The existence of nonexpansive retractions in Banach spaces
Let C be a closed convex subset of a uniformly convex Banach space E, and T a nonexpansive mapping of C into itself. Assume that T has fixed points. Then we show the existence of a nonexpansive retraction P of C into the set of fixed points of T such that \(PT=TP=P\) and Px is implied in the convex closure of \(\{T^ nx:\) \(n=0,1,2,...\}\) for each \(x\in C\). If E has a Fréchet differentiable norm, then such retraction P is unique and the Cesàro means \(\frac{1}{n}\sum^{n-1}_{i=0}T^ ix\) converge, as \(n\to \infty\), weakly to Px for each \(x\in C\). This is a simple proof of a mean ergodic theorem for a nonexpansive mapping in a Banach space.
mean ergodic theorem for a nonexpansive mapping in a Banach space
scientific article; zbMATH DE number 3955812