Iterative approximation to common fixed points of nonexpansive mapping sequences in reflexive Banach spaces (Q858635)

Let \(E\) be a real reflexive Banach space with a weakly sequentially continuous duality mapping, \(K\) be a nonempty closed convex subset of \(E\). Suppose that \(\{T_n\}\) is a uniformly asymptotically regular sequence of nonexpansive mappings from \(K\) into itself such that \(F\), the set of common fixed points of \(\{T_n\}\), is nonempty. It is proved that the iterative sequence \(x_{n+1}=\lambda_{n+1}f(x_n)+(1-\lambda_{n+1})T_{n+1}x_n\), with \(x_0\in K\) and \(f\) a contractive mapping, converges to \(p\in F\), where \(p\) is the unique solution of the variational inequality \(\langle(I-f)p,j(p-u)\rangle \leq 0\) for all \(u\in F\).