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)

 
 

 

An abstract fixed point theorem for nonexpansive mappings


Author: W. A. Kirk
Journal: Proc. Amer. Math. Soc. 82 (1981), 640-642
MSC: Primary 54H25
DOI: https://doi.org/10.1090/S0002-9939-1981-0614894-6
MathSciNet review: 614894
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: A class $ \mathcal{S}$ of subsets of a bounded metric space is said to be normal if each member of $ \mathcal{S}$ contains a nondiametral point. An induction proof is given for the following. Suppose $ M$ is a nonempty bounded metric space which contains a class $ \mathcal{S}$ of subsets which is countably compact, normal, stable under arbitrary intersections, and which contains the closed balls in $ M$. Then every nonexpansive self-mapping of $ M$ has a fixed point.


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

  • [1] B. Fuchssteiner, Iterations and fixpoints, Pacific J. Math. 68 (1977), 73-80. MR 0513055 (58:23795)
  • [2] A. A. Gillespie and B. B. Williams, Fixed point theorem for nonexpansive mappings on Banach spaces with uniformly normal structure, Applicable Anal. 9 (1979), 121-124. MR 539537 (81b:47068)
  • [3] W. A. Kirk, A fixed point theorem for mappings which do not increase distances, Amer. Math. Monthly 72 (1965), 1004-1006. MR 0189009 (32:6436)
  • [4] -, Mappings of generalized contractive type, J. Math. Anal. Appl. 32 (1970), 567-572. MR 0271794 (42:6675)
  • [5] T. C. Lim, A constructive proof of the infinite version of the Belluce-Kirk theorem, Pacific J. Math. 8 (1979), 467-469. MR 547612 (80j:47070)
  • [6] -, Asymptotic centers and nonexpansive mappings in some conjugate spaces, Pacific J. Math. (to appear).
  • [7] J. P. Penot, Fixed point theorems without convexity, Analyse Non Convex (Pau, 1977), Bull. Soc. Math. France Mém. 60 (1979), 129-152. MR 562261 (81c:47061)

Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC: 54H25

Retrieve articles in all journals with MSC: 54H25


Additional Information

DOI: https://doi.org/10.1090/S0002-9939-1981-0614894-6
Keywords: Fixed point theorem, nonexpansive mappings
Article copyright: © Copyright 1981 American Mathematical Society

American Mathematical Society