Almost all 1-set contractions have a fixed point

Butler, G. J.

doi:10.1090/S0002-9939-1979-0524316-2

Almost all $1$-set contractions have a fixed point
HTML articles powered by AMS MathViewer

by G. J. Butler

Proc. Amer. Math. Soc. 74 (1979), 353-357

DOI: https://doi.org/10.1090/S0002-9939-1979-0524316-2

PDF | Request permission

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

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
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 425703
Gabriele Darbo, Punti uniti in trasformazioni a codominio non compatto, Rend. Sem. Mat. Univ. Padova 24 (1955), 84–92 (Italian). MR 70164
Dietrich Göhde, Zum Prinzip der kontraktiven Abbildung, Math. Nachr. 30 (1965), 251–258 (German). MR 190718, DOI 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 10.2307/2313345

Sur les espaces complets

15

Roger D. Nussbaum, The fixed point index for local condensing maps, Ann. Mat. Pura Appl. (4) 89 (1971), 217–258. MR 312341, DOI 10.1007/BF02414948
Giovanni Vidossich, Existence, uniqueness and approximation of fixed points as a generic property, Bol. Soc. Brasil. Mat. 5 (1974), no. 1, 17–29. MR 397710, DOI 10.1007/BF02584769