On the strong ergodic theorems for commutative semigroups in Banach spaces (Q1321749)

Let \(T\) be an asymptotically nonexpansive mapping on a closed convex \(C\) in a space (i.e. for some sequence \(\alpha_n\to 0\) \(|T^nx- T^ny|\leq (1+\alpha_n)|x-y|\), \(n\geq 1\), \(x,y\in C\)) and assume that the fixed-point set \(F(C)\) is non-empty. Then \({1\over n}\sum_{i=0}^{n-1} T^{i+k}x\to y\) for \(x\in C\), \(y\in F(T)\), uniformly in \(k\in\mathbb{Z}_+\) [see \textit{J. B. Baillon}, C. R. Acad. Sci., Paris, Sér. A 283, 587-590 (1976; Zbl 0343.47046) and \textit{R. E. Bruck}, Isr. J. Math. 29, 1-16 (1978; Zbl 0367.47037)]. The aim of the paper under review is to prove an analogous result for one-parameter semigroups \(T(\cdot)\) acting on \(C\) (contained in a uniformly convex Banach space) and to provide a unified approach for the discrete-parameter and the continuous-parameter case. Starting with preliminaries and technical lemmas on commutative topological semigroups acting in \(C\), the main results are proved in 4. In 5. the author generalizes previous results to affine semigroups acting on general Banach spaces.