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 *n*th element
and the solution to the problem is proportional to the

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