Approximation for the Hierarchical Constrained Variational Inequalities over the Fixed Points of Nonexpansive Semigroups

Li-Jun Zhu

doi:10.1155/2013/604369

Abstract

The purpose of the present paper is to study the hierarchical constrained variational inequalities of finding a pointx*such thatx*∈Ω,〈(A-γf)x*-(I-B)Sx*,x-x*〉≥0, ∀x∈Ω, whereΩis the set of the solutions of the following variational inequality:x*∈Ϝ,〈(A-S)x*,x-x*〉≥0, ∀x∈Ϝ, whereA,Bare two strongly positive bounded linear operators,fis aρ-contraction,Sis a nonexpansive mapping, andϜis the fixed points set of a nonexpansive semigroup{T(s)}s≥0. We present a double-net convergence hierarchical to some elements inϜwhich solves the above hierarchical constrained variational inequalities.

Highlights

Let H be a real Hilbert space with inner product ⟨⋅, ⋅⟩ and norm ‖ ⋅ ‖, respectively
We consider the following hierarchical variational inequality with the variational inequality constraint over the fixed points set of nonexpansive semigroups {T(s)}s≥0
Let f : C → H be a ρ-contraction with coefficient ρ ∈ [0, 1) and let S : C → C be a nonexpansive mapping

Summary

Introduction

Let H be a real Hilbert space with inner product ⟨⋅, ⋅⟩ and norm ‖ ⋅ ‖, respectively. For each (s, t) ∈ (0, κ) × (0, 1), we define a double net {xs,t} implicitly by xs,t = PC [s (tγf (xs,t) + (I − tB) Sxs,t) We define the mapping x 󳨃󳨀→ Ws,t (x) := PC [s (tγf (x) + (I − tB) Sx)

Objectives

Results

Conclusion