  This paper is concerned with the weak solvability of fully nonlinear
parabolic variational inequalities with time dependent convex constraints. As
possible approaches to such problems, there are for instance the
time-discretization method and the fixed point method of Schauder type with
appropriate compactness theorems. In this paper, our attention is paid to the
latter approach. However, there has not been prepared any appropriate
compactness theorem up to date that enables us the direct application of fixed
point method to variational inequalities of parabolic type. In order to
establish it we have to start on the set up of a new compactness theorem for a
wide class of parabolic variational inequalities.
