixed point theorem function f(x graph Hence Hex game homeomorphic homotopic implies inequalities inside integer intermediate value theorem intersection inverse images iteration ith row king's fields Let f Let ƒ lying mapping f mapping ƒ mapping g moves neighborhood number of edges number of inverse orientation P₁ P₂ piecewise linear mapping plane point f(x point x point xo polygon problem proof real line retraction rook rook's fields satisfies side Sperner's lemma sphere square Q subdivision Suppose T₂ three different numbers total number triangular faces triangulation vector vertex vertices x₁