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)

 
 

 

Weak compactness is equivalent to the fixed point property in $c_0$


Authors: P. N. Dowling, C. J. Lennard and B. Turett
Translated by:
Journal: Proc. Amer. Math. Soc. 132 (2004), 1659-1666
MSC (2000): Primary 47H10, 47H09, 46E30
DOI: https://doi.org/10.1090/S0002-9939-04-07436-2
Published electronically: January 29, 2004
MathSciNet review: 2051126
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: A nonempty, closed, bounded, convex subset of $c_0$ has the fixed point property if and only if it is weakly compact.


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

  • 1. D. Alspach, A fixed point free nonexpansive map, Proc. Amer. Math. Soc. 82(3) (1981), 423-424. MR 82j:47070
  • 2. Jon M. Borwein and Brailey Sims, Nonexpansive mappings on Banach lattices and related topics, Houston J. Math. 10(3) (1984), 339-356. MR 86e:47059
  • 3. T. Domínguez Benavides, M. A. Japón Pineda and S. Prus, Weak compactness and fixed point property for affine mappings, to appear, J. Functional Analysis.
  • 4. P. N. Dowling, C. J. Lennard, and B. Turett, Characterizations of weakly compact sets and new fixed point free maps in $c_0$, Studia Math. 154 (2003), no. 3, 277-293. MR 2003m:46016
  • 5. K. Goebel and T. Kuczumow, Irregular convex sets with fixed-point property for non-expansive mappings, Colloq. Math. 40 (1979), 259-264. MR 80j:47068
  • 6. R. Haydon, E. Odell, and Y. Sternfeld, A fixed point theorem for a class of star-shaped sets in ${c}\sb{0}$, Israel J. Math. 38(1-2) (1981), 75-81. MR 82c:47070
  • 7. Maria A. Japón Pineda, The fixed-point property in Banach spaces containing a copy of $c_0$, Abstract and Applied Analysis 2003 (2003), 183-192, and in ``Proceedings of the International Conference on Fixed-Point Theory and Applications'', Haifa, June 13-19, 2001 (S. Reich, editor), Hindawi Publishing Corporation, 2003, pp. 183-192.
  • 8. Enrique Llorens-Fuster and Brailey Sims, The fixed point property in $c\sb 0$, Canad. Math. Bull. 41(4) (1998), 413-422. MR 99i:47097
  • 9. B. Maurey, Points fixes des contractions de certains faiblement compacts de ${L}\sp{1}$, Seminaire d'Analyse Fonctionelle, 1980-1981, Centre de Mathématiques, École Polytech., Palaiseau, 1981, Exp. No. VIII, 19 pp. MR 83h:47041
  • 10. P. Meyer-Nieberg, Banach Lattices, Springer-Verlag, New York-Berlin-Heidelberg, 1991. MR 93f:46025
  • 11. E. Odell and Y. Sternfeld, A fixed point theorem in $c\sb{0}$, Pacific J. Math. 95(1) (1981), 161-177. MR 83b:47060
  • 12. P. M. Soardi, Existence of fixed points on nonexpansive mappings in certain Banach lattices, Proc. Amer. Math. Soc. 73 (1979), 25-29. MR 80c:47051

Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC (2000): 47H10, 47H09, 46E30

Retrieve articles in all journals with MSC (2000): 47H10, 47H09, 46E30


Additional Information

P. N. Dowling
Affiliation: Department of Mathematics and Statistics, Miami University, Oxford, Ohio 45056
Email: dowlinpn@muohio.edu

C. J. Lennard
Affiliation: Department of Mathematics, University of Pittsburgh, Pittsburgh, Pennsylvania 15260
Email: lennard@pitt.edu

B. Turett
Affiliation: Department of Mathematics and Statistics, Oakland University, Rochester, Michigan 48309
Email: turett@oakland.edu

DOI: https://doi.org/10.1090/S0002-9939-04-07436-2
Received by editor(s): June 11, 2002
Published electronically: January 29, 2004
Additional Notes: The second author thanks Paddy Dowling and the Department of Mathematics and Statistics at Miami University for their hospitality during part of the preparation of this paper. He also acknowledges the financial support of Miami University
Communicated by: Jonathan M. Borwein
Article copyright: © Copyright 2004 American Mathematical Society

American Mathematical Society