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

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.