A Method of the Study of the Cauchy Problem for a Singularly Perturbed Linear Inhomogeneous Differential Equation
We construct a sequence that converges to a solution of the Cauchy problem for a singularly perturbed linear inhomogeneous differential equation of an arbitrary order. This sequence is also an asymptotic sequence in the following sense: the deviation (in the norm of the space of continuous functions) of its $n$th element from the solution of the problem is proportional to the $(n+1)$th power of the parameter of perturbation. This sequence can be used for justification of asymptotics obtained by the method of boundary functions.
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