A Method to Study the Cauchy Problem for a Singularly Perturbed Homogeneous Linear Differential Equation of Arbitrary Order / E. E. Bukzhalev. // Vestnik Moskovskogo Universiteta. Seriya 1. Matematika. Mekhanika. 2018. № 2. P. 3-12 [Moscow Univ. Math. Bulletin. Vol. 72, N 2, 2017. P. 41-49].

We construct a sequence converging to the solution to the Cauchy problem for a singularly perturbed linear homogeneous differential equation of any order. This sequence is asymptotic in the following sense: the distance (with respect to the norm of the space of continuous functions) between its nth element and the solution to the problem is proportional to the (n+1)th power of the perturbation parameter.

Key words: singular perturbations, Banach fixed-point theorem, asymptotic iteration method, boundary function method.