Set-Valued Symmetric Generalized Strong Vector Quasi-Equilibrium Problems with Variable Ordering Structures

Jing-Nan Li,San-Hua Wang,Yu-Ping Xu

doi:10.3390/math8091604

Abstract

In this paper, two types of set-valued symmetric generalized strong vector quasi-equilibrium problems with variable ordering structures are discussed. By using the concept of cosmically upper continuity rather than the one of upper semicontinuity for cone-valued mapping, some existence theorems of solutions are established under suitable assumptions of cone-continuity and cone-convexity for the equilibrium mappings. Moreover, the results of compactness for solution sets are proven. As applications, some existence results of strong saddle points are obtained. The main results obtained in this paper unify and improve some recent works in the literature.

Highlights

Throughout this paper, without special statements, we always assume that D and K are two nonempty closed convex subsets of locally convex Hausdorff topological vector spaces X and Y, respectively
The main purpose of this paper is to investigate the existence of solutions for set-valued symmetric generalized strong vector quasi-equilibrium problems with variable ordering structures (VMSVEP)
The greatest difficulty is dealing with the variable ordering structures, which are characterized by a family of cones or a cone-valued mapping

Summary

Introduction

Throughout this paper, without special statements, we always assume that D and K are two nonempty closed convex subsets of locally convex Hausdorff topological vector spaces X and Y, respectively. Let Z1 and Z2 be two real normed vector spaces. Let C : D → 2Z1 and P : K → 2Z2 be two set-valued mappings such that, for any x ∈ D, y ∈ K, C ( x ) and P(y) are two closed, convex, and pointed cones of Z1 and Z2 , respectively. Given set-valued mappings S : D × K → 2D , T : D × K → ( (V MSVEP 1). The second one is to find ( x, ȳ) ∈ D × K such that x ∈ S( x, ȳ), ȳ ∈ T ( x, ȳ) and: (V MSVEP 2). G ( x, ȳ, y) ⊆ G ( x, ȳ, ȳ) + P(ȳ), Special cases: Mathematics 2020, 8, 1604; doi:10.3390/math8091604 www.mdpi.com/journal/mathematics (1) If F and G are single-valued mappings, (VMSVEP 1)reduces to the following symmetric generalized strong vector quasi-equilibrium problem with variable ordering structures (VSVEP): find ( x, ȳ) ∈ D × K such that x ∈ S( x, ȳ), ȳ ∈ T ( x, ȳ) and: (VSVEP)

Objectives

Results

Conclusion