The failure of the fixed point property for unbounded sets in $c_0$
Author:
T. Domínguez Benavides
Journal:
Proc. Amer. Math. Soc. 140 (2012), 645-650
MSC (2010):
Primary 47H09, 47H10; Secondary 46B20
DOI:
https://doi.org/10.1090/S0002-9939-2011-10938-9
Published electronically:
June 17, 2011
MathSciNet review:
2846334
Full-text PDF Free Access
Abstract | References | Similar Articles | Additional Information
Abstract: In this paper we prove that for every unbounded convex closed set $C$ in $c_0$ there exists a nonexpansive mapping $T:C\to C$ which is fixed point free. This result solves in a negative sense a question that has remained open for some time in Metric Fixed Point Theory.
- Felix E. Browder, Fixed-point theorems for noncompact mappings in Hilbert space, Proc. Nat. Acad. Sci. U.S.A. 53 (1965), 1272–1276. MR 178324, DOI https://doi.org/10.1073/pnas.53.6.1272
- P. N. Dowling, C. J. Lennard, and B. Turett, The fixed point property for subsets of some classical Banach spaces, Nonlinear Anal. 49 (2002), no. 1, Ser. A: Theory Methods, 141–145. MR 1887917, DOI https://doi.org/10.1016/S0362-546X%2801%2900104-3
- P. N. Dowling, C. J. Lennard, and B. Turett, Weak compactness is equivalent to the fixed point property in $c_0$, Proc. Amer. Math. Soc. 132 (2004), no. 6, 1659–1666. MR 2051126, DOI https://doi.org/10.1090/S0002-9939-04-07436-2
- 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
- W. A. Kirk, Some questions in metric fixed point theory, Recent advances on metric fixed point theory (Seville, 1995) Ciencias, vol. 48, Univ. Sevilla, Seville, 1996, pp. 73–97. MR 1440220
- William A. Kirk and Brailey Sims (eds.), Handbook of metric fixed point theory, Kluwer Academic Publishers, Dordrecht, 2001. MR 1904271
- Enrique Llorens-Fuster and Brailey Sims, The fixed point property in $c_0$, Canad. Math. Bull. 41 (1998), no. 4, 413–422. MR 1658231, DOI https://doi.org/10.4153/CMB-1998-055-2
- 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
- William O. Ray, Nonexpansive mappings on unbounded convex domains, Bull. Acad. Polon. Sci. Sér. Sci. Math. Astronom. Phys. 26 (1978), no. 3, 241–245 (English, with Russian summary). MR 493551
- William O. Ray, The fixed point property and unbounded sets in Hilbert space, Trans. Amer. Math. Soc. 258 (1980), no. 2, 531–537. MR 558189, DOI https://doi.org/10.1090/S0002-9947-1980-0558189-1
- Simeon Reich, The almost fixed point property for nonexpansive mappings, Proc. Amer. Math. Soc. 88 (1983), no. 1, 44–46. MR 691276, DOI https://doi.org/10.1090/S0002-9939-1983-0691276-4
- Itai Shafrir, The approximate fixed point property in Banach and hyperbolic spaces, Israel J. Math. 71 (1990), no. 2, 211–223. MR 1088815, DOI https://doi.org/10.1007/BF02811885
- Robert Sine, On the converse of the nonexpansive map fixed point theorem for Hilbert space, Proc. Amer. Math. Soc. 100 (1987), no. 3, 489–490. MR 891152, DOI https://doi.org/10.1090/S0002-9939-1987-0891152-1
Retrieve articles in Proceedings of the American Mathematical Society with MSC (2010): 47H09, 47H10, 46B20
Retrieve articles in all journals with MSC (2010): 47H09, 47H10, 46B20
Additional Information
T. Domínguez Benavides
Affiliation:
Facultad de Matemáticas, Universidad de Sevilla, P.O. Box 1160, 41080 Sevilla, Spain
Email:
tomasd@us.es
Received by editor(s):
November 9, 2010
Received by editor(s) in revised form:
December 3, 2010
Published electronically:
June 17, 2011
Additional Notes:
The author was partially supported by MCIN, Grant MTM 2009-10696-C02-01, and Andalusian Regional Government Grant FQM-127
Communicated by:
Thomas Schlumprecht
Article copyright:
© Copyright 2011
American Mathematical Society