Taiwanese Journal of Mathematics

Ergodic Retractions for Semigroups in Strictly Convex Banach Spaces

Abstract

We study the existence of ergodic retractions for semigroups of mappings in strictly convex Banach spaces. We prove, for instance, the following theorem. Let $(X,\|\cdot\|)$ be a strictly convex Banach space and let $\Gamma$ be a norming set for $X$. Let $C$ be a bounded and convex subset of $X$, and suppose $C$ is compact in the $\Gamma$-topology. If $\mathcal S$ is a right amenable semigroup, $\varphi=\{T_s:s\in\mathcal S\}$ is a semigroup on $C$ with a nonempty set $F=F(\varphi)$ of common fixed points, and each $T_s$ is ($F$-quasi-) nonexpansive, then there exists an ($F$-quasi-) nonexpansive retraction $R$ from $C$ onto $F$ such that $RT_s=T_sR=R$ for each $s\in \mathcal S$, and every $\Gamma$-closed, convex and $\varphi$-invariant subset of $C$ is also $R$-invariant.

Article information

Source
Taiwanese J. Math., Volume 15, Number 4 (2011), 1447-1456.

Dates
First available in Project Euclid: 18 July 2017

https://projecteuclid.org/euclid.twjm/1500406356

Digital Object Identifier
doi:10.11650/twjm/1500406356

Mathematical Reviews number (MathSciNet)
MR2848966

Zentralblatt MATH identifier
1244.47049

Citation

Kaczor, Wieslawa; Reich, Simeon. Ergodic Retractions for Semigroups in Strictly Convex Banach Spaces. Taiwanese J. Math. 15 (2011), no. 4, 1447--1456. doi:10.11650/twjm/1500406356. https://projecteuclid.org/euclid.twjm/1500406356

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