Convergence theorems by extragradient method in Banach spaces

TL;DR

This paper introduces a new extragradient method in Banach spaces that combines generalized metric projections to find common solutions of equilibrium problems, variational inequalities, and fixed points, with proven strong convergence.

Contribution

It proposes a novel extragradient algorithm utilizing generalized metric projections for multiple problem types in Banach spaces, with strong convergence results.

Findings

Proved strong convergence theorems under suitable conditions.

Provided a numerical example demonstrating practical usability.

Abstract

In this paper, using generalized metric projection, we propose a new extragradient method for finding a common element of the solutions set of a generalized equilibrium problem and a variational inequality for an $\alpha$-inverse-strongly monotone operator and fixed points of two relatively nonexpansive mappings in Banach spaces. We prove strong convergence theorems by this method under suitable conditions. A numerical example is given to illustrate the usability of our results.