The Solvability of a New System of Nonlinear Variational-Like Inclusions

Zeqing Liu,Jeong Sheok Ume,Min Liu,Shin Min Kang

doi:10.1155/2009/609353

Abstract

We introduce and study a new system of nonlinear variational-like inclusions involving - -maximal monotone operators, strongly monotone operators, -strongly monotone operators, relaxed monotone operators, cocoercive operators, -relaxed cocoercive operators, - -relaxed cocoercive operators and relaxed Lipschitz operators in Hilbert spaces. By using the resolvent operator technique associated with - -maximal monotone operators and Banach contraction principle, we demonstrate the existence and uniqueness of solution for the system of nonlinear variational-like inclusions. The results presented in the paper improve and extend some known results in the literature.

Highlights

It is well known that the resolvent operator technique is an important method for solving various variational inequalities and inclusions 1–20
Lan 10 studied a system of general mixed quasivariational inclusions involving A, η -accretive mappings in q-uniformly smooth Banach spaces
Lan 11 analyzed and established an existence theorem for nonlinear parametric multivalued variational inclusion systems involving A, η -accretive mappings in Banach spaces

Summary

Introduction

It is well known that the resolvent operator technique is an important method for solving various variational inequalities and inclusions 1–20. Lan 10 studied a system of general mixed quasivariational inclusions involving A, η -accretive mappings in q-uniformly smooth Banach spaces. Lan 11 analyzed and established an existence theorem for nonlinear parametric multivalued variational inclusion systems involving A, η -accretive mappings in Banach spaces. By using the random resolvent operator technique associated with A, η accretive mappings, Lan 13 established an existence result for nonlinear random multivalued variational inclusion systems involving A, η -accretive mappings in Banach spaces. Lan and Verma 15 studied a class of nonlinear Fuzzy variational inclusion systems with A, η -accretive mappings in Banach spaces. By virtue of the Banach’s fixed point theorem and the resolvent operator technique, we prove the existence and uniqueness of solution for the system of nonlinear variational-like inclusions. The results presented in the paper generalize some known results in the field

Preliminaries

Existence and Uniqueness Theorems