Nonlinear mean ergodic theorems for nonexpansive semigroups in Banach spaces (Q926616)

The authors prove the following nonlinear ergodic theorem. Let \(C\) be a compact and convex subset of a strictly convex Banach space \(E\), let \(S\) be a semigroup, let \(S=\{T(s): s\in S\}\) be a representation of \(S\) by nonexpansive mappings of \(C\) into itself, let \(X\) be a closed, left translation invariant and admissible subspace of \(l^\infty(S)\) which contains constants, and let \(\{\mu_\alpha\}\) be an asymptotically invariant net of means on \(X\). Then, for each \(x\in C\), \(T(r(h)^*\mu_\alpha)x\) converges strongly to a common fixed point of \(S\) uniformly in \(h\in S\). Using this result, the authors extend the result of \textit{A.\,Atsushiba}, \textit{A.\,T.--M.\thinspace Lau} and \textit{W.\,Takahasi} [J.~Nonlinear Convex Anal 1, No.\,2, 213--231 (2000; Zbl 0968.47028)] to amenable semigroups of nonexpansive mappings in a strictly convex Banach space and prove some nonlinear ergodic theorems for discrete and one-parameter semigroups of nonexpansive mappings.