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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
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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.

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  • [1] F. E. Browder, Nonexpansive nonlinear operators in a Banach space, Proc. Nat. Acad. Sci. U.S.A. 54 (1965), 1041-1044. MR 0187120 (32:4574)
  • [2] F. S. De Blasi and J. Myjak, Sur la convergence des approximations successives pour les contractions non linéaires dans un espace de Banach, C. R. Acad. Sci. Paris Sér. A 283 (1976), 185-187. MR 0425703 (54:13656)
  • [3] G. Darbo, Punti uniti in transformazione a condiminio non compatto, Rend. Sem. Mat. Univ. Padova 24 (1955), 84-92. MR 0070164 (16:1140f)
  • [4] D. Göhde, Zum Prinzip der kontraktiven Abbildung, Math. Nachr. 30 (1966), 251-258. MR 0190718 (32:8129)
  • [5] W. A. Kirk, A fixed point theorem for mappings which do not increase distances, Amer. Math. Monthly 72 (1965), 1004-1006. MR 0189009 (32:6436)
  • [6] C. Kuratowski, Sur les espaces complets, Fund. Math. 15 (1930), 301-309.
  • [7] R. D. Nussbaum, The fixed point index for local condensing maps, Ann. Mat. Pura Appl. 89 (1971), 217-258. MR 0312341 (47:903)
  • [8] G. Vidossich, Existence, uniqueness and approximation of fixed points as a generic property, Bol. Soc. Brasil. Mat. 5 (1974), 17-29. MR 0397710 (53:1568)

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Keywords: Fixed points, k-set contractions, genericity
Article copyright: © Copyright 1979 American Mathematical Society

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