Abstract
The article is devoted to research the Cauchy problem for singularly perturbed higher-order linear integro-differential equation with a small parameter at the highest derivatives, provided that the roots of additional characteristic equation have negative signs. The aim of this paper is to bring asymptotic estimation of the solution of a singularly perturbed Cauchy problem and the asymptotic convergence of the solution of a singularly perturbed initial value problem to the solution of an unperturbed initial value problem. In this paper the fundamental system of solutions, initial functions of a singularly perturbed homogeneous differential equation are constructed and their asymptotic estimates are obtained. By using the initial functions, we obtain an explicit analytical formula of the solution. The theorem about asymptotic estimate of a solution of the initial value problem is proved. The unperturbed Cauchy problem is constructed. We find the solution of the unperturbed Cauchy problem. An estimate difference of the solution of a singularly perturbed and unperturbed initial value problems. The asymptotic convergence of solution of a singularly perturbed initial value problem to the solution of the unperturbed initial value problem is proved