ON THE SOLUTION EXISTENCE OF GENERALIZED VECTOR QUASI-EQUILIBRIUM PROBLEMS WITH DISCONTINUOUS MULTIFUNCTIONS

Abstract

In this paper we deal with the following generalized vector quasi-equilibrium problem: given a closed convex set $K$ in a normed space $X$, a subset $D$ in a Hausdorff topological vector space $Y$, and a closed convex cone $C$ in $R^n$. Let $\Gamma: K\to 2^K$, $\Phi : K\rightarrow 2^{D}$ be two multifunctions and $f : K\times D\times K\to R^n$ be a single-valued mapping. Find a point $(\hat x, \hat y)\in K\times D$ such that \begin{gather} (\hat x, \hat y)\in \Gamma(\hat x)\times\Phi(\hat x),\,\, {\rm and}\,\, \{f(\hat x, \hat y, z): z\in\Gamma(\hat x)\}\cap (-{\rm Int }C)=\emptyset. \notag \end{gather} We prove some existence theorems for the problem in which $\Phi$ can be discontinuous and $K$ can be unbounded.