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

Abstract

Let E E be a topological vector space and X X be a non-empty subset of E E . Let S : X → 2 X S:X\rightarrow 2^{X} and T : X → 2 E ∗ T:X\rightarrow 2^{E^{*}} be two maps. Then the generalized quasi-variational inequality (GQVI) problem is to find a point y ^ ∈ S ( y ^ ) \hat y\in S(\hat y) and a point w ^ ∈ T ( y ^ ) \hat w\in T(\hat y) such that R e ⟨ w ^ , y ^ − x ⟩ ≤ 0 Re\langle \hat w,\hat y-x\rangle \leq 0 for all x ∈ S ( y ^ ) 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 X in a Hausdorff locally convex space where the set-valued operator T T is either strongly pseudo-monotone or pseudo-monotone and is upper semicontinuous from c o ( A ) co(A) to the weak ∗ ^{*} -topology on E ∗ E^{*} for each non-empty finite subset A A of X X .