Let an integro-differential equation be given (see article). In this paper, we investigate the solution of equation (1) for initial conditions (2) by the method we applied for in [1]. This method, with slight modifications, is easily transferred to the case of m≤n as well.

In [1], [2], [3] we have investigated solutions of some classes of linear integro-differential equations. At the same time, both general solutions and solutions of the Cauchy problem were found for the mentioned types of equations. The method we used in the cited papers can be successfully applied to the solution of the boundary value problem for linear integro-differential equations. Part of the eigennumbers of the integro-differential system (2) at m≤n. are the eigennumbers of the integral equation (32), and another part of them are the roots of the equation D (X) = 0. Together they form the spectrum of eigennumbers of the integro-differential system (2). It follows that the spectrum of eigennumbers of an integro-differential system is a discrete set.

In this note, we study solutions of the integro-differential equation (see article). This solution depends on q arbitrary parameters. If for λ = λ' equation (51) has no solutions, then the Cauchy problem under consideration has no solutions. Finally, we note that if the determinant (40) on the manifold (39) vanishes, then system (43) is not solvable, or is solvable uniquely with respect to S and tk. Therefore, equation (46) will include arbitrary parameters. Consequently, if the initial manifold (39) is characteristic, then equation (1) has none, or has an infinite number of solutions.

In connection with the study of branching points of nonlinear integro-differential equations, to the solution of which the problem of longitudinal bending of a rod is reduced, N. N. Nazarov proposed to study the integro-differential equation (1) (see article). ... This article proposes a simple method for solving equation (1), which leads to the need to solve one Fredholm integral equation and one algebraic system, the number of unknowns of which does not exceed m (see article).