A generalised quasi-variational inequality without upper semicontinuity

In this note we deal with the following problem: given a nonempty closed convex subset X of R(n) and two multifunctions Gamma:X --> 2(X) and Phi:X --> 2(Rn), to find ((x) over cap, (z) over cap), is an element of X x R(n) such that (x) over cap is an element of Gamma (x) over cap, (z) over cap is an element of Phi((x) over cap) and (z) over cap, (x) over cap - y) less than or equal to 0 for all y is an element of Gamma((x) over cap. We prove a very general existence result where neither Gamma nor Phi are assumed to be upper semicontinuous. In particular, our result give a positive answer to an open problem raised by the first author recently.