Arcwise Connectedness of the Solution Sets for Generalized Vector Equilibrium Problems

Qiuying Li,Sanhua Wang

doi:10.3390/math9202532

Abstract

In this research, by means of the scalarization method, arcwise connectedness results were established for the sets of globally efficient solutions, weakly efficient solutions, Henig efficient solutions and superefficient solutions for the generalized vector equilibrium problem under suitable assumptions of natural quasi cone-convexity and natural quasi cone-concavity.

Highlights

By applying the scalarization technique, Gong [6] discussed the connectedness of the sets of Henig efficient solutions and weakly efficient solutions for vector equilibrium problems in normed spaces
Han and Huang [9] extended the density method used in [8] to investigate the connectedness of the efficient solution set for the generalized vector quasi-equilibrium problem
We prove a conclusion of lower semicontinuity in regard to the f -efficient solution mapping

Summary

Introduction

Let K be a nonempty subset of a Banach space X. By applying the scalarization technique, Gong [6] discussed the connectedness of the sets of Henig efficient solutions and weakly efficient solutions for vector equilibrium problems in normed spaces. Han and Huang [9] extended the density method used in [8] to investigate the connectedness of the efficient solution set for the generalized vector quasi-equilibrium problem. By virtue of a density result, Peng et al [13] proved some connectedness theorems for the set of approximate efficient points and the set of approximate efficient solutions to generalized semi-infinite vector optimization problems. We prove a conclusion of lower semicontinuity in regard to the f -efficient solution mapping All of this helps us to establish the arcwise connectedness of the efficient solution sets for (GVEP). Examples are given to illustrate our main results of arcwise connectedness

Preliminaries

Arcwise Connectedness

Conclusions