Iterative algorithm for common fixed points of infinite family of nonexpansive mappings in Banach spaces (Q443092)

Summary: Let \(C\) be a nonempty closed convex subset of a real uniformly smooth Banach space \(X\), let \(\{T_k\}^\infty_{k=1} : C \rightarrow C\) be an infinite family of nonexpansive mappings with the nonempty set of common fixed points \(\bigcap^\infty_{k=1} \text{Fix}(T_k)\), and \(f : C \rightarrow C\) be a contraction. We introduce an explicit iterative algorithm \(x_{n+1} = \alpha_n f(x_n) + (1 - \alpha_n)L_nx_n\), where \(L_n = \sum^n_{k=1}(\omega_k/s_n) T_k, S_n = \sum^n_{k=1} \omega_k\), and \(w_k > 0\) with \(\sum^\infty_{k=1} \omega_k = 1\). Under certain appropriate conditions on \(\{\alpha_n\}\), we prove that \(\{x_n\}\) converges strongly to a common fixed point \(x^\ast\) of \(\{T_k\}^\infty_{k=1}\), which solves the following variational inequality: \[ \langle x^\ast - f(x^\ast), J(x^\ast - p)\rangle \leq 0, \quad p \in \bigcap^\infty_{k=1} \text{Fix}(T_k), \] where \(J\) is the (normalized) duality mapping of \(X\). This algorithm is brief and needs less computational work, since it does not involve \(W\)-mappings.