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 weak compactness
and countable weak compactness
in fixed point theory


Authors: John Kulesza and Teck-Cheong Lim
Journal: Proc. Amer. Math. Soc. 124 (1996), 3345-3349
MSC (1991): Primary 47H10, 47H09; Secondary 54E50, 54D30
DOI: https://doi.org/10.1090/S0002-9939-96-03390-4
MathSciNet review: 1328357
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: We prove that weak compactness and countable weak compactness in metric spaces are not equivalent. However, if the metric space has normal structure, they are equivalent. It follows that some fixed point theorems proved recently are consequences of a classical theorem of Kirk.


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

  • 1. H. H. Corson and J. Lindenstrauss, On weakly compact subsets of Banach spaces, Proc. Amer. Math. Soc. 17 (1966), 407--412. MR 33:7812
  • 2. G. Godefroy and N. Kalton, The ball topology and its applications, Banach Space Theory
    (B. L. Lin, ed.), Contemp. Math., vol. 85, Amer. Math. Soc., Providence, RI, 1989, pp. 195--237. MR 90c:46022
  • 3. M. A. Khamsi, On metric spaces with uniform normal structure, Proc. Amer. Math. Soc. 106 (1989), 723--726. MR 90c:54028
  • 4. W. A. Kirk, A fixed point theorem for mappings which do not increase distances, Amer. Math. Monthly 72 (1965), 1004--1006. MR 32:6436
  • 5. ------, Nonexpansive mappings in metric and Banach spaces, Estratto Dai, Rend. Sem. Mat. Fis. Milano, 51 (1981), 133--144. MR 84j:47091
  • 6. ------, An abstract fixed point theorem for nonexpansive mappings, Proc. Amer. Math. Soc. 82 (1981), 640--642. MR 82f:54075

Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC (1991): 47H10, 47H09, 54E50, 54D30

Retrieve articles in all journals with MSC (1991): 47H10, 47H09, 54E50, 54D30


Additional Information

John Kulesza
Affiliation: Department of Mathematics, George Mason University, 4400 University Drive, Fairfax, Virginia 22030
Email: jkulesza@gmu.edu

Teck-Cheong Lim
Affiliation: Department of Mathematics, George Mason University, 4400 University Drive, Fairfax, Virginia 22030
Email: tlim@gmu.edu

DOI: https://doi.org/10.1090/S0002-9939-96-03390-4
Keywords: Nonexpansive mappings, fixed point, normal structure, weak compact, countable weak compact, uniform normal structure
Received by editor(s): January 3, 1995
Received by editor(s) in revised form: March 27, 1995
Communicated by: Palle E. T. Jorgensen
Article copyright: © Copyright 1996 American Mathematical Society

American Mathematical Society