Generalized vector quasi-equilibrium problems with set-valued mappings

Jian Wen Peng,Dao Li Zhu

doi:10.1155/jia/2006/69252

Abstract

A new mathematical model of generalized vector quasiequilibrium problem with set-valued mappings is introduced, and several existence results of a solution for the generalized vector quasiequilibrium problem with and without -condensing mapping are shown. The results in this paper extend and unify those results in the literature.

Highlights

Throughout this paper, let Z, E, and F be topological vector spaces, let X ⊆ E and Y ⊆ F be nonempty, closed, and convex subsets
(i) If Ψ is replaced by a single-valued function f : X × Y × X → Z and C(x) = C for all x ∈ X, the generalized vector quasi-equilibrium problem with set-valued mappings (GVQEP) reduces to finding (x, y) in X × Y such that x ∈ D(x), y ∈ T(x), f (x, y, z) ∈/ − int C, ∀z ∈ D(x)
If Ψ is replaced by a scalar function f : X × Y × X → R and C(x) = {r ∈ R : r ≥ 0} for all x ∈ X, the GVQEP reduces to finding (x, y) in X × Y such that x ∈ D(x), y ∈ T(x), f (x, y, z) ≥ 0, ∀z ∈ D(x)

Summary

Introduction

Throughout this paper, let Z, E, and F be topological vector spaces, let X ⊆ E and Y ⊆ F be nonempty, closed, and convex subsets. (i) If Ψ is replaced by a single-valued function f : X × Y × X → Z and C(x) = C for all x ∈ X, the GVQEP reduces to finding (x, y) in X × Y such that x ∈ D(x), y ∈ T(x), f (x, y, z) ∈/ − int C, ∀z ∈ D(x). It was investigated by Chiang et al in [7].

Generalized vector quasi-equilibrium problem

Preliminaries

Existence results