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Suppose that E is a topological vector space and X is a non-empty subset of E. Let S: X --> 2(X) and T:X --> 2(E)* be two maps. Then the generalized quasi-variational inequality problem (GQVI) is to find a point (y) over cap is an element of S((y) over cap) and a point (w) over cap is an element of T((y) over cap) such that Re[(w) over cap, (y) over cap - x] less than or equal to 0 far all x is an element of S((y) over cap). We shall use Chowdhury and Tan's generalized version [4] of Ky Fan's minimax inequality [7] as a tool to obtain some general theorems on solutions of the GQVI in locally convex Hausdorff topological vector spaces. We obtain the existence theorems of GQVI on paracompact sets X where the set-valued operators T are demi operators [3] and are upper hemi-continuous [5] along line segments in X to the weak * -topology on E*.

Existence theorems of generalized quasi-variational inequalities with upper hemi-continuous and demi operators on non-compact sets