Remote Access Proceedings of the American Mathematical Society
Green Open Access

Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)



A fixed point free nonexpansive map

Author: Dale E. Alspach
Journal: Proc. Amer. Math. Soc. 82 (1981), 423-424
MSC: Primary 47H10; Secondary 54H25
MathSciNet review: 612733
Full-text PDF

Abstract | References | Similar Articles | Additional Information

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 [Enhancements On Off] (What's this?)

  • [1] L. P. Belluce and W. A. Kirk, Nonexpansive mappings and fixed points in Banach spaces, Illinois J. Math. 11 (1967), 474-479. MR 0215145 (35:5988)
  • [2] F. E. Browder, Nonexpansive nonlinear operators in Banach spaces, Proc. Nat. Acad. Sci. U.S.A. 54 (1965), 1041-1044. MR 0187120 (32:4574)
  • [3] N. Dunford and J. T. Schwartz, Linear operators: General theory, Pure and Appl. Math., vol. 7, Interscience, New York, 1958.
  • [4] E. Hopf, Ergodentheorie, Ergebnisse der Math., Vol. 5, Springer-Verlag, Berlin, 1937.
  • [5] L. A. Karlovitz, On nonexpansive mappings, Proc. Amer. Math. Soc. 35 (1975), 321-325. MR 0405182 (53:8976)
  • [6] E. Odell and Y. Sternfeld, A fixed point theorem in $ {c_0}$, preprint.
  • [7] S. Reich, The fixed point property for nonexpansive mappings, Amer. Math. Monthly 87 (1980), 292-294. MR 1539350

Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC: 47H10, 54H25

Retrieve articles in all journals with MSC: 47H10, 54H25

Additional Information

Keywords: Fixed point theory
Article copyright: © Copyright 1981 American Mathematical Society

American Mathematical Society