Abstract
In this paper, several kinds of generalized vector quasi-equilibrium problems are introduced and studied in abstract convex spaces. Using the properties of Γ-convex and \(\mathfrak{KC}\)-maps, some sufficient conditions are given to guarantee the existence of solutions in connection with these generalized vector quasi-equilibrium problems. As applications, some existence theorems of solutions for the generalized semi-infinite programs with vector quasi-equilibrium constraints are also given.
Highlights
It is well known that the vector quasi-equilibrium problem is an important generalization of the vector equilibrium problem which provides a unified model for vector quasi-variational inequalities, vector quasi-complementarity problems, vector optimization problems and vector saddle point problems
It is important and interesting to study the existence of solutions concerned with some generalized semiinfinite programs with vector quasi-equilibrium constraints in abstract convex spaces
The main purpose of this paper is to study several classes of generalized vector quasiequilibrium problems in abstract convex spaces with applications to generalized semiinfinite programs
Summary
It is well known that the vector quasi-equilibrium problem is an important generalization of the vector equilibrium problem which provides a unified model for vector quasi-variational inequalities, vector quasi-complementarity problems, vector optimization problems and vector saddle point problems. Yang and Huang [ ] studied the existence of solutions for the generalized vector equilibrium problems in abstract convex spaces. It is important and interesting to study the existence of solutions concerned with some generalized semiinfinite programs with vector quasi-equilibrium constraints in abstract convex spaces. We will consider the following generalized vector quasi-equilibrium problems in abstract convex spaces. Assume that h : X ⇒ L is a set-valued mapping, where L is a real topological vector space ordered by a closed convex pointed cone H ⊆ L with int H = ∅. ([ ]) Assume that A is a nonempty compact subset of a real topological vector space V and D is a closed convex cone in V with D = V. There exists x ∈ E such that x ∈ S(x) and F(x, y, z) ⊂ C(x) for all y ∈ S(x) and z ∈ B(x)
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