Handbook of Topological Fixed Point Theory (Hardcover, 1st ed. 2005)

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.

General

Imprint:	Springer-Verlag New York
Country of origin:	United States
Release date:	December 2011
First published:	July 2005
Editors:	Robert F. Brown • Massimo Furi • L. Gorniewicz • Boju Jiang
Dimensions:	235 x 155 x 46mm (L x W x T)
Format:	Hardcover
Pages:	972
Edition:	1st ed. 2005
ISBN-13:	978-1-4020-3221-9
Categories:	Books > Science & Mathematics > Mathematics > Calculus & mathematical analysis > Differential equations Books > Science & Mathematics > Mathematics > Topology > Algebraic topology
LSN:	1-4020-3221-8
Barcode:	9781402032219

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