Integro-differential equations with Fredholm operator by the derivative of the higest order in Banach spaces and it's applications

In this paper the Cauchy problem for integro-differential equation in Banach spaces with Fredholm operator in main part is investigated by the methods of the theory of fundamental operator-functions. The fundamental operator-function is constructed, and constructiv formula for the generalized solution in the class of distributions with left-bounded support is obtained. The conditions for the coincidence of classical and generalized solutions are described. The abstract results are illustrated by examples of the Cauchy problem for a system of integro-differential equations of two-contour circuit and the Cauchy–Dirichlet problem of the mathematical theory of viscoelasticity.