On existence of a solution for the system of generalized vector quasi-equilibrium problems with upper semicontinuous set-valued maps

Throughout this paper, we use intA and CoA to denote the interior and the convex hull of a set A, respectively. Let I be an index set. For each i ∈ I , let Yi, Ei be two Hausdorff topological vector spaces. Consider a family of nonempty convex subsets {Xi}i∈I with Xi ⊆ Ei. Let X = ∏ i∈I Xi and E = ∏ i∈I Ei. An element of the set Xi = ∏ j∈I\i Xi will be denoted by xi; therefore, x ∈ X will be written as x = (x,xi) ∈ Xi×Xi. For each i ∈ I , let Di : X → 2Xi and Fi : X ×Xi → 2Yi be two set-valued maps, let Ci : X → 2Yi be a set-valued map such that Ci(x) is a convex, pointed, and closed cone with intCi(x) = ∅ for all x ∈ X . Then the system of generalized vector quasi-equilibrium problems with set-valued maps (in short, SGVQEP) is to find x̄ = (x̄i, x̄i) in X such that for each i∈ I , x̄i ∈Di(x̄), Fi ( x̄, yi ) ⊆ − intCi(x̄), ∀yi ∈Di(x̄). (1.1)