# Projection methods for solving split equilibrium problems

## TL;DR

This paper introduces a new projection-based algorithm for solving split equilibrium problems in Hilbert spaces, demonstrating weak convergence and improved performance over existing methods.

## Contribution

A novel projection method for split equilibrium problems is proposed, differing from traditional proximal and extragradient approaches, with proven convergence.

## Key findings

- Algorithm is weakly convergent under mild conditions
- Numerical results confirm convergence and efficiency
- Comparison shows advantages over existing methods

## Abstract

The paper considers a split inverse problem involving component equilibrium problems in Hilbert spaces. This problem therefore is called the split equilibrium problem (SEP). It is known that almost solution methods for solving problem (SEP) are designed from two fundamental methods as the proximal point method and the extended extragradient method (or the two-step proximal-like method). Unlike previous results, in this paper we introduce a new algorithm, which is only based on the projection method, for finding solution approximations of problem (SEP), and then establish that the resulting algorithm is weakly convergent under mild conditions. Several of numerical results are reported to illustrate the convergence of the proposed algorithm and also to compare with others.