A fixed point free nonexpansive map
HTML articles powered by AMS MathViewer
- by Dale E. Alspach PDF
- Proc. Amer. Math. Soc. 82 (1981), 423-424 Request permission
Abstract:
In this note we give an example of a weakly compact convex subset of ${L_1}[0,1]$ that fails to have the fixed point property for nonexpansive maps. This answers a long-standing question which was recently raised again by S. Reich [7].References
- L. P. Belluce and W. A. Kirk, Nonexpansive mappings and fixed-points in Banach spaces, Illinois J. Math. 11 (1967), 474–479. MR 215145
- Felix E. Browder, Nonexpansive nonlinear operators in a Banach space, Proc. Nat. Acad. Sci. U.S.A. 54 (1965), 1041–1044. MR 187120, DOI 10.1073/pnas.54.4.1041 N. Dunford and J. T. Schwartz, Linear operators: General theory, Pure and Appl. Math., vol. 7, Interscience, New York, 1958. E. Hopf, Ergodentheorie, Ergebnisse der Math., Vol. 5, Springer-Verlag, Berlin, 1937.
- L. A. Karlovitz, On nonexpansive mappings, Proc. Amer. Math. Soc. 55 (1976), no. 2, 321–325. MR 405182, DOI 10.1090/S0002-9939-1976-0405182-X E. Odell and Y. Sternfeld, A fixed point theorem in ${c_0}$, preprint.
- Simeon Reich, Research Problems: The Fixed Point Property for Non-Expansive Mappings, II, Amer. Math. Monthly 87 (1980), no. 4, 292–294. MR 1539350, DOI 10.2307/2321568
Additional Information
- © Copyright 1981 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 82 (1981), 423-424
- MSC: Primary 47H10; Secondary 54H25
- DOI: https://doi.org/10.1090/S0002-9939-1981-0612733-0
- MathSciNet review: 612733