The failure of the fixed point property for unbounded sets in 𝑐₀

Benavides, T.

doi:10.1090/S0002-9939-2011-10938-9

The failure of the fixed point property for unbounded sets in $c_0$
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by T. Domínguez Benavides PDF

Proc. Amer. Math. Soc. 140 (2012), 645-650 Request permission

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.

References

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 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 10.1016/S0362-546X(01)00104-3
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 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, DOI 10.1017/CBO9780511526152
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, DOI 10.1007/978-94-017-1748-9
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 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 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 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 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 10.1090/S0002-9939-1987-0891152-1