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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

 

Nonexpansive actions of topological semigroups on strictly convex Banach spaces and fixed points


Author: Wojciech Bartoszek
Journal: Proc. Amer. Math. Soc. 104 (1988), 809-811
MSC: Primary 47H20; Secondary 47H09, 47H10
MathSciNet review: 964861
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Abstract: Let $ C$ be a closed convex subset of a strictly convex Banach space $ X$ and $ \left\{ {{T_s}:s \in S} \right\}$ be a continuous representation of a semitopological semigroup $ S$ as nonexpansive mappings of $ C$ into itself. The main result establishes the fact that if for some $ x \in C$ the trajectory $ \left\{ {{T_s}x:s \in S} \right\}$ is relatively compact and $ AP(S)$ has a left invariant mean then $ K = \overline {\operatorname{conv} \{ {T_s}x:s \in S\} } $ contains a common fixed point for $ {\left\{ {{T_s}} \right\}_{s \in S}}$.


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DOI: http://dx.doi.org/10.1090/S0002-9939-1988-0964861-4
PII: S 0002-9939(1988)0964861-4
Article copyright: © Copyright 1988 American Mathematical Society