Iterative methods for triple hierarchical variational inequalities with mixed equilibrium problems, variational inclusions, and variational inequalities constraints

Yen-Cherng Lin,Lu-Chuan Ceng,Ching-Feng Wen

doi:10.1186/s13660-014-0535-x

Abstract

In this paper, we introduce and analyze a multi-step hybrid steepest-descent extragradient algorithm and multi-step composite Mann-type viscosity iterative algorithm for finding a solution of triple hierarchical variational inequalities defined over the common set of solutions of mixed equilibrium problems, variational inclusions, variational inequalities, and fixed point problems. Under appropriate assumptions, we prove that the proposed algorithms converge strongly to a common element of the fixed point set of a strict pseudocontractive mapping, a solution set of finitely many generalized mixed equilibrium problems, a solution set of finitely many variational inclusions, and a solution set of a general system of variational inequalities. Such an element is a unique solution of a triple hierarchical variational inequality problem. In addition, we also consider as an application the proposed algorithm to solve a hierarchical variational inequality problem defined over the set of common solutions of finitely many generalized mixed equilibrium problems, finitely many variational inclusions, and a general system of variational inequalities. The results obtained in this paper improve and extend the corresponding results announced by many other authors.

Highlights

Introduction and formulations LetC be a nonempty, closed, and convex subset of a real Hilbert space H and A : C → H be a nonlinear mapping on C
Yao et al [ ] considered the following hierarchical variational inequality problem (HVIP): find hierarchically a fixed point of T which is a solution to the VIP for the monotone mapping I – S, namely, find x ∈ Fix(T) such that (I – S)x, p – x ≥, ∀p ∈ Fix(T)
We introduce and study the following triple hierarchical variational inequality problem (THVIP) with constraints of finitely many generalized mixed equilibrium problem (GMEP), finitely many variational inclusions, and a general system of variational inequalities

Summary

Introduction

Introduction and formulations LetC be a nonempty, closed, and convex subset of a real Hilbert space H and A : C → H be a nonlinear mapping on C. Yao et al [ ] considered the following hierarchical variational inequality problem (HVIP): find hierarchically a fixed point of T which is a solution to the VIP for the monotone mapping I – S, namely, find x ∈ Fix(T) such that (I – S)x, p – x ≥ , ∀p ∈ Fix(T).

Results

Conclusion