Applications of pseudo-monotone operators with some kind of upper semicontinuity in generalized quasi-variational inequalities on non-compact sets

Abstract:

Let $E$ be a topological vector space and $X$ be a non-empty subset of $E$. Let $S:X\rightarrow 2^{X}$ and $T:X\rightarrow 2^{E^{*}}$ be two maps. Then the generalized quasi-variational inequality (GQVI) problem is to find a point $\hat y\in S(\hat y)$ and a point $\hat w\in T(\hat y)$ such that $Re\langle \hat w,\hat y-x\rangle \leq 0$ for all $x\in S(\hat y)$. We shall use Chowdhury and Tan’s 1996 generalized version of Ky Fan’s minimax inequality as a tool to obtain some general theorems on solutions of the GQVI on a paracompact set $X$ in a Hausdorff locally convex space where the set-valued operator $T$ is either strongly pseudo-monotone or pseudo-monotone and is upper semicontinuous from $co(A)$ to the weak$^{*}$-topology on $E^{*}$ for each non-empty finite subset $A$ of $X$.