Convergence Analysis of a Three-Step Iterative Algorithm for Generalized Set-Valued Mixed-Ordered Variational Inclusion Problem

Praveen Agarwal,M Dilshad,Doaa Filali,M Akram

doi:10.3390/sym13030444

Abstract

This manuscript aims to study a generalized, set-valued, mixed-ordered, variational inclusion problem involving H(·,·)-compression XOR-αM-non-ordinary difference mapping and relaxed cocoercive mapping in real-ordered Hilbert spaces. The resolvent operator associated with H(·,·)-compression XOR-αM-non-ordinary difference mapping is defined, and some of its characteristics are discussed. We prove existence and uniqueness results for the considered generalized, set-valued, mixed-ordered, variational inclusion problem. Further, we put forward a three-step iterative algorithm using a ⊕ operator, and analyze the convergence of the suggested iterative algorithm under some mild assumptions. Finally, we reconfirm the existence and convergence results by an illustrative numerical example.

Highlights

The theory of variational inequalities was studied in the early 1960s to solve a problem which appeared in a mechanical system
Set-valued, mixed-ordered, variational inclusion with H(·, ·)-compression XOR-αM-non-ordinary difference mapping and relaxed cocoercive mapping in real ordered Hilbert spaces
This section begins with the designing of a generalized ordered variational inclusion problem involving H(·, ·)-compression XOR-αM-non-ordinary difference mapping and relaxed cocoercive mapping

Summary

Introduction

The theory of variational inequalities was studied in the early 1960s to solve a problem which appeared in a mechanical system. In 1994, Hassouni and Moudafi [16] evolved a class of mixed-type variational inequalities with single-valued mappings using the technique of a resolvent operator for monotone mapping, namely- variational inclusion problem. They developed a perturbed algorithm to estimate the solution of mixed variational inequalities. Li et al [25] presented the convergence of an Ishikawa-type iterative method for the general nonlinear ordered variational inclusion with (γG, λ)-weak-GRD set-valued mappings, and exhibited the stability of the algorithm. A numerical example is given to show that the considered three-step iterative algorithm converges to the unique solution of a generalized, set-valued, mixed-ordered, variational inclusion

Preliminaries and Auxiliary Results

Formulation of the Problem and Existence Result

Convergence Analysis

Numerical Example

Conclusions