Generalized solutions of singular integro-differential equations in Banach spaces and their applications

We investigate special classes of linear integro-differential equations with a Noether operator at the highest derivative and convolution integral terms of Volterra type. We obtain sufficient conditions for the solvability of the Cauchy problem for such equations both in generalized functions and in classes of functions of finite smoothness and investigate the connection between these types of solutions. The investigation uses the fundamental operator function techniques. Abstract results are illustrated by examples of initial-boundary value problems that appear in the mathematical theory of viscoelasticity.