Singular perturbations of two-point boundary problems for systems of ordinary differential equations

Abstract

Asymptotic solutions of linear systems of ordinary differential equations are employed to discuss the relationship of the solution of a certain “complete” boundary problem.

$$\begin{gathered} \left\{ \begin{gathered} {\text{ }}\frac{{d{\text{ }}x_1 }}{{d{\text{ }}t}} = A_{11} (t,\varepsilon ){\text{ }}x_1 (t,\varepsilon ){\text{ }} + \cdots + A_{1p} (t,\varepsilon ){\text{ }}x_p (t,\varepsilon ) \hfill \\ \varepsilon ^{h_2 } \frac{{d{\text{ }}x_2 }}{{d{\text{ }}t}} = A_{21} (t,\varepsilon ){\text{ }}x_1 (t,\varepsilon ){\text{ }} + \cdots + A_{2p} (t,\varepsilon ){\text{ }}x_p (t,\varepsilon ) \hfill \\ {\text{ }} \vdots {\text{ }} \vdots {\text{ }} \vdots \hfill \\ \varepsilon ^{h_p } \frac{{d{\text{ }}x_2 }}{{d{\text{ }}t}} = A_{p1} (t,\varepsilon ){\text{ }}x_1 (t,\varepsilon ){\text{ }} + \cdots + A_{pp} (t,\varepsilon ){\text{ }}x_p (t,\varepsilon ) \hfill \\ \end{gathered} \right\} \hfill \\ {\text{ }}R(\varepsilon ){\text{ }}x(a,{\text{ }}\varepsilon ){\text{ }} + {\text{ }}S(\varepsilon ){\text{ }}x(b,{\text{ }}\varepsilon ) = c(\varepsilon ){\text{ }} \hfill \\ \end{gathered}$$

as ɛ→0⁺ and the elated “degenerate” problem obtained by setting ɛ = 0. Here the h _i are integers, 0<h ₂<⋯<h _p, x _i is a vector of dimension x _i, A_ijt, ɛ are matrices of appropriate orders with asymptotic expansions, x is the vector $\left( \begin{gathered} x_1 \hfill \\ {\text{ }} \vdots \hfill \\ x_p \hfill \\ \end{gathered} \right)$, R and S are square matrices of order $\sum\limits_{i = 1}^p {n_i }$ and ɛ > 0.

It is shown that under certain conditions the solution of the “complete” problem as ɛ→0⁺ approaches a solution of the “degenerate” differential system and satisfies n ₁ appropriate “degenerate” boundary conditions.