The fixed point property and unbounded sets in Hilbert space
Author:
William O. Ray
Journal:
Trans. Amer. Math. Soc. 258 (1980), 531-537
MSC:
Primary 47H09; Secondary 47H10
DOI:
https://doi.org/10.1090/S0002-9947-1980-0558189-1
MathSciNet review:
558189
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Abstract | References | Similar Articles | Additional Information
Abstract: It is shown that a closed convex subset K of a real Hilbert space H has the fixed point property for nonexpansive mappings if and only if K is bounded.
- Felix E. Browder, Nonexpansive nonlinear operators in a Banach space, Proc. Nat. Acad. Sci. U.S.A. 54 (1965), 1041–1044. MR 187120, DOI https://doi.org/10.1073/pnas.54.4.1041
- K. Goebel and T. Kuczumow, A contribution to the theory of nonexpansive mappings, Bull. Calcutta Math. Soc. 70 (1978), no. 6, 355–357. MR 584472
- Dietrich Göhde, Zum Prinzip der kontraktiven Abbildung, Math. Nachr. 30 (1965), 251–258 (German). MR 190718, DOI https://doi.org/10.1002/mana.19650300312
- W. A. Kirk, A fixed point theorem for mappings which do not increase distances, Amer. Math. Monthly 72 (1965), 1004–1006. MR 189009, DOI https://doi.org/10.2307/2313345
- 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
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Additional Information
Keywords:
Fixed points,
nonexpansive operators
Article copyright:
© Copyright 1980
American Mathematical Society
