Generalized vector variational inequalities with star-pseudomonotone and discontinuous operators

Abstract

Abstract. In this paper we deal with the following generalized vector variational inequality problem: let X, Y and Z be topological vector spaces, K be a convex set in X, D be a nonempty set in Y , and C be a closed convex cone in Z. Let T : K → 2 D be a multifunction andWe prove some existence theorems in which T can be discontinuous, or K can be unbounded, and a existence theorem in which T is pseudomonotone.

Intersection theorems for generalized weak KKM set‐valued mappings with applications in optimization

“…Problem (SVRP) has been investigated in [6] and, as is mentioned there, it contains as special cases several variational inequalities and equilibrium problems studied, for instance, in [11, 16 19, 21, 22] and [31].…”

New existence theorems for vector equilibrium problems with set-valued mappings

“…(Weak) Vector equilibrium problem is to findx ∈ X 0 such that f (x, y) ∈ −S\{θ }(−intS), ∀y ∈ X 0 , where θ is the zero element of V and f : X 0 × X 0 → V is a vector-valued mapping. This problem provides a unified framework of many important problems, such as, vector optimization problems, vector saddle point problems, vector non-zero sum game problems and vector variational inequality problems and has wide applications in mathematical economics and operational research; see, e.g., [1,2,3,4,5,6] and the references therein. In view of these important applications, a lot of authors have devoted themselves to the study of existence results of vector equilibrium problems and weak vector equilibrium problems.…”

“…Under convex and nonconvex domains, Gong [5] investigated existence problems for weak vector equilibrium problems. Kien, Wong and Yao [6] investigated existence results for variational inequality problems under some pseudomonotone and discontinuity assumptions.…”

General set-valued vector variational inequality problems

“…In this pape, inspired by the works [1,4,5,9,11,14,17,18,21,22,23,24,25], we give some existence results for the solutions of the problems (2)-(4) and also discuss the coincidence point results.…”