A New System of Generalized Mixed Quasivariational Inclusions with Relaxed Cocoercive Operators and Applications

Zhongping Wan,Jia-Wei Chen,Liuyang Yuan,Hai Sun

doi:10.1155/2011/961038

Zhongping Wan, Jia-Wei Chen + Show 2 more

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https://doi.org/10.1155/2011/961038

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Abstract

A new system of generalized mixed quasivariational inclusions (for short, SGMQVI) with relaxed cocoercive operators, which develop some preexisting variational inequalities, is introduced and investigated in Banach spaces. Next, the existence and uniqueness of solutions to the problem (SGMQVI) are established in real Banach spaces. From fixed point perspective, we propose some new iterative algorithms for solving the system of generalized mixed quasivariational inclusion problem (SGMQVI). Moreover, strong convergence theorems of these iterative sequences generated by the corresponding algorithms are proved under suitable conditions. As an application, the strong convergence theorem for a class of bilevel variational inequalities is derived in Hilbert space. The main results in this paper develop, improve, and unify some well‐known results in the literature.

Highlights

Generalized mixed quasivariational inclusion problems, which are extensions of variational inequalities introduced by Stampacchia 1 in the early sixties, are among the most interesting and extensively investigated classes of mathematics problems and have many applications in the fields of optimization and control, abstract economics, electrical networks, game theory, auction, engineering science, and transportation equilibria see, e.g., 2–5 and the references therein
The aim of this paper is to introduce and study a new system of generalized mixed quasivariational inclusion problem SGMQVI in uniformly smooth Banach spaces which includes some previous variational inequalities as special cases
We study the properties for the lower-level variational inequalities of a class of bilevel variational inequalities for short, BVI in Hilbert spaces and suggest a reasonable iterative algorithm for BVI

Summary

Introduction

Generalized mixed quasivariational inclusion problems, which are extensions of variational inequalities introduced by Stampacchia 1 in the early sixties, are among the most interesting and extensively investigated classes of mathematics problems and have many applications in the fields of optimization and control, abstract economics, electrical networks, game theory, auction, engineering science, and transportation equilibria see, e.g., 2–5 and the references therein. In , Kamraksa and Wangkeeree gave a general iterative method for a general system of variational inclusions and proved a strong convergence theorem in strictly convex and 2 uniformly smooth Banach spaces. Wangkeeree and Kamraksa introduced an iterative algorithm for finding a common element of the set of solutions of a mixed equilibrium problem, the set of fixed points of an infinite family of nonexpansive mappings and the set of solutions of a general system of variational inequalities and obtained the strong convergence of the iterative in Hilbert spaces. The aim of this paper is to introduce and study a new system of generalized mixed quasivariational inclusion problem SGMQVI in uniformly smooth Banach spaces which includes some previous variational inequalities as special cases. The results presented in this paper improve, develop, and extend the results of 8, 23, 24, 27

Preliminaries

Existence Theorems

Convergence Analysis

An Application