Cauchy problem and inverse problem for integro-differential equations of Gerasimov type with regular kernel

Abstract

We study the unique solvability of the Cauchy problem for a linear regular integro-differential equation of Gerasimov type in a Banach space. This allows us to obtain a well-posedness criterion for the corresponding linear inverse problem with a constant unknown coefficient in the right-hand side.

The abstract results are used to consider direct and inverse initial boundary value problems for a class of equations with a Gerasimov type integro-differential operator in time and polynomials of the Laplace operator in spatial variables, as well as to study the unique solvability of the Cauchy problem and the linear inverse problem for a system of ordinary integro-differential equations. The regular kernel of the integral operator in the system under consideration is essentially operator-valued and defines linear combinations of various integro-differential operators in the equations of the system.