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