Weak convergence to common fixed points of countable nonexpansive mappings and its applications (Q2770406)

Let \(C\) be a nonempty closed convex subset of a uniformly convex Banach space \(E\) such that either \(E\) satisfies Opial's condition or \(E\) has a Fréchet differentiable norm. Moreover, let \(\{T_n\}\) be a sequence of nonexpansive mappings of \(C\) into itself such that \(\bigcap_{n=1}^{\infty} F(T_n)\neq\emptyset\), where \(F(T_n)\) denotes the set of fixed points of \(T_n\) for \(n\in\mathbb{N}\). In this paper, the authors introduce an iteration \(\{x_n\}\) and prove that this iteration converges weakly to \(z\in\bigcap_{n=1}^{\infty} F(T_n)\). As an application, the authors consider the feasibility problem of finding a solution of the countable convex inequality system and a problem of finding a common fixed point for a commuting countable family of nonexpansive mappings which map \(C\) into itself.