General System of -Monotone Nonlinear Variational Inclusions Problems with Applications

Jian-Wen Peng,Lai-Jun Zhao

doi:10.1155/2009/364615

Abstract

We introduce and study a new system of nonlinear variational inclusions involving a combination of -Monotone operators and relaxed cocoercive mappings. By using the resolvent technique of the -monotone operators, we prove the existence and uniqueness of solution and the convergence of a new multistep iterative algorithm for this system of variational inclusions. The results in this paper unify, extend, and improve some known results in literature.

Highlights

Fang and Huang 1 introduced a new class of H-monotone mappings in the context of solving a system of variational inclusions involving a combianation of Hmonotone and strongly monotone mappings based on the resolvent operator techniques
Verma 2 introduced the notion of A-monotone mappings and its applications to the solvability of a system of variational inclusions involving a combination of A-monotone and strongly monotone mappings
On the top of that, A-monotonicity originates from hemivariational inequalities, and emerges as a major contributor to the solvability of nonlinear variational problems on nonconvex settings.” and as a matter of fact, some nice examples on A-monotone or generalized maximal monotone mappings can be found in Naniewicz and Panagiotopoulos 3 and Verma 4

Summary

Introduction

Fang and Huang 1 introduced a new class of H-monotone mappings in the context of solving a system of variational inclusions involving a combianation of Hmonotone and strongly monotone mappings based on the resolvent operator techniques. Verma 2 introduced the notion of A-monotone mappings and its applications to the solvability of a system of variational inclusions involving a combination of A-monotone and strongly monotone mappings. Verma 6 studied the solvability of a system of variational inclusions involving a combination of A-monotone and relaxed cocoercive mappings using resolvent operator techniques of A-monotone mappings. Inspired and motivated by recent works in 1, 2, 6 , the purpose of this paper is to introduce a new mathematical model, which is called a general system of A-monotone nonlinear variational inclusion problems, that is, a family of A-monotone nonlinear variational inclusion problems defined on a product set. The result in this paper unifies, extends, and improves some results in [1, 2, 6,7,8] and the references therein

Preliminaries

A System of Set-Valued Variational Inclusions

Existence of Solutions and Convergence of an Iterative Algorithm

Iterative Algorithm and Convergence