Remote Access Proceedings of the American Mathematical Society
Green Open Access

Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)



On asymptotically nonexpansive mappings in hyperconvex metric spaces

Author: M. A. Khamsi
Journal: Proc. Amer. Math. Soc. 132 (2004), 365-373
MSC (2000): Primary 47H09, 47H10
Published electronically: August 28, 2003
MathSciNet review: 2022357
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: Since bounded hyperconvex metric spaces have the fixed point property for nonexpansive mappings, it is natural to extend such a powerful result to asymptotically nonexpansive mappings. Our main result states that the approximate fixed point property holds in this case. The proof is based on the use, for the first time, of the ultrapower of a metric space.

References [Enhancements On Off] (What's this?)

  • [AK] A. G. Aksoy and M. A. Khamsi, Nonstandard Methods in Fixed Point Theory, Springer-Verlag, New York, Berlin (1990). MR 91i:47073
  • [AP] N. Aronszajn and P. Panitchpakdi, Extension of uniformly continuous transformations and hyperconvex metric spaces, Pacific J. Math. 6 (1956), 405-439. MR 18:917c
  • [ADL] J. M. Ayerbe Toledano, T. Dominguez Benavides, and G. Lopez Acedo, Measures of Noncompactness in Metric Fixed Point Theory, Operator Theory: Advances and Applications 99, Birkhäuser-Verlag, Basel, 1997. MR 99e:47070
  • [Ba] J. B. Baillon, Nonexpansive mapping and hyperconvex spaces, Contemp. Math. 72 (1988), 11-19. MR 89k:54068
  • [EK] R. Espinola and M. A. Khamsi, Introduction to hyperconvex spaces, Handbook of Metric Fixed Point Theory, W. A. Kirk and B. Sims, Editors, Kluwer Academic Publishers, Dordrecht, 2001. MR 2003g:47099
  • [GK] K. Goebel and W. A. Kirk, Topics in Metric Fixed Point Theory, Cambridge Univ. Press, Cambridge (1990). MR 92c:47070
  • [Is] J. R. Isbell, Six theorems about injective metric spaces, Comment. Math. Helvetici 39 (1964), 65-76. MR 32:431
  • [JMP] E. Jawhari, D. Misane and M. Pouzet, Retracts: graphs and ordered sets from the metric point of view, Contemp. Math. 57 (1986), 175-226. MR 88i:54022
  • [KK] M. A. Khamsi, and W. A. Kirk, An Introduction to Metric Spaces and Fixed Point Theory, Pure and Applied Mathematics, Wiley Interscience, New York (2001). MR 2002b:46002
  • [KX] T. H. Kim and H. K. Xu, Remarks on asymptotically nonexpansive mappings, Nonlinear Analysis 41 (2000), 405-415. MR 2001b:47089
  • [Ki1] W. A. Kirk, A fixed point theorem for mappings which do not increase distances, Amer. Math. Monthly 72 (1965), 1004-1006. MR 32:6436
  • [Ki2] W. A. Kirk, Personal Communication.
  • [La] H. E. Lacey, The Isometric Theory of Classical Banach Spaces, Die Grundlehren der mathematischen Wissenschaften, Band 208, Springer-Verlag, Berlin, Heidelberg, New York (1974). MR 58:12308
  • [Pe] J. P. Penot, Fixed point theorems without convexity, Bull. Soc. Math. France Mémoire 60 (1979), 129-152. MR 81c:47061
  • [Sm] B. Sims, ``Ultra''-techniques in Banach Space Theory, Queen's Papers in Pure and Appl. Math., 60, Queen's University, Kingston, Ontario (1982). MR 86h:46032
  • [Sn1] R. C. Sine, On linear contraction semigroups in sup norm spaces, Nonlinear Anal. 3 (1979), 885-890. MR 80i:47082
  • [Sn2] R. C. Sine, Hyperconvexity and approximate fixed points, Nonlinear Anal. 13 (1989), 863-869. MR 90g:54041
  • [So] P. Soardi, Existence of fixed points of nonexpansive mappings in certain Banach lattices, Proc. Amer. Math. Soc. 73 (1979), 25-29. MR 80c:47051

Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC (2000): 47H09, 47H10

Retrieve articles in all journals with MSC (2000): 47H09, 47H10

Additional Information

M. A. Khamsi
Affiliation: Department of Mathematical Sciences, The University of Texas at El Paso, El Paso, Texas 79968-0514

Keywords: Nonexpansive mappings, asymptotically nonexpansive mappings, fixed point, hyperconvex
Received by editor(s): March 12, 2002
Published electronically: August 28, 2003
Communicated by: Jonathan M. Borwein
Article copyright: © Copyright 2003 American Mathematical Society

American Mathematical Society