Extragradient and linesearch algorithms for solving equilibrium problems and fixed point problems in Banach spaces

TL;DR

This paper introduces new extragradient and linesearch algorithms in Banach spaces for solving equilibrium and fixed point problems, with proven convergence under certain conditions and demonstrated practical applicability.

Contribution

It develops novel algorithms using sunny generalized nonexpansive retraction, extending existing methods with convergence guarantees in Banach spaces.

Findings

Strong convergence of extragradient method under $4$-Lipschitz-type condition

Linesearch method achieves convergence without the Lipschitz condition

Numerical example confirms the effectiveness of the proposed algorithms

Abstract

In this paper, using sunny generalized nonexpansive retraction, we propose new extragradient and linesearch algorithms for finding a common element of the set of solutions of an equilibrium problem and the set of fixed points of a relatively nonexpansive mapping in Banach spaces. To prove strong convergence of iterates in the extragradient method, we introduce a $\phi$-Lipschitz-type condition and assume that the equilibrium bifunction satisfies in this condition. This condition is unnecessary when the linesearch method is used instead of the extragradient method. A numerical example is given to illustrate the usability of our results. Our results generalize, extend and enrich some existing results in the literature.