We study the following generalized quasivariational inequality problem: given a closed convex set <i>X</i> in a normed space <i>E</i> with the dual <i>E</i> <sup>*</sup>, a multifunction \Phi :Xightarrow 2^{E^{*}} and a multifunction :<i>X</i>2<sup> <i>X</i> </sup>, find a point \Phi :Xightarrow 2^{E^{*}} such that \Phi :Xightarrow 2^{E^{*}} , \Phi :Xightarrow 2^{E^{*}} . We prove some existence theorems in which may be discontinuous, <i>X</i> may be unbounded, and is not assumed to be Hausdorff lower semicontinuous.