The 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.
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.