Hybrid Extragradient Methods for Finding Zeros of Accretive Operators and Solving Variational Inequality and Fixed Point Problems in Banach Spaces

Abstract

We introduce and analyze hybrid implicit and explicit extragradient methods for finding a zero of an accretive operator and solving a general system of variational inequalities and a fixed point problem of an infinite family of nonexpansive self-mappings in a uniformly convex Banach space $X$ which has a uniformly Gateaux differentiable norm. We establish some strong convergence theorems for hybrid implicit and explicit extra-gradient algorithms under suitable assumptions. Furthermore, we derive the strong convergence of hybrid implicit and explicit extragradient algorithms for finding a common element of the set of zeros of an accretive operator and the common fixed point set of an infinite family of nonexpansive self-mappings and a self-mapping whose complement is strictly pseudocontractive and strongly accretive in $X$. The results presented in this paper improve, extend, supplement, and develop the corresponding results announced in the earlier and very recent literature.