Abstract
It follows from Banach’s fixed point theorem that every nonexpansive self-mapping of a bounded, closed and convex set in a Banach space has approximate fixed points. This is no longer true, in general, if the set is unbounded. Nevertheless, as we show in the present paper, there exists an open and everywhere dense set in the space of all nonexpansive self-mappings of any closed and convex (not necessarily bounded) set in a Banach space (endowed with the natural metric of uniform convergence on bounded subsets) such that all its elements have approximate fixed points.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Banach S.: Sur les opérations dans les ensembles abstraits et leur application aux équations intégrales. Fund. Math. 3, 133–181 (1922)
de Blasi F.S., Myjak J.: Sur la convergence des approximations successives pour les contractions non linéaires dans un espace de Banach. C. R. Acad. Sci. Paris 283, 185–187 (1976)
de Blasi F. S., Myjak J.: Sur la porosité de l’ensemble des contractions sans point fixe. C. R. Acad. Sci. Paris 308, 51–54 (1989)
Goebel K., Kirk W.A.: Topics in Metric Fixed Point Theory. Cambridge University Press, Cambridge (1990)
Goebel K., Reich S.: Uniform Convexity, Hyperbolic Geometry, and Nonexpansive Mappings. Marcel Dekker, New York (1984)
W. A. Kirk, Contraction mappings and extensions. In: Handbook of Metric Fixed Point Theory, Kluwer, Dordrecht, 2001, 1–34.
Kuczumow T.: A remark on the approximate fixed point property. Abstr. Appl. Anal. 2003, 93–99 (2003)
Matoušková E., Reich S.: Reflexivity and approximate fixed points. Studia Math. 159, 403–415 (2003)
Rakotch E.: A note on contractive mappings. Proc. Amer. Math. Soc. 13, 459–465 (1962)
Reich S.: The almost fixed point property for nonexpansive mappings. Proc. Amer. Math. Soc. 88, 44–46 (1983)
Reich S.: The alternating algorithm of von Neumann in the Hilbert ball. Dynam. Systems Appl. 2, 21–25 (1993)
Reich S., Shafrir I.: Nonexpansive iterations in hyperbolic spaces. Nonlinear Anal. 15, 537–558 (1990)
Reich S., Zaslavski A. J.: Convergence of Krasnoselskii-Mann iterations of nonexpansive operators. Math. Comput. Modelling 32, 1423–1431 (2000)
Shafrir I.: The approximate fixed point property in Banach and hyperbolic spaces. Israel J. Math. 71, 211–223 (1990)
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Reich, S., Zaslavski, A.J. Approximate fixed points of nonexpansive mappings in unbounded sets. J. Fixed Point Theory Appl. 13, 627–632 (2013). https://doi.org/10.1007/s11784-013-0121-8
Published:
Issue Date:
DOI: https://doi.org/10.1007/s11784-013-0121-8