Convergence analysis of the extragradient method for vector quasi-equilibrium problems

TL;DR

This paper analyzes the extragradient method for vector quasi-equilibrium problems in Banach spaces, introducing a regularization procedure that guarantees strong convergence without requiring monotonicity or weak continuity assumptions.

Contribution

It proposes a new regularization approach ensuring convergence of the extragradient method for vector quasi-equilibrium problems without standard monotonicity or continuity assumptions.

Findings

The regularization procedure guarantees strong convergence.

Boundedness of sequences implies nonemptiness of the solution set.

Numerical experiments demonstrate the method's effectiveness.

Abstract

We study the extragradient method for solving vector quasi-equilibrium problems in Banach spaces, which generalizes the extragradient method for vector equilibrium problems and scalar quasi-equilibrium problems. We propose a regularization procedure which ensures strong convergence of the generated sequence to a solution of the vector quasi-equilibrium problem, under standard assumptions on the problem without assuming neither any monotonicity assumption on the vector valued bifunction nor any weak continuity assumption of $f$ in its arguments that in the many well-known methods have been used. Also, we show that the boundedness of the generated sequences implies that the solution set of the vector quasi-equilibrium problem is nonempty, and prove the strong convergence of the generated sequences to a solution of the problem. Finally, we give some examples of vector quasi-equilibrium problems in several Banach spaces to which our main theorem can be applied. We also present some numerical experiments.