Extragradient algorithms for split equilibrium problem and nonexpansive mapping

TL;DR

This paper introduces new extragradient algorithms combining cutting techniques to solve split equilibrium problems involving nonexpansive mappings in Hilbert spaces, with proven convergence under specific conditions.

Contribution

The paper develops novel extragradient algorithms with convergence proofs for split equilibrium problems involving nonexpansive mappings and pseudomonotone bifunctions.

Findings

Algorithms converge weakly and strongly to solutions.

Proposed methods handle complex split equilibrium problems.

Convergence is established under certain parameter conditions.

Abstract

In this paper, we propose new extragradient algorithms for solving a split equilibrium and nonexpansive mapping SEPNM($C, Q, A, f, g, S, T)$ where $C, Q$ are nonempty closed convex subsets in real Hilbert spaces $\mathcal{H}_1, \mathcal{H}_2 $ respectively, $A : \mathcal{H}_1 \to \mathcal{H}_2$ is a bounded linear operator, $f$ is a pseudomonotone bifunction on $C$ and $g$ is a monotone bifunction on $Q$, $S, T$ are nonexpansive mappings on $C$ and $Q$ respectively. By using extragradient method combining with cutting techniques, we obtain algorithms for the problem. Under certain conditions on parameters, the iteration sequences generated by the algorithms are proved to be weakly and strongly convergent to a solution of this problem.