Two-Step Viscosity Approximation Scheme for Variational Inequality in Banach Spaces

Abstract

This paper introduces and analyzes a viscosity iterative algorithm for an infinite family of nonexpansive mappings ${\{T_i\}}_{i=1}^{\infty }$ in the framework of a strictly convex and uniformly smooth Banach space. It is shown that the proposed iterative method converges strongly to a common fixed point of ${\{T_i\}}_{i=1}^{\infty }$, which solves specific variational inequalities. Necessary and sufficient convergence conditions of the iterative algorithm for an infinite family of nonexpansive mappings are given. Results shown in this paper represent an extension and refinement of the previously known results in this area.