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Proceedings of the American Mathematical Society

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A common fixed-point theorem for compact convex semigroups of nonexpansive mappings


Author: Ronald E. Bruck
Journal: Proc. Amer. Math. Soc. 53 (1975), 113-116
MSC: Primary 47H10
DOI: https://doi.org/10.1090/S0002-9939-1975-0383164-3
MathSciNet review: 0383164
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Abstract: Let $ C$ be a bounded closed convex subset of a strictly convex Banach space and let $ S$ be a semigroup of nonexpansive self-mappings of $ C$ which is convex and compact in the topology of weak point-wise convergence. If $ S$ has the property that $ \overline {\operatorname{co} \,} \mathcal{R}({s_1}) \cap \overline {\operatorname{co} \,} \mathcal{R}({s_2}\;) \ne \emptyset $ whenever $ {s_1},\;{s_2}\epsilon S$, then $ S$ has a common fixed point and $ F(S)$ is a nonexpansive retract of $ C$.


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DOI: https://doi.org/10.1090/S0002-9939-1975-0383164-3
Keywords: Common fixed point, nonexpansive retract, semigroup
Article copyright: © Copyright 1975 American Mathematical Society

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