This paper deals with a class of
functions which are defined in an n-dimensional rectangle and which possess there,
only the generalized partial derivatives of mixed type. It is shown that (i) this class
contains as a proper subset the usual Sobolev class of order n, the dimeniion of the
domain and (ii) this class can be imbedded in the space of continuous functions.
In addition to the compactness of the imbedding operator, the closedness
of certain nonlinear partial integro differential operators is also studied.
Finally, a system of partial integro differential equations with Darboux type
boundary data in a rectangle, is shown to have solutions in this class. The
results of this paper are used in certain existence theorems of optimal control
theory.
