Direct and Inverse Problems for Evolution Equations with Regular Integro-Differential Operators

We establish the unique solvability of a Cauchy type problem for equations in Banach spaces with a bounded operator at the unknown function and an integro-differential operator representing the sequential action of the convolution operator and the first order differentiation. Under natural conditions on the Laplace transform of the kernel, we get the form of the solution. We obtain a criterion for the well-posedness of the inverse problem with time-independent unknown and sufficient conditions for the unique solvability of the inverse problem in the case where the unknown depends on time. The results are illustrated by direct and inverse problems for some class of partial differential equations.