Linear equations of the Sobolev type with integral delay operator

We establish sufficient conditions for the local and global solvability of initial problems for a class of linear operator-differential equations of the first order in a Banach space. Equations are assumed to have a degenerate operator at the derivative and an integral delay operator. We apply methods of the theory of degenerate semigroups of operators and the contraction mapping theorem. As examples illustrating the general results we consider the evolution equation for a free surface of a filtered liquid with a delay and a linearized quasistationary system of equations for a phase field with a delay.