Degenerated integro-differential equations of special kind in Banach spaces and it's applications

In this paper a special class of degenerated linear integro-differential equations in Banach spaces is investigated by the methods of the theory of fundamental operator-functions of singular integro-differential operators. Sufficient conditions of existence and uniqueness of Cauchy problem of classical solution are obtained. Abstract results are illustrated by two initial boundary value problems, arised in mathematical theory of viscoelasticity.