Generalized Vector Quasivariational Inclusion Problems with Moving Cones

This paper deals with the generalized vector quasivariational inclusion Problem (P1) (resp. Problem (P2)) of finding a point (z0,x0) of a set E×K such that (z0,x0)∈B(z0,x0)×A(z0,x0) and, for all η∈A(z0,x0), \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{array}{l}F(z_0,x_0,\eta)\subset G(z_0,x_0,x_0)+C(z_0,x_0)\cr \mathrm{[resp.}F(z_0,x_0,x_0)\subset G(z_0,x_0,\eta)+C(z_0,x_0)],\end{array}$$\end{document} where A:E×K→2K, B:E×K→2E, C:E×K→2Y, F,G:E×K×K→2Y are some set-valued maps and Y is a topological vector space. The nonemptiness and compactness of the solution sets of Problems (P1) and (P2) are established under the verifiable assumption that the graph of the moving cone C is closed and that the set-valued maps F and G are C-semicontinuous in a new sense (weaker than the usual sense of semicontinuity).