On solutions to abstract Volterra integro-differential equations with iterated deviating arguments

We study the existence and uniqueness of solutions for a non-autonomous Volterra integro-differential equation of parabolic type with iterated deviating arguments in an arbitrary Banach space. The existence of a solution is established in a Banach space with smooth source terms in a Banach space. While the existence and uniqueness of a solution are proved in a Hilbert space without smoothness conditions. The continuation of a solution is also discussed of the problem. As an application, we illustrate the theory by an example.