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Let E be a topological vector space and X be a non-empty subset of E. Let S : X → 2X and T : X → 2E be two maps. Then the generalized quasi-variational inequality (GQVI) problem is to find a point ŷ ∈ S(ŷ) and a point ŵ ∈ T (ŷ) such that Re〈ŵ, ŷ − x〉 ≤ 0 for all x ∈ S(ŷ). 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 nonempty finite subset A of X.

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