A mean ergodic theorem for asymptotically quasi-nonexpansive affine mappings in Banach spaces satisfying Opial's condition (Q841414)

If \(C\) is a nonempty closed convex subset of a Hilbert space \(H\) and \(T:C\rightarrow C\) is nonexpansive, then by the nonlinear ergodic theorem established by \textit{J.-B.\thinspace Baillon} [C.\ R.\ Acad.\ Sci., Paris, Sér.\ I Math.\ 280, 1511--1514 (1975; Zbl 0307.47006)], it is known that, if the set Fix\,\((T)\) of fixed points of \(T\) is nonempty, then for each \(x\in C\), the Cesàro means \(S_n(x)=\frac{1}{n}\sum_{k=0}^{n}T^k x\) converge weakly to a fixed point of \(T\). The main aim of the paper under review is to prove the following ergodic result: if the set \(C\) is a weakly compact convex subset of a Banach space \(E\) satisfying Opial's condition and \(T:C\rightarrow C\) is an asymptotically quasi-nonexpansive affine mapping with Fix\,\((T)\neq \emptyset\), then for each \(x\in C\), \(\{T^n x\}\) converges weakly to a fixed point of \(T\).