Handbook of Topological Fixed Point Theory

Robert F Brown

ISBN 10: 1402032218 / ISBN 13: 9781402032219

Published by Springer Jun 2005, 2005

Neuware - Fixed point theory concerns itself with a very simple, and basic, mathematical setting. For a functionf that has a setX as bothdomain and range, a xed point off isa pointx ofX for whichf(x)=x. Two fundamental theorems concerning xed points are those of Banach and of Brouwer. In Banach s theorem, X is a complete metric space with metricd andf:X X is required to be a contraction, that is, there must existL 1 such thatd(f(x),f(y)) Ld(x,y) for allx,y X. Theconclusion is thatf has a xed point, in fact exactly one of them. Brouwer stheorem requiresX to betheclosed unit ball in a Euclidean space and f:X X to be a map, that is, a continuous function. Again we can conclude that f has a xed point. But in this case the set of xed points need not be a single point, in fact every closed nonempty subset of the unit ball is the xed point set for some map. ThemetriconX in Banach stheorem is used in the crucialhypothesis about the function, that it is a contraction. The unit ball in Euclidean space is also metric, and the metric topology determines the continuity of the function, but the focus of Brouwer s theorem is on topological characteristics of the unit ball, in particular that it is a contractible nite polyhedron. The theorems of Banach and Brouwer illustrate the di erence between the two principal branches of xed point theory: metric xed point theory and topological xed point theory. Seller Inventory # 9781402032219

Synopsis:

Fixed point theory concerns itself with a very simple, and basic, mathematical setting. For a functionf that has a setX as bothdomain and range, a ?xed point off isa pointx ofX for whichf(x)=x. Two fundamental theorems concerning ?xed points are those of Banach and of Brouwer. In Banach's theorem, X is a complete metric space with metricd andf:X?X is required to be a contraction, that is, there must existL< 1 such thatd(f(x),f(y))?Ld(x,y) for allx,y?X. Theconclusion is thatf has a ?xed point, in fact exactly one of them. Brouwer'stheorem requiresX to betheclosed unit ball in a Euclidean space and f:X?X to be a map, that is, a continuous function. Again we can conclude that f has a ?xed point. But in this case the set of?xed points need not be a single point, in fact every closed nonempty subset of the unit ball is the ?xed point set for some map. ThemetriconX in Banach'stheorem is used in the crucialhypothesis about the function, that it is a contraction. The unit ball in Euclidean space is also metric, and the metric topology determines the continuity of the function, but the focus of Brouwer's theorem is on topological characteristics of the unit ball, in particular that it is a contractible ?nite polyhedron. The theorems of Banach and Brouwer illustrate the di?erence between the two principal branches of ?xed point theory: metric ?xed point theory and topological ?xed point theory.

Synopsis: This book will be especially useful for post-graduate students and researchers interested in the fixed point theory, particularly in topological methods in nonlinear analysis, differential equations and dynamical systems. The content is also likely to stimulate the interest of mathematical economists, population dynamics experts as well as theoretical physicists exploring the topological dynamics.

Bibliographic Details

Title: Handbook of Topological Fixed Point Theory
Publisher: Springer Jun 2005
Publication Date: 2005
Binding: Buch
Condition: Neu