A nonlinear strong ergodic theorem for families of asymptotically nonexpansive mappings with compact domains (Q2758087)

The authors deal with families of asymptotically nonexpansive mappings acting on a compact convex subset of a strictly convex Banach space. Let \(C\) be such a space, \(x\in C\) and let \(S=\{S(t) \mid t\geq 0\) be an asymptotically nonexpansive semigroup on \(C\). Denote by \(F(S)\) the set of all common fixed points of \(S\). The main result of this paper states that under these assumptions \(\frac{1}{t}\int_0^t S(\tau+h)x d\tau\) converges strongly to a common fixed point of \(S\), uniformly in \(h\geq 0\). Moreover, if \(Qx=\lim_{t\to\infty}\frac{1}{t}\int_0^t S(\tau)x d\tau\) for \(x\in C\), then \(Q\) is a nonexpansive mapping from \(C\) onto \(F(S)\) such that \(QS(t)=S(t)Q=Q\) for every \(t\geq 0\) and \(Qx\in\overline{\text{co}}\{S(t)x \mid t\geq 0\}\) for every \(x\in C\). NEWLINENEWLINENEWLINEThe above ergodic theorem extends earlier result of \textit{S. Atsushiba} and \textit{W. Takahashi} [``Strong convergence theorems for one-parameter nonexpansive semigroups with compact domains'', to appear] for nonexpansive semigroups with a compact convex domain in a Banach space.