Projector approach to the Butuzov–Nefedov algorithm for finding asymptotic solutions for a class of discrete problems with a small step

V.F. Butuzov and N.N. Nefedov proposed an algorithm for constructing asymptotics with boundary functions of two types for solving a discrete initial value problem with a small step $\varepsilon^2$ and a nonlinear term of order $\varepsilon$ in the critical case, i.e., when the degenerate equation with $\varepsilon=0$ is not solvable uniquely for the unknown variable. In this paper, an asymptotic solution of the same problem is constructed by applying a new approach based on orthogonal projectors onto $\ker(B(t) - I)$ and $\ker(B(t) - I)'$, where $B(t)$ is the matrix premultiplying the unknown variable in the linear part of the equation, $I$ is the identity matrix of suitable size, and the prime denotes transposition. This approach considerably simplifies the understanding of the asymptotics-constructing algorithm and makes it possible to represent the problems of finding asymptotic terms of any order in explicit form, which is convenient for researchers applying asymptotic methods for real-world problems.