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)



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
MathSciNet review: 1403125
Full-text PDF Free Access

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.

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

  • D. van Dulst and Brailey Sims, Fixed points of nonexpansive mappings and Chebyshev centers in Banach spaces with norms of type (KK), Banach space theory and its applications (Bucharest, 1981) Lecture Notes in Math., vol. 991, Springer, Berlin-New York, 1983, pp. 35–43. MR 714171
  • G. A. Edgar and R. F. Wheeler, Topological properties of Banach spaces, Pacific J. Math. 115 (1984), no. 2, 317–350. MR 765190
  • Kazimierz Goebel and W. A. Kirk, Topics in metric fixed point theory, Cambridge Studies in Advanced Mathematics, vol. 28, Cambridge University Press, Cambridge, 1990. MR 1074005
  • A. Jiménez-Melado and E. Llorens Fuster, Stability of the fixed point property for nonexpansive mappings, Houston J. Math. 18 (1992), no. 2, 251–257. MR 1164107
  • N. J. Kalton, $M$-ideals of compact operators, Illinois J. Math. 37 (1993), no. 1, 147–169. MR 1193134
  • Kalton, N. J. and Werner D., [1993], Property (M), M-ideals and almost isometric structure of Banach spaces, preprint, pp47.
  • Åsvald Lima, On $M$-ideals and best approximation, Indiana Univ. Math. J. 31 (1982), no. 1, 27–36. MR 642613, DOI
  • Pei-Kee Lin, Unconditional bases and fixed points of nonexpansive mappings, Pacific J. Math. 116 (1985), no. 1, 69–76. MR 769823
  • B. Maurey, Points fixes des contractions de certains faiblement compacts de $L^{1}$, Seminar on Functional Analysis, 1980–1981, École Polytech., Palaiseau, 1981, pp. Exp. No. VIII, 19 (French). MR 659309
  • Sims, B., [1982], Fixed points of nonexpansive maps on weak and weak$^{*}$ compact convex sets, Queen’s University seminar notes, pp34.
  • Brailey Sims, Orthogonality and fixed points of nonexpansive maps, Workshop/Miniconference on Functional Analysis and Optimization (Canberra, 1988) Proc. Centre Math. Anal. Austral. Nat. Univ., vol. 20, Austral. Nat. Univ., Canberra, 1988, pp. 178–186. MR 1009604
  • Brailey Sims, The weak${}^\ast $ Karlovitz lemma for dual lattices, Bull. Austral. Math. Soc. 45 (1992), no. 1, 171–176. MR 1147256, DOI

Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC (1991): 47H09, 47H10, 46B20

Retrieve articles in all journals with MSC (1991): 47H09, 47H10, 46B20

Additional Information

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

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

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