An analogue of the Cauchy formula for a linear integro-differential equation of the Fredholm type

The Cauchy problem is considered for one class of linear inhomogeneous Fredholm integro-differential equations, which is a generalization of the E.A. Barbashin integro-differential equation. Such equations describe the dynamics of some complex processes. In particular, the Cauchy problem considered in this paper for a system of Fredholm-type integro-differential equations describes the dynamics of a number of populations. Therefore, the development of a qualitative theory of such integro-differential equations, which is a generalization of the integro-differential equation of E.A. Barbashin, is very relevant. An integral representation of the solution of the Cauchy problem under consideration is obtained. The resulting representation of the solution can later be used to study the qualitative theory of optimal control of the dynamics of some populations. Using this representation, it is possible to obtain both necessary and sufficient optimality conditions, and to investigate problems related to controllability and observability in optimal control problems described by the system of integro-differential equations under consideration. The result obtained is a nontrivial generalization of a similar result established in the work of E. A. Barbashin and L. P. Bisyarina. The relationship of the obtained result with the close result of E.A. Barbashin and L.P. Bisyarina is studied, or in another way only for a scalar equation with a constant coefficient. The resulting representation for the general problem under consideration is constructive in nature.