Implicit Iterative Scheme for a Countable Family of Nonexpansive Mappings in 2-Uniformly Smooth Banach Spaces

Abstract

Implicit Mann process and Halpern-type iteration have been extensively studied by many others. In this paper, in order to find a common fixed point of a countable family of nonexpansive mappings in the framework of Banach spaces, we propose a new implicit iterative algorithm related to a strongly accretive and Lipschitzian continuous operator $F:x_n={\alpha }_n\gamma V(x_n)+{\beta }_n{x}_{n-\mathrm{1}}+((\mathrm{1}-{\beta }_n)I-{\alpha }_n\mu F)T_n{x}_n$ and get strong convergence under some mild assumptions. Our results improve and extend the corresponding conclusions announced by many others.