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Nonexpansive, $\mathcal{T}$-continuous antirepresentations have common fixed points

Author: Wojciech Bartoszek
Journal: Proc. Amer. Math. Soc. 127 (1999), 1051-1055
MSC (1991): Primary 47H10, 22A25; Secondary 28D05
MathSciNet review: 1469398
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Abstract: Let $C$ be a closed convex subset of a Banach (dual Banach) space $\mathfrak{X}$. By $\mathcal{S}$ we denote an antirepresentation $\{ T_{s} \,:\, s \in S \}$ of a semitopological semigroup $S$ as nonexpansive mappings on $C$. Suppose that the mapping $S \times C \ni (s,x) \to T_{s}x \in C$ is jointly continuous when $C$ has the weak (weak*) topology and the Banach space $RUC(S)$ of bounded right uniformly continuous functions on $S$ has a right invariant mean. If $C$ is weakly compact (for some $x \in C$ the set ${\overline{\{ T_{s}x \,:\, s \in S \}}} ^{\text{weak*}}$ is weakly* compact) and norm separable, then $\{ T_{s} \,:\, s \in S \}$ has a common fixed point in $C$.

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Additional Information

Wojciech Bartoszek
Affiliation: Department of Mathematics, University of South Africa, P.O. Box 392, Pretoria 0003, South Africa

Keywords: Fixed point, topological semigroup, nonexpansive mapping
Received by editor(s): July 14, 1997
Communicated by: Dale Alspach
Article copyright: © Copyright 1999 American Mathematical Society