An explicit iterative method to solve generalized mixed equilibrium problem, variational inequality problem and hierarchical fixed point problem for a nearly nonexpansive mapping

TL;DR

This paper introduces a new iterative method for solving multiple complex problems involving nearly nonexpansive mappings, proving strong convergence without the usual demiclosedness condition, and extending recent methods.

Contribution

The paper presents a novel iterative approach that converges strongly to a common solution for generalized equilibrium, variational inequality, and fixed point problems without requiring the demiclosedness condition.

Findings

Method converges strongly under suitable conditions.

Main theorem does not require demiclosedness condition.

Results extend and improve recent related methods.

Abstract

In this paper, we introduce a new iterative method to find a common solution of a generalized mixed equilibrium problem, a variational inequality problem and a hierarchical fixed point problem for a demicontinuous nearly nonexpansive mapping. We prove that the proposed method converges strongly to a common solution of above problems under the suitable conditions. It is also noted that the main theorem is proved without usual demiclosedness condition. Also, under the appropriate assumptions on the control sequences and operators, our iterative method can be reduced to recent methods. So, the results here improve and extend some recent corresponding results given by many other authors.