Iterative methods for triple hierarchical variational inequalities and common fixed point problems

Vidya Sagar,Satyendra Kumar,Shin Min Kang,DR Sahu

doi:10.1186/1687-1812-2014-244

Abstract

AbstractThe purpose of this paper is to introduce a new iterative scheme for approximating the solution of a triple hierarchical variational inequality problem. Under some requirements on parameters, we study the convergence analysis of the proposed iterative scheme for the considered triple hierarchical variational inequality problem which is defined over the set of solutions of a variational inequality problem defined over the intersection of the set of common fixed points of a sequence of nearly nonexpansive mappings and the set of solutions of the classical variational inequality. Our strong convergence theorems extend and improve some known corresponding results in the contemporary literature for a wider class of nonexpansive type mappings in Hilbert spaces.MSC:47J20, 47J25.

Highlights

The classical variational inequality problem initially studied by Stampacchia [ ] for a nonlinear operator A : C → H is a problem which provides us such x∗ ∈ D which satisfies
The equivalence relation between the variational inequality and fixed point problems can be seen by projection technique which plays an important role in developing an important role in developing some efficient methods for solving variational inequality problems and related optimization problems
Motivated and inspired by the works mentioned above, we introduce an explicit iterative scheme that generates a sequence and prove that this sequence converges strongly to a unique solution of the considered triple hierarchical variational inequality problem defined over the set of solutions of a variational inequality problem which is defined over the intersection of the set of common fixed points of a sequence of nearly nonexpansive mappings and the set of solutions of the classical variational inequality problem

Summary

Introduction

The classical variational inequality problem initially studied by Stampacchia [ ] for a nonlinear operator A : C → H is a problem which provides us such x∗ ∈ D which satisfiesAx∗, y – x∗ ≥ , ∀y ∈ D, ( . )where C is a nonempty closed convex subset of a real Hilbert space H and D is a nonempty closed convex subset of C. . the sequence {xn} converges strongly to a unique solution x∗ of the variational inequality of finding x∗ ∈ such that Fx∗, x – x∗ ≥ , ∀x ∈ .

Objectives

Results

Conclusion