Some Mann-Type Implicit Iteration Methods for Triple Hierarchical Variational Inequalities, Systems Variational Inequalities and Fixed Point Problems

Lu-Chuan Ceng,Xiaoye Yang

doi:10.3390/math7030218

Abstract

This paper discusses a monotone variational inequality problem with a variational inequality constraint over the common solution set of a general system of variational inequalities (GSVI) and a common fixed point (CFP) of a countable family of nonexpansive mappings and an asymptotically nonexpansive mapping in Hilbert spaces, which is called the triple hierarchical constrained variational inequality (THCVI), and introduces some Mann-type implicit iteration methods for solving it. Norm convergence of the proposed methods of the iteration methods is guaranteed under some suitable assumptions.

Highlights

Let C be a convex closed nonempty subset of a real Hilbert space H with norm k · k and inner product h·, ·i
The purpose of this paper is to introduce and analyze some Mann-type implicit iteration methods for treating a monotone variational inequality with a inequality constraint over the common solution set of the general system of variational inequalities (GSVI) (1) for two inverse-strongly monotone mappings and a common fixed point problem (CFPP) of a countable family of nonexpansive mappings and an asymptotically nonexpansive mapping in Hilbert spaces, which is called the triple hierarchical constrained variational inequality (THCVI)
We prove strong convergence of the proposed methods to the unique solution of the THCVI

Summary

Introduction

Let C be a convex closed nonempty subset of a real Hilbert space H with norm k · k and inner product h·, ·i. Let PC be the metric (or nearest point) projection from H onto C, that is, for all x ∈ H, PC x ∈ C and k x − PC x k = infy∈C k x − yk. Let T : C → C be a possible nonlinear mapping. Denote by Fix( T ) the set of fixed points of T, i.e., Fix( T ) = { x ∈ C : x = Tx }. We use the notations R, * and → to indicate the set of real numbers, weak convergence and strong convergence, respectively. A mapping T : C → C is said to be asymptotically nonexpansive (see [1]), if there exists a sequence

Objectives

Results

Conclusion