Asymptotic Expansions of Solutions to $$n \times n$$ Systems of Ordinary Differential Equations with a Large Parameter

The $$n \times n$$ system of ordinary differential equations $$y' - \sum_{l=0}^{m}\lambda^{-l}B_l(x)y - \lambda^{-m}C(x, \lambda)y=\lambda A(x)y, \qquad x \in [0, 1], \quad m \in \mathbb{N},$$ is considered, where $$A=\operatorname{diag}\{a_1, \dots, a_n\}, \qquad B_l=\{b_{jk}^l\}, \qquad C=\{c_{jk}(\cdot, \lambda)\}, \qquad y=(y_1, \dots, y_n)^\top.$$ It is assumed that, for some integer $$m\ge 1$$ , the entries of the matrices $$A(x)$$ and $$B_l(x)$$ are complex-valued functions satisfying the conditions $$\begin{gathered} \, a_i \in W_1^m[0, 1], \quad b^{0}_{ii} \in W_1^{m-1}[0, 1], \quad b^{0}_{jk} \in W_1^{m}[0, 1], \qquad j \ne k, \quad i, j, k=1, \dots, n, \\ b_{jk}^{l} \in W_1^{m-l}[0, 1], \qquad j, k=1, \dots, n, \quad l=1, \dots, m, \end{gathered}$$ where the $$W^k_1$$ are the Sobolev spaces, and the entries of the matrix $$C(\cdot, \lambda)$$ are integrable functions on the interval $$[0, 1]$$ such that $$\|c_{ij}(\cdot, \lambda)\|_{1} \to 0$$ in the metric of the space $$L_1[0, 1]$$ uniformly as $$\lambda \to \infty$$ , $$\lambda \in \mathbb{C}$$ . The main results of the paper refine and supplement classical results of Birkhoff–Tamarkin–Langer theory concerning asymptotic expansions of the fundamental solutions of the system under consideration in sectors of the complex plane. Special attention is given to minimal smoothness requirements on the matrix entries and to explicit expressions for matrices in asymptotic expansions.