Hybrid Mann Viscosity Implicit Iteration Methods for Triple Hierarchical Variational Inequalities, Systems of Variational Inequalities and Fixed Point Problems

Lu-Chuan Ceng,Qing Yuan

doi:10.3390/math7020142

Abstract

In the present work, we introduce a hybrid Mann viscosity-like implicit iteration to find solutions of a monotone classical variational inequality with a variational inequality constraint over the common solution set of a general system of variational inequalities and a problem of common fixed points of an asymptotically nonexpansive mapping and a countable of uniformly Lipschitzian pseudocontractive mappings in Hilbert spaces, which is called the triple hierarchical constrained variational inequality. Strong convergence of the proposed method to the unique solution of the problem is guaranteed under some suitable assumptions. As a sub-result, we provide an algorithm to solve problem of common fixed points of pseudocontractive, nonexpansive mappings, variational inequality problems and generalized mixed bifunction equilibrium problems in Hilbert spaces.

Highlights

We suppose that H is a real or complex Hilbert space and let H be with inner product h·, ·i and norm k · k
B2 is a nonself mapping from convex nonempty closed set C to entire space H, respectively. we study the system of approximating ( x ∗, y∗ ) ∈ C × C such that hμ1 B1 y∗ − y∗ + x ∗, x − x ∗ i ≥ 0, hμ2 B2 x ∗ − x ∗ + y∗, x − y∗ i ≥ 0
The main aim of this paper is to introduce and analyze a hybrid Mann viscosity implicit iteration method for solving a monotone variational inequality with a variational inequality constraint over the common solution set of the general system of variational inequalities (GSVI) (4) for two inverse-strongly monotone mappings and a common fixed point problem (CFPP) of a countable family of uniformly Lipschitzian pseudocontractive mappings and an asymptotically nonexpansive mapping in Hilbert spaces, which is called the triple hierarchical constrained variational inequality (THCVI)

Summary

Introduction

We suppose that H is a real or complex Hilbert space and let H be with inner product h·, ·i and norm k · k. Suppose that A is a nonself mapping from convex nonempty closed set C to entire space H. B2 is a nonself mapping from convex nonempty closed set C to entire space H, respectively. The main aim of this paper is to introduce and analyze a hybrid Mann viscosity implicit iteration method for solving a monotone variational inequality with a variational inequality constraint over the common solution set of the GSVI (4) for two inverse-strongly monotone mappings and a common fixed point problem (CFPP) of a countable family of uniformly Lipschitzian pseudocontractive mappings and an asymptotically nonexpansive mapping in Hilbert spaces, which is called the triple hierarchical constrained variational inequality (THCVI). We list an algorithm to solve problems of common fixed point of pseudocontractive and nonexpansive mappings, classical variational inequalities and generalized mixed equilibrium problems in Hilbert setting

Preliminaries

Main Results

Applications to Finite Generalized Mixed Equilibria