Projector approach to the Butuzov–Nefedov algorithm for asymptotic solution of a class of singularly perturbed problems in a critical case

Under some conditions, an asymptotic solution containing boundary functions of two types was constructed by V.F. Butuzov and N.N. Nefedov for initial value problems for differential equations involving the second power of a small parameter multiplying the derivative with a right-hand side consisting of a singular matrix $A(t)$ times the unknown function (as a linear part of the equation) plus the same small parameter multiplying a nonlinear function. In the present paper, an algorithm for constructing asymptotics using the orthogonal projectors onto $\text{ker}A(t)$ and $\text{ker}A(t)'$ (the prime denotes transposition) is given. This approach can be useful for understanding the algorithm underlying the construction of asymptotics. It allows us to present formulas for finding asymptotic terms of any order in an explicit form.