Hostname: page-component-77c89778f8-cnmwb Total loading time: 0 Render date: 2024-07-22T05:34:02.390Z Has data issue: false hasContentIssue false

Moons, bridges, birds … and nonexpansive mappings in Hilbert space

Published online by Cambridge University Press:  17 April 2009

Kasimierz Goebel
Affiliation:
Instytut Matematyki, Uniwersytet Marii Curie − Sklodowskiej, Lublin, Poland;
Rainald Schöneberg
Affiliation:
Lehrstuhl C für Mathematik, Rheinisch-Westfälische Technische, Hochschule, Aachen, Germany.
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

In recent years some fixed point theorems have been proved for nonexpansive mappings in Hilbert space, which have non-convex domains. The purpose of this paper is to present a simple but very useful new result of that kind and to indicate some of its consequences.

Type
Research Article
Copyright
Copyright © Australian Mathematical Society 1977

References

[1]Baillon, Jean-Bernard, “Un théorème de type ergodique pour les contractions non linéaires dans un espace de Hilbert”, C.R. Acad. Sci. Paris Sér. A 280 (1975), 15111514.Google Scholar
[2]Browder, Felix E., “Nonexpansive nonlinear operators in a Banach space”, Proc. Nat. Acad. Sol. U.S.A. 54 (1965), 10411044.CrossRefGoogle Scholar
[3]Göhde, Dietrich, “Zum Prinzip der kontraktiven Abbildung”, Math. Nachr. 30 (1965), 251258.CrossRefGoogle Scholar
[4]Göhde, Dietrich, “Elementare Bemerkungen zu nichtexpansiven Selbst-abbildungen nicht konvexer Mengen im Hilbertraum”, Math. Nachr. 63 (1974), 331335.CrossRefGoogle Scholar
[5]Kirk, W.A., “A fixed point theorem for mappings which do not increase distances”, Amer. Math. Monthly 72 (1965), 10041006.Google Scholar
[6]Rèinermann, J. and Schöneberg, R., “Some results in fixed point theory for nonexpansive and pseudocontractive mappings in Hilbert-space”, Fixed point theory and its applications, 187196 (Proc. Seminar Fixed Point Theory and its Applications, Dalhousie University, Halifax, 1975. Academic Press [Harcourt Brace Jovanovich], New York, San Francisco, London, 1976).Google Scholar
[7]Schoenberg, I.J., “On a theorem of Kirzbraun and Valentine”, Amer. Math. Monthly 60 (1953), 620622.CrossRefGoogle Scholar
[8]Staples, John, “Fixed point theorems in uniformly rotund metric spaces”, Bull. Austral. Math. Soc. 14 (1976), 181192.CrossRefGoogle Scholar