Modified subgradient extragradient method for system of variational inclusion problem and finite family of variational inequalities problem in real Hilbert space

Araya Kheawborisut,Atid Kangtunyakarn

doi:10.1186/s13660-021-02583-1

Abstract

For the purpose of this article, we introduce a modified form of a generalized system of variational inclusions, called the generalized system of modified variational inclusion problems (GSMVIP). This problem reduces to the classical variational inclusion and variational inequalities problems. Motivated by several recent results related to the subgradient extragradient method, we propose a new subgradient extragradient method for finding a common element of the set of solutions of GSMVIP and the set of a finite family of variational inequalities problems. Under suitable assumptions, strong convergence theorems have been proved in the framework of a Hilbert space. In addition, some numerical results indicate that the proposed method is effective.

Highlights

Throughout this paper, let H be a real Hilbert space and C be a nonempty closed convex subset of H with the inner product ·, · and norm ·
3 Main result we prove the strong convergence of the sequence acquired from the proposed iterative methods for finding a common element of the set of finite family variational inequalities problems and the set of solutions of the proposed problem
6 Conclusion In this paper, we have proposed a new problem, called a generalized system of modified variational inclusion problems (GSMVIP)

Summary

Introduction

Throughout this paper, let H be a real Hilbert space and C be a nonempty closed convex subset of H with the inner product ·, · and norm ·. Lemma 2.4 ([14]) Let C be a nonempty closed convex subset of a real Hilbert space H and let A,B: C → H be α- and β-inverse strongly monotone mappings, respectively, with α, β > 0 and VI(C, A) ∩ VI(C, B) = ∅. Lemma 2.7 ([17]) Let C be a nonempty closed and convex subset of a real Hilbert space H. Lemma 2.11 Let H be a real Hilbert space and let AG : H → H be an α-inverse strongly monotone mapping. Let y ∈ , we have θ ∈ AGy + MAy and θ ∈ AGy + MBy. From Lemma 2.10, it implies that y ∈ F(JMA,λA (I – λAAG)) ∩ F(JMB,λB (I – λBAG)).

Application

Conclusion