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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
Published electronically: June 17, 2011
MathSciNet review: 2846334
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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.

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  • 1. F.E. Browder, Fixed point theorems for noncompact mappings in Hilbert spaces, Proc. Nat. Acad. Sci. USA 53 (1965) 1272-1276. MR 0178324 (31:2582)
  • 2. P.N. Dowling; C.J. Lennard; B. Turett, The fixed point for subsets of some classical Banach spaces, Nonlinear Analysis 49 (2002) 141-145. MR 1887917 (2002k:47115)
  • 3. P.N. Dowling; C.J. Lennard; B. Turett, Weak compactness is equivalent to the fixed point property in $ c_0$, Proc. Amer. Math. Soc. 132 (2004) 1659-1666. MR 2051126 (2004m:46024)
  • 4. K. Goebel, W.A. Kirk, Topics in Metric Fixed Point Theory. Cambridge Studies in Advanced Mathematics, 28. Cambridge University Press, Cambridge, 1990. MR 1074005 (92c:47070)
  • 5. W.A. Kirk, Some questions in metric fixed point theory, Recent advances on metric fixed point theory (Seville, 1995), 73-97, Ciencias, 48, Univ. Sevilla, Seville, 1996. MR 1440220 (98f:46010)
  • 6. Handbook of Metric Fixed Point Theory. Edited by William A. Kirk and Brailey Sims. Kluwer Academic Publishers, Dordrecht, 2001. MR 1904271 (2003b:47002)
  • 7. E. Llorens-Fuster; B. Sims, The fixed point property in $ c_0$, Canad. Math. Bull. 41 (1998) 413-422. MR 1658231 (99i:47097)
  • 8. B. Maurey, Points fixes des contractions de certains faiblement compacts de $ L^{1}$, Seminar on Functional Analysis, 1980-1981, Exp. No. VIII, 19 pp., École Polytech., Palaiseau, 1981. MR 659309 (83h:47041)
  • 9. W.O. Ray, Nonexpansive mappings on unbounded convex domains, Bull. Acad. Polon. Sci. Sér. Sci. Math. Astronom. Phys. 26 (1978), no. 3, 241-245. MR 0493551 (58:12545)
  • 10. W.O. Ray, The fixed point property and unbounded sets in Hilbert space, Trans. Amer. Math. Soc. 258 (1980) 531-537. MR 558189 (81e:47044)
  • 11. S. Reich, The almost fixed point property for nonexpansive mappings, Proc. Amer. Math. Soc. 88 (1983), no. 1, 44-46. MR 691276 (84g:47052)
  • 12. I. Shafrir, The approximate fixed point property in Banach and hyperbolic spaces, Israel J. Math. 71 (1990), no. 2, 211-223. MR 1088815 (92b:47096)
  • 13. R. 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 (88g:47122)

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Additional Information

T. Domínguez Benavides
Affiliation: Facultad de Matemáticas, Universidad de Sevilla, P.O. Box 1160, 41080 Sevilla, Spain

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

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