Weak convergence theorems for a symmetric generalized hybrid mapping and an equilibrium problem

TL;DR

This paper introduces three new iterative algorithms combining existing methods to find common solutions of fixed points and equilibrium problems in Hilbert spaces, with proven weak convergence.

Contribution

The paper presents novel iterative methods that extend previous algorithms by integrating Ishikawa's process with proximal and extragradient methods, ensuring weak convergence.

Findings

Iteration sequences are weakly convergent under certain conditions.

The methods successfully find common solutions in Hilbert spaces.

Numerical example demonstrates effectiveness of the algorithms.

Abstract

In this paper, we introduce three new iterative methods for finding a common point of the set of fixed points of a symmetric generalized hybrid mapping and the set of solutions of an equilibrium problem in a real Hilbert space. Each method can be considered as an combination of Ishikawa's process with the proximal point algorithm, the extragradient algorithm with or without linesearch. Under certain conditions on parameters, the iteration sequences generated by the proposed methods are proved to be weakly convergent to a solution of the problem. These results extend the previous results given in the literature. A numerical example is also provided to illustrate the proposed algorithms.