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

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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.

**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 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