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On weak compactness and countable weak compactness in fixed point theory
Author(s):
John
Kulesza;
Teck-Cheong
Lim
Journal:
Proc. Amer. Math. Soc.
124
(1996),
3345-3349.
MSC (1991):
Primary 47H10, 47H09;
Secondary 54E50, 54D30
MathSciNet review:
1328357
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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:
- 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
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Additional Information:
John
Kulesza
Affiliation:
Department of Mathematics, George Mason University, 4400 University Drive, Fairfax, Virginia 2203
Email:
jkulesza@gmu.edu
Teck-Cheong
Lim
Affiliation:
Department of Mathematics, George Mason University, 4400 University Drive, Fairfax, Virginia 2203
Email:
tlim@gmu.edu
DOI:
10.1090/S0002-9939-96-03390-4
PII:
S 0002-9939(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
Copyright of article:
Copyright
1996,
American Mathematical Society
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