Handbook of Topological Fixed Point Theory - Hardcover

Hardcover

ISBN 10: 1402032218 ISBN 13: 9781402032219

Verlag: Springer, 2005

Inhaltsangabe

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.

Details zur Ausgabe

• Verlag: Springer
• Erscheinungsdatum: 2005
• Sprache: Englisch
• ISBN 10: 1402032218
• ISBN 13: 9781402032219
• Einband: Gebundene Ausgabe
• Anzahl der Seiten: 982
• Herausgeber: Brown Robert F., Furi Massimo, Gorniewicz L., Jiang Boju
• Kontakt zum Hersteller: Springer Nature Customer Service Center GmbH; ProductSafety@springernature.com; ; Tiergartenstr. 17; Heidelberg; 69121; Deutschland