Convergences of nonexpansive iteration processes in Banach spaces (Q1856825)

Let \(T_i : E \rightarrow E\), \(i = 1, \dots, k\), be nonexpansive mappings on a reflexive Banach space \(E\) with a uniformly \(G\)-differentiable norm. The author considers the iteration \(x_{n+1} = (\alpha_0 I + \alpha_1 T_1 + \cdots \alpha_k T_k) x_n\) with \(\alpha_i \geq 0\), \(\alpha_0 > 0\), \(\alpha_0 + \cdots + \alpha_k = 1\), and proves a dual weak almost convergence result which is then used to obtain the weak convergence of the process.