Asymptotics of the solutions of boundary-value problems for linear singularly perturbed systems of differential equations in the case of multiple spectrum of the boundary matrix (Q2202624)

In this paper, the author deals with a two-point boundary value problem for a system of linear ordinary differential equations \[ \varepsilon\frac{dx}{dt}=A(t,\varepsilon)x+f(t,\varepsilon), \ \varepsilon\in(0,\varepsilon_0), \] \[ Mx(0,\varepsilon)+Nx(T,\varepsilon)=d(\varepsilon), \] where the square matrix \(A(t,\varepsilon)\) of the dimension \(n\) and the vector \(f(t,\varepsilon)\) admit uniform asymptotic expansions in powers of a small parameter \(\varepsilon\) on the interval \([0,T],\) \(A(t,\varepsilon)\approx\sum\limits_{k=0}^{\infty}\varepsilon^k A_k(t),\) \(f(t,\varepsilon)\approx\sum\limits_{k=0}^{\infty}\varepsilon^k f_k(t),\) respectively, and the vector \(d(\varepsilon)\) can be represented in the form of an asymptotic expansion \(d(\varepsilon)\approx\sum\limits_{k=0}^{\infty}\varepsilon^kd_k\) as \(\varepsilon\to 0.\) Moreover, it is assumed that the leading matrix \(A_0(t)\) has for all \(t\in[0,T]\) an unique eigenvalue \(\lambda_0(t)\) of the multiplicity \(n.\) By using a formal asymptotic expansion technique the unique solution of the boundary value problem under consideration in the form of the sum of a linear combination of the solutions of the homogeneous system and a particular solution of the inhomogeneous system was constructed.