Existence of Solutions for Generalized Vector Quasi-Equilibrium Problems by Scalarization Method with Applications

De-Ning Qu,Cao-Zong Cheng

doi:10.1155/2013/916089

Abstract

The aim of this paper is to study generalized vector quasi-equilibrium problems (GVQEPs) by scalarization method in locally convex topological vector spaces. A general nonlinear scalarization function for set-valued mappings is introduced, its main properties are established, and some results on the existence of solutions of the GVQEPs are shown by utilizing the introduced scalarization function. Finally, a vector variational inclusion problem is discussed as an application of the results of GVQEPs.

Highlights

Various vector equilibrium problems were investigated by adopting many different methods, such as the scalarization method (e.g., [1, 2]), the recession method (e.g., [3]), and duality method (e.g., [4]).The scalarization method is an important and efficacious tool of translating the vector problems into the scalar problems
The aim of this paper is to study generalized vector quasi-equilibrium problems (GVQEPs) by scalarization method in locally convex topological vector spaces
Gerth and Weidner [6] solved a vector optimization problem by introducing a scalarization function with a variable and Gong [1] dealt with vector equilibrium problems by using the scalarization function defined in [6]

Summary

Introduction

Various vector equilibrium problems were investigated by adopting many different methods, such as the scalarization method (e.g., [1, 2]), the recession method (e.g., [3]), and duality method (e.g., [4]). By constructing new nonlinear scalarization functions with two variables, Chen and Yang [7] discussed a vector variational inequality and Chen et al [2] investigated a generalized vector quasi-equilibrium problem (GVQEP), respectively. The authors in [8, 9] studied the systems of vector equilibrium problems by the scalarization method since the gap functions, established by the nonlinear scalarization function defined in [2], were adopted. Every TVS is Hausdorff (see [12])

Preliminaries

A General Nonlinear Scalarization Function

Existence Results on Solutions of the GVQEPs

An Application of GVQEP1: A VVIP