Composite extragradient implicit rule for solving a hierarch variational inequality with constraints of variational inclusion and fixed point problems

Lu-Chuan Ceng,Meijuan Shang

doi:10.1186/s13660-020-2306-1

Abstract

Let X be a uniformly convex and q-uniformly smooth Banach space with 1< qleq 2. In the framework of this space, we are concerned with a composite gradient-like implicit rule for solving a hierarchical monotone variational inequality with the constraints of a system of monotone variational inequalities, a variational inclusion and a common fixed point problem of a countable family of nonlinear operators {S_{n}}^{infty }_{n=0}. Our rule is based on the Korpelevich extragradient method, the perturbation mapping, and the W-mappings constructed by {S_{n}}^{infty }_{n=0}.

Highlights

Throughout this work, one always supposes that C is a nonempty convex set in a Banach space X whose dual is denoted by X∗
This common problem is called the convex feasibility problem, which can be characterized via the following model: x ∈ i∈I Ci, where I denotes some index set, Ci is a convex set in X
We study, in the framework of Banach spaces, a convex feasibility problem with the constraints of the generalized system of monotone variational inequalities, a variational inclusion and a countable family of nonexpansive operators

Summary

Introduction

Throughout this work, one always supposes that C is a nonempty convex set in a Banach space X whose dual is denoted by X∗. Lemma 2.1 ([25, 26]) Suppose that {Sn}∞ n=0 is a countable family of nonexpansive mappings defined on a subset C of a strictly convex space X.

Results

Conclusion