Extragradient algorithms for equilibrium problems and symmetric generalized hybrid mappings

TL;DR

This paper introduces new extragradient algorithms to find common solutions for equilibrium problems and fixed points of symmetric generalized hybrid mappings in Hilbert spaces, ensuring convergence under specific conditions.

Contribution

The paper develops novel extragradient algorithms that incorporate Armijo linesearch, improving convergence analysis for equilibrium and hybrid fixed point problems.

Findings

Convergence of the proposed algorithms is established under weaker assumptions.

Inclusion of Armijo linesearch eliminates the need for Lipschitz-type conditions.

Algorithms effectively find common points of equilibrium and fixed point sets.

Abstract

In this paper, we propose new algorithms for finding a common point of the solution set of a pseudomonotone equilibrium problem and the set of fixed points of a symmetric generalized hybrid mapping in a real Hilbert space. The convergence of the iterates generated by each method is obtained under assumptions that the fixed point mapping is quasi-nonexpansive and demiclosed at $0$, and the bifunction associated with the equilibrium problem is weakly continuous. The bifunction is assumed to be satisfying a Lipschitz-type condition when the basic iteration comes from the extragradient method. It becomes unnecessary when an Armijo back tracking linesearch is incorporated in the extragradient method.