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.