Iterative Algorithms for New General Systems of Set-Valued Variational Inclusions Involving(A,η)-Maximal Relaxed Monotone Operators

Ting-Jian Xiong,Heng-You Lan

doi:10.1155/2014/698593

Abstract

We introduce and study a class of new general systems of set-valued variational inclusions involving(A,η)-maximal relaxed monotone operators in Hilbert spaces. By using the general resolvent operator technique associated with(A,η)-maximal relaxed monotone operators, we construct some new iterative algorithms for finding approximation solutions to the general system of set-valued variational inclusion problem and prove the convergence of this algorithm. Our results improve and extend some known results.

Highlights

It is well known that variational inequalities and variational inclusions, which have been extended and generalized in different directions by using novel and innovative techniques and ideas, provide mathematical models to some problems arising in economics, mechanics, engineering science, and other pure and applied sciences
We introduce and study a class of new general systems of set-valued variational inclusions involving (A, η)-maximal relaxed monotone operators in Hilbert spaces
Fang et al [1], Yan et al [2], Fang and Huang [3], and Cao [4] considered some new systems of variational inclusions involving H-monotone operators and (H, η)monotone operators in Hilbert space, respectively

Summary

Introduction

It is well known that variational inequalities and variational inclusions, which have been extended and generalized in different directions by using novel and innovative techniques and ideas, provide mathematical models to some problems arising in economics, mechanics, engineering science, and other pure and applied sciences. By using the concept and properties of A-monotone operators and the resolvent operator technique associated with A-monotone operators due to Verma [8], the author constructed a new iterative algorithm for solving this system of nonlinear multivalued variational inclusions associated with A-monotone operators in Hilbert spaces and proved the existence of solutions for the nonlinear multivalued variational inclusion systems and the convergence of iterative sequences generated by the algorithm. Inspired and motivated by the above works, the purpose of this paper is to consider the following new general system of set-valued variational inclusions involving relative (A, η)-maximal monotone operators in Hilbert spaces: find In this paper, we will construct some new iterative algorithms to approximate the solution of the general system of set-valued variational inclusions and prove the convergence of the sequences generated by the algorithms in Hilbert spaces

Preliminaries

Iterative Algorithm and Convergence