System of Extended General Variational Inequalities for Relaxed Cocoercive Mappings in Hilbert Space

Kyung Kim

doi:10.3390/math6100198

Abstract

In this manuscript, we study a system of extended general variational inequalities (SEGVI) with several nonlinear operators, more precisely, six relaxed ( α , r ) -cocoercive mappings. Using the projection method, we show that a system of extended general variational inequalities is equivalent to the nonlinear projection equations. This alternative equivalent problem is used to consider the existence and convergence (or approximate solvability) of a solution of a system of extended general variational inequalities under suitable conditions.

Highlights

In recent years, many theories of variational inequality types and its special forms have been extended and generalized to research a variety of applications and problems arising from several fields such as applied mathematics, optimization, control theory, equilibrium problems and nonlinear programming problems, etc
In 2016, Noor [2] introduced and researched the existence of solution by using fixed point theory for a system of extended general variational inequalities with six strongly monotone operators. We intend in this manuscript to consider a system of extended general variational inequalities with nonlinear operators, more precisely, relaxed cocoercive operators which are more generalized than strongly monotone operators
We show that a system of extended general variational inequalities include general variational inequality and several other classes of variational inequalities as special cases

Summary

Introduction

Many theories of variational inequality types and its special forms have been extended and generalized to research a variety of applications and problems arising from several fields such as applied mathematics, optimization, control theory, equilibrium problems and nonlinear programming problems, etc. In 2016, Noor [2] introduced and researched the existence of solution by using fixed point theory for a system of extended general variational inequalities with six strongly monotone operators. We intend in this manuscript to consider a system of extended general variational inequalities with nonlinear operators, more precisely, relaxed cocoercive operators which are more generalized than strongly monotone operators. It is shown that a system of extended general variational inequalities (SEGVI) are equivalent to the nonlinear projection equations. This alternative equivalent problem is used to consider the existence and convergence of a solution of a system of extended general variational inequalities under appropriate conditions

Preliminaries

Results

Conclusions