Modified Hybrid Steepest-Descent Methods for General Systems of Variational Inequalities with Solutions to Zeros of m -Accretive Operators in Banach Spaces

Abstract

The purpose of this paper is to introduce and analyze modified hybrid steepest-descent methods for a general system of variational inequalities (GSVI), with solutions being also zeros of an $m$-accretive operator $A$ in the setting of real uniformly convex and 2-uniformly smooth Banach space $X$. Here the modified hybrid steepest-descent methods are based on Korpelevich's extragradient method, hybrid steepest-descent method, and viscosity approximation method. We propose and consider modified implicit and explicit hybrid steepest-descent algorithms for finding a common element of the solution set of the GSVI and the set $A^{-\mathrm{1}}(\mathrm{0})$ of zeros of $A$ in $X$. Under suitable assumptions, we derive some strong convergence theorems. The results presented in this paper improve, extend, supplement, and develop the corresponding results announced in the earlier and very recent literature.