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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)

 

 

Properties of fixed-point sets of nonexpansive mappings in Banach spaces


Author: Ronald E. Bruck
Journal: Trans. Amer. Math. Soc. 179 (1973), 251-262
MSC: Primary 47H10
DOI: https://doi.org/10.1090/S0002-9947-1973-0324491-8
MathSciNet review: 0324491
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Abstract: Let C be a closed convex subset of the Banach space X. A subset F of C is called a nonexpansive retract of C if either $ F = \emptyset $ or there exists a retraction of C onto F which is a nonexpansive mapping. The main theorem of this paper is that if $ T:C \to C$ is nonexpansive and satisfies a conditional fixed point property, then the fixed-point set of T is a nonexpansive retract of C. This result is used to generalize a theorem of Belluce and Kirk on the existence of a common fixed point of a finite family of commuting nonexpansive mappings.


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DOI: https://doi.org/10.1090/S0002-9947-1973-0324491-8
Keywords: Fixed-point set, metrically convex, nonexpansive mapping, nonexpansive retract, normal structure
Article copyright: © Copyright 1973 American Mathematical Society