Property (M) and the weak fixed point property

Authors:
Jesús Garcia Falset and Brailey Sims

Journal:
Proc. Amer. Math. Soc. **125** (1997), 2891-2896

MSC (1991):
Primary 47H09, 47H10, 46B20

DOI:
https://doi.org/10.1090/S0002-9939-97-03953-1

MathSciNet review:
1403125

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Abstract | References | Similar Articles | Additional Information

Abstract: It is shown that in Banach spaces with the property (M) of Kalton, nonexpansive self mappings of nonempty weakly compact convex sets necessarily have fixed points. The stability of this conclusion under renormings is examined and conditions for such spaces to have weak normal structure are considered.

**1.**van Dulst, D. and Sims, B., [1983],*Fixed points of nonexpansive mappings and Chebyshev centers in Banach spaces with norms of type (KK)*, Proceedings of the first Romanian-GDR seminar on Banach space theory and its applications - Bucharest 1981, Springer-Verlag Lecture Notes in Mathematics**991**, 35-43. MR**84i:46027****2.**Edgar, G. A. and Wheeler, R. F., [1984],*Topological properties of Banach spaces*, Pacific J. Math.**115**, 317-350. MR**86e:46013****3.**Goebel, K. and Kirk, W. A., [1990], Topics in metric fixed point theory,*Cambridge Studies in advanced mathematics***28**, Cambridge University Press, pp244. MR**92c:47070****4.**Jiménez-Melado, A. and Lloréns-Fuster, E., [1992],*Stability of the fixed point property for nonexpansive mappings*, Houston J. Math.**18**, 251-257. MR**93d:47097****5.**Kalton, N. J., [1993],*M-ideals of compact operators*, Illinois J. of Math.**37**, 147-169. MR**94b:46028****6.**Kalton, N. J. and Werner D., [1993],*Property (M), M-ideals and almost isometric structure of Banach spaces*, preprint, pp47.**7.**Lima, Å., [1982],*On M-ideals and best approximation*, Indiana Univ. Math. J.**31**, 27-36. MR**83b:46021****8.**Lin, Pei-Kee, [1985],*Unconditional bases and fixed points of nonexpansive mappings*, Pacific J. Math.**116**, 69-76. MR**86c:47075****9.**Maurey, B., [1980],*Points fixes des contractions de certains faiblement compacts de*, Seminaire d'Analyse Fonctionnelle, Exposé No.**VIII**, pp18 MR**83h:47041****10.**Sims, B., [1982],*Fixed points of nonexpansive maps on weak and weak compact convex sets*, Queen's University seminar notes, pp34.**11.**Sims, B., [1988],*Orthogonality and fixed points of nonexpansive maps*, Proc. Centre for Math. Anal.,**20**, Australian National University, 178-186. MR**90i:46045****12.**Sims, B., [1992],*The weak Karlovitz lemma for dual lattices*, Bull. Austral. Math. Soc.**45**, 171-176. MR**92k:47105**

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Additional Information

**Jesús Garcia Falset**

Affiliation:
Departament d’Anàlisi Matematica, Facultat de Matematiques, Universitat de València, Doctor Moliner 50, 46100 Burjassot, Spain

Email:
Jesus.Garcia@uv.es

**Brailey Sims**

Affiliation:
Department of Mathematics, The University of Newcastle, New South Wales 2308, Australia

Email:
bsims@frey.newcastle.edu.au

DOI:
https://doi.org/10.1090/S0002-9939-97-03953-1

Received by editor(s):
January 3, 1996

Received by editor(s) in revised form:
April 19, 1996

Additional Notes:
Partially supported by grant DGICYT PB-1177-c02-02 and a travel grant from the University of Newcastle.

Communicated by:
Palle E. T. Jorgensen

Article copyright:
© Copyright 1997
American Mathematical Society