A modified projection method for a common solution of a system of variational inequalities, a split equilibrium problem and a hierarchical fixed-point problem

Abdellah Bnouhachem

doi:10.1186/1687-1812-2014-22

Abstract

In this paper, we suggest and analyze a modified projection method for finding a common solution of a system of variational inequalities, a split equilibrium problem, and a hierarchical fixed-point problem in the setting of real Hilbert spaces. We prove the strong convergence of the sequence generated by the proposed method to a common solution of a system of variational inequalities, a split equilibrium problem, and a hierarchical fixed-point problem. Several special cases are also discussed. The results presented in this paper extend and improve some well-known results in the literature. MSC: 49J30, 47H09, 47J20.

Highlights

Let H be a real Hilbert space, whose inner product and norm are denoted by ·, · and ·
As pointed out in [ ] the system of variational inequalities is used as a tool to study the Nash equilibrium problem, see, for example, [ – ] and the references therein
We believe that the problem ( . ) could be used to study the Nash equilibrium problem for a two players game

Summary

Introduction

Let H be a real Hilbert space, whose inner product and norm are denoted by ·, · and ·. Ceng et al [ ] introduced and analyzed an extragradient method with regularization for finding a common element of the solution set of the split feasibility problem and the set of fixed points of a nonexpansive mapping S in the setting of infinitedimensional Hilbert spaces. By combining Mann’s iterative method and the extragradient method, Ceng et al [ ] proposed three different kinds of Mann type iterative methods for finding a common element of the solution set of the split feasibility problem and the set of fixed points of a nonexpansive mapping S in the setting of infinite-dimensional Hilbert spaces. In , Ceng et al [ ] investigated the following iterative method: xn+ = PC αnρU(xn) + (I – αnμF) T(yn) , ∀n ≥ , where U is a Lipschitzian mapping, and F is a Lipschitzian and strongly monotone mapping They proved that under some approximate assumptions on the operators and parameters, the sequence {xn} generated by {xn} is weakly convergent to a point in C

The proposed method and some properties

Conclusions