Degenerated integro-differential equation in Banach spaces and its application

This paper is devoted to the study Cauchy problem for a linear integro-differential operator equation with convolutional type Volterra integral part and Fredholm operator by the derivative of highest order. Sufficient conditions of classical ($N$ times strongly continuously differentiable) solution existence and uniqueness, as well as explicit formulas for its restoration are obtained. These results are applyed to the investigation of initial boundary value problem, arising in mathematical theory of viscoelacity.