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)

 

 

Almost all $ 1$-set contractions have a fixed point


Author: G. J. Butler
Journal: Proc. Amer. Math. Soc. 74 (1979), 353-357
MSC: Primary 47H10
MathSciNet review: 524316
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: The 1-set contractions and strict set contractions of a bounded, closed, convex subset C of a Banach space X are generalizations of the nonexpansive mappings and the Banach contractions of C, defined in terms of the measure of noncompactness of bounded subsets of X. Vidossich has shown that ``almost all'' nonexpansive mappings of C into itself have fixed points. In this note we establish a similar generic result for the 1-set contractions of C.


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

  • [1] Felix E. Browder, Nonexpansive nonlinear operators in a Banach space, Proc. Nat. Acad. Sci. U.S.A. 54 (1965), 1041–1044. MR 0187120
  • [2] Francesco S. De Blasi and Józef Myjak, Sur la convergence des approximations successives pour les contractions non linéaires dans un espace de Banach, C. R. Acad. Sci. Paris Sér. A-B 283 (1976), no. 4, Aiii, A185–A187 (French, with English summary). MR 0425703
  • [3] Gabriele Darbo, Punti uniti in trasformazioni a codominio non compatto, Rend. Sem. Mat. Univ. Padova 24 (1955), 84–92 (Italian). MR 0070164
  • [4] Dietrich Göhde, Zum Prinzip der kontraktiven Abbildung, Math. Nachr. 30 (1965), 251–258 (German). MR 0190718
  • [5] W. A. Kirk, A fixed point theorem for mappings which do not increase distances, Amer. Math. Monthly 72 (1965), 1004–1006. MR 0189009
  • [6] C. Kuratowski, Sur les espaces complets, Fund. Math. 15 (1930), 301-309.
  • [7] Roger D. Nussbaum, The fixed point index for local condensing maps, Ann. Mat. Pura Appl. (4) 89 (1971), 217–258. MR 0312341
  • [8] Giovanni Vidossich, Existence, uniqueness and approximation of fixed points as a generic property, Bol. Soc. Brasil. Mat. 5 (1974), no. 1, 17–29. MR 0397710

Similar Articles

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

Retrieve articles in all journals with MSC: 47H10


Additional Information

DOI: https://doi.org/10.1090/S0002-9939-1979-0524316-2
Keywords: Fixed points, k-set contractions, genericity
Article copyright: © Copyright 1979 American Mathematical Society