Generalized quasi-variational inequalities without continuities

Given a nonempty set X subset of or equal to R(n) and two multifunctions beta:X --> 2(X), phi:X --> 2(Rn), we consider the following generalized quasivariational inequality problem associated with X, beta, phi: Find ((x) over bar, (z) over bar) is an element of X x R(n) such that (x) over bar is an element of beta ((x) over bar), (z) over bar is an element of phi ((x) over bar), and sup(y is an element of beta((x) over bar)) [(z) over bar, (x) over bar - y] less than or equal to 0. We prove several existence results in which the multifunction phi is not supposed to have any continuity property. Among others, we extend the results obtained in Ref. 1 for the case beta(x) = X.