Stability and Convergence Analysis for Set-Valued Extended Generalized Nonlinear Mixed Variational Inequality Problems and Generalized Resolvent Dynamical Systems

Iqbal Ahmad,Zahoor Ahmad Rather,Ching-Feng Wen,Rais Ahmad,Efthymios G Tsionas

doi:10.1155/2021/5573833

Abstract

In this paper, we study a set-valued extended generalized nonlinear mixed variational inequality problem and its generalized resolvent dynamical system. A three-step iterative algorithm is constructed for solving set-valued extended generalized nonlinear variational inequality problem. Convergence and stability analysis are also discussed. We have shown the globally exponential convergence of generalized resolvent dynamical system to a unique solution of set-valued extended generalized nonlinear mixed variational inequality problem. In support of our main result, we provide a numerical example with convergence graphs and computation tables. For illustration, a comparison of our three-step iterative algorithm with Ishikawa-type algorithm and Mann-type algorithm is shown.

Highlights

Variational inequality theory was introduced by Hartmann and Stampacchia [1] in 1966 as a tool for the study of partial differential equations with applications principally drawn from mechanics
It is well known that the subdifferential of a proper, convex, and lower-semicontinuous functional is a maximal monotone operator. is characterization enables researchers to define the resolvent operator associated with the maximal monotone operator; see, for example, [7,8,9] and the references therein. e resolvent operator technique is used to establish the equivalence between the variational inequalities and fixed-point problems; see, for example, [10,11,12] and the references therein
We introduce and study a set-valued extended generalized nonlinear mixed variational inequality problem including many existing problems studied by several authors

Summary

Introduction

Variational inequality theory was introduced by Hartmann and Stampacchia [1] in 1966 as a tool for the study of partial differential equations with applications principally drawn from mechanics. We would like to emphasize that the projection method cannot be applied for solving the mixed variational inequalities involving the nonlinear term. E resolvent operator technique is used to establish the equivalence between the variational inequalities and fixed-point problems; see, for example, [10,11,12] and the references therein. Glowinski and Tallec [17] suggested and analyzed some three-step splitting methods for solving variational inequality problems by using the Lagrange multipliers technique. Ey studied the convergence of these splitting methods under the assumption that the underlying operator is monotone and Lipschitz continuous. We introduce and study a set-valued extended generalized nonlinear mixed variational inequality problem including many existing problems studied by several authors. (iv) T is said to be strongly monotone with respect to g, if there exists a constant λT > 0 such that

Preliminaries

Formulation of the Problem and Existence of Solution

Convergence and Stability Analysis

Generalized Resolvent Dynamical System

Numerical Example