Nonexpansive, $\mathcal T$-continuous antirepresentations have common fixed points
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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 $C$.References
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Additional Information
- Wojciech Bartoszek
- Affiliation: Department of Mathematics, University of South Africa, P.O. Box 392, Pretoria 0003, South Africa
- Email: bartowk@alpha.unisa.ac.za
- Received by editor(s): July 14, 1997
- Communicated by: Dale Alspach
- © Copyright 1999 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 127 (1999), 1051-1055
- MSC (1991): Primary 47H10, 22A25; Secondary 28D05
- DOI: https://doi.org/10.1090/S0002-9939-99-04567-0
- MathSciNet review: 1469398