Multi-step hybrid viscosity method for systems of variational inequalities defined over sets of solutions of an equilibrium problem and fixed point problems

Abdul Latif,Qamrul Hasan Ansari,Lu-Chuan Ceng

doi:10.1186/1687-1812-2012-186

Abstract

In this paper, we consider a system of variational inequalities defined over the intersection of the set of solutions of an equilibrium problem, the set of common fixed points of a finite family of nonexpansive mappings, and the solution set of a nonexpansive mapping. We also consider a triple hierarchical variational inequality problem, that is, a variational inequality problem defined over a set of solutions of another variational inequality problem which is defined over the intersection of the set of solutions of an equilibrium problem, the set of common fixed points of a finite family of nonexpansive mappings, and the solution set of a nonexpansive mapping. These two problems are very general and include, as special cases, several problems studied in the literature. We propose a multi-step hybrid viscosity method to compute the approximate solutions of our system of variational inequalities and a triple hierarchical variational inequality problem. The convergence analysis of the sequences generated by the proposed method is also studied. In addition, the nontrivial examples of two systems are presented and our results are applied to these examples.MSC:49J40, 47H05, 47H19.

Highlights

Introduction and formulations LetH be a real Hilbert space whose inner product and norm are denoted by ·, · and ·, respectively
If C is the set of fixed points of a nonexpansive mapping T, denoted by Fix(T), and if S is another nonexpansive mapping, variational inequality problem (VIP) ( . ) becomes the following problem: find x* ∈ Fix(T) such that (I – S)x*, x – x* ≥, ∀x ∈ Fix(T)
It is worth mentioning that many practical problems can be written in the form of a hierarchical variational inequality problem; see for example [ – ] and the references therein

Summary

Introduction

Introduction and formulations LetH be a real Hilbert space whose inner product and norm are denoted by ·, · and · , respectively. ) is nonempty and the following conditions hold for two sequences {λn}, {αn} ⊂ ( , ): (i) < lim infn→∞ αn ≤ lim supn→∞ αn < ; (ii) limn→∞ λn = and αn– λn– |

Results

Conclusion