On a Class of Generalized Vector Quasiequilibrium Problems with Set-Valued Maps

This paper considers the following generalized vector quasiequilibrium problem: find a point (z0,x0) of a set E×K such that x0∈A(z0,x0) and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\forall \eta \in A(z_{0},x_{0}),\quad \exists v\in B(z_{0},x_{0},\eta),\quad (F(v,x_{0},\eta),C(v,x_{0},\eta))\in \alpha ,$$\end{document} where α is a subset of 2Y×2Y, A:E×K→2K, B:E×K×K→2E, C:E×K×K→2Y, F:E×K×K→2Y are set-valued maps and Y is a topological vector space. Existence theorems are established under suitable assumptions, one of which is the requirement of the openness of the lower sections of some set-valued maps which can be satisfied with maps B,C, F being discontinuous. It is shown that, in some special cases, this requirement can be verified easily by using the semicontinuity property of these maps. Another assumption in the obtained existence theorems is assured by appropriate notions of diagonal quasiconvexity.