On weak compactness and countable weak compactness in fixed point theory

Kulesza, John; Lim, Teck-Cheong

doi:10.1090/S0002-9939-96-03390-4

On weak compactness and countable weak compactness in fixed point theory
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by John Kulesza and Teck-Cheong Lim PDF

Proc. Amer. Math. Soc. 124 (1996), 3345-3349 Request permission

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

H. H. Corson and J. Lindenstrauss, On weakly compact subsets of Banach spaces, Proc. Amer. Math. Soc. 17 (1966), 407–412. MR 199669, DOI 10.1090/S0002-9939-1966-0199669-9
G. Godefroy and N. J. Kalton, The ball topology and its applications, Banach space theory (Iowa City, IA, 1987) Contemp. Math., vol. 85, Amer. Math. Soc., Providence, RI, 1989, pp. 195–237. MR 983386, DOI 10.1090/conm/085/983386
M. A. Khamsi, On metric spaces with uniform normal structure, Proc. Amer. Math. Soc. 106 (1989), no. 3, 723–726. MR 972234, DOI 10.1090/S0002-9939-1989-0972234-4
W. A. Kirk, A fixed point theorem for mappings which do not increase distances, Amer. Math. Monthly 72 (1965), 1004–1006. MR 189009, DOI 10.2307/2313345
W. A. Kirk, Nonexpansive mappings in metric and Banach spaces, Rend. Sem. Mat. Fis. Milano 51 (1981), 133–144 (1983) (English, with Italian summary). MR 708039, DOI 10.1007/BF02924816
W. A. Kirk, An abstract fixed point theorem for nonexpansive mappings, Proc. Amer. Math. Soc. 82 (1981), no. 4, 640–642. MR 614894, DOI 10.1090/S0002-9939-1981-0614894-6