New general systems of set-valued variational inclusions involving relative ( A , η ) -maximal monotone operators in Hilbert spaces

Ting-Jian Xiong,Heng-You Lan

doi:10.1186/1029-242x-2014-407

Abstract

The purpose of this paper is to introduce and study a class of new general systems of set-valued variational inclusions involving relative -maximal monotone operators in Hilbert spaces. By using the generalized resolvent operator technique associated with relative -maximal monotone operators, we also construct some new iterative algorithms for finding approximation solutions to the general systems of set-valued variational inclusions and prove the convergence of the sequences generated by the algorithms. The results presented in this paper improve and extend some known results in the literature.

Highlights

1 Introduction Recently, some systems of variational inequalities, variational inclusions, complementarity problems, and equilibrium problems have been studied by many authors because of their close relations to some problems arising in economics, mechanics, engineering science and other pure and applied sciences
Agarwal and Verma [ ] introduced and studied relative (A, η)-maximal monotone operators and discussed the approximation solvability of a new system of nonlinear variational inclusions involving (A, η)-maximal relaxed monotone and relative (A, η)-maximal monotone operators in Hilbert spaces based on a generalized hybrid iterative algorithm and the general (A, η)-resolvent operator method
By using the generalized resolvent operator technique associated with relative (A, η)-maximal monotone operators, we construct some new iterative algorithms for finding approximation solutions to the general systems of set-valued variational inclusions and prove convergence of the sequences generated by the algorithms

Summary

Introduction

Some systems of variational inequalities, variational inclusions, complementarity problems, and equilibrium problems have been studied by many authors because of their close relations to some problems arising in economics, mechanics, engineering science and other pure and applied sciences. Agarwal and Verma [ ] introduced and studied relative (A, η)-maximal monotone operators and discussed the approximation solvability of a new system of nonlinear (set-valued) variational inclusions involving (A, η)-maximal relaxed monotone and relative (A, η)-maximal monotone operators in Hilbert spaces based on a generalized hybrid iterative algorithm and the general (A, η)-resolvent operator method. By using the generalized resolvent operator technique associated with relative (A, η)-maximal monotone operators, we construct some new iterative algorithms for finding approximation solutions to the general systems of set-valued variational inclusions and prove convergence of the sequences generated by the algorithms. A set-valued operator U : H → H is said to be D-γ -Lipschitz continuous, if there exists a constant γ > such that. + λ RAMii,η,ρii Ai gi xni – ρiFi uni , . . . , unii– , unii, unii+ , . . . , unim – RAMii,η,ρii Ai gi xni – – ρiFi uni – , . . . , unii–– , unii– , unii–+ , . . . , unim– ≤ ( – λ) xni – xni – + λ xni – xni – – gi xni – gi xni –

Ai gi xni

Ai gi x i