On System of Generalized Vector Quasiequilibrium Problems with Applications

Jian-Wen Peng,Lun Wan

doi:10.1155/2009/430930

Abstract

We introduce a new system of generalized vector quasiequilibrium problems which includes system of vector quasiequilibrium problems, system of vector equilibrium problems, and vector equilibrium problems, and so forth in literature as special cases. We prove the existence of solutions for this system of generalized vector quasi-equilibrium problems. Consequently, we derive some existence results of a solution for the system of generalized quasi-equilibrium problems and the generalized Debreu-type equilibrium problem for both vector-valued functions and scalar-valued functions.

Highlights

Introduction and FormulationsIn the resent years, the vector equilibrium problems have been studied in 1–7 and the references therein which is a unified model of several problems, for instance, vector variational inequality, vector variational-like inequality, vector complementarity problems, vector optimization problems
We introduce a new system of generalized vector quasiequilibrium problems which includes system of vector quasiequilibrium problems, system of vector equilibrium problems, and vector equilibrium problems, and so forth in literature as special cases
We prove the existence of solutions for this system of generalized vector quasi-equilibrium problems

Summary

Introduction and Formulations

The vector equilibrium problems have been studied in 1–7 and the references therein which is a unified model of several problems, for instance, vector variational inequality, vector variational-like inequality, vector complementarity problems, vector optimization problems. We define a function φi : X × Xi → Zi and a function hi : X × Y → Zi as φi x, zi fi x, y, zi , for all x, zi ∈ X × Xi, and hi x φi x, y , for all x ∈ X, SGVQEP and G-Debreu VEP , respectively, reduce to the system of vector quasi-equilibrium problems and the Debreu VEP introduced by Ansari et al which contain those mathematical in 18, as special cases.

Preliminaries

Existence Results

Applications