Taiwanese Journal of Mathematics

Ergodic Retractions for Semigroups in Strictly Convex Banach Spaces

Wieslawa Kaczor and Simeon Reich

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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.

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

First available in Project Euclid: 18 July 2017

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Zentralblatt MATH identifier

Primary: 47H09: Contraction-type mappings, nonexpansive mappings, A-proper mappings, etc. 47H10: Fixed-point theorems [See also 37C25, 54H25, 55M20, 58C30] 47H20: Semigroups of nonlinear operators [See also 37L05, 47J35, 54H15, 58D07]

$\Gamma$-topology mean nonexpansive ergodic retraction nonexpansive mapping nonexpansive semigroup quasi-nonexpansive ergodic retraction quasi-nonexpansive mapping quasi-nonexpansive semigroup


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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