Asymptotic representations of solutions of $n\times n$ systems of ordinary differential equations with a large parameter

A. P. Kosarev ; A. A. Shkalikov

Matematičeskie zametki, Tome 116 (2024) no. 2, pp. 266-289

Résumé

The paper considers $n \times n$ systems of ordinary differential equations of the form $$ y'-By-C(\cdot, \lambda)y=\lambda Ay, \qquad y=y(x), \quad x \in [0, 1], $$ where $A=\operatorname{diag}\{a_1(x), \dots, a_n(x)\}$, $B=\{b_{jk}(x)\}_{j, k=1}^n$, and $C= \{c_{jk}(x, \lambda)\}_{j, k=1}^n$. All functions in these matrices are complex-valued and integrable over $x \in [0, 1]$, and $\|c_{jk}(\cdot, \lambda)\|_{L_1} \to 0$ as $\lambda \to \infty$. The theorems proved in the paper generalize the results of the classical Birkhoff–Tamarkin–Langer theory concerning asymptotic representations of fundamental solutions in sectors and half-strips of the complex plane as $\lambda \to \infty$. The focus is on the minimality of the smoothness requirements on the coefficients.

Détail

Keywords: asymptotics of solutions of ordinary differential equations and systems, spectral asymptotics, Birkhoff asymptotics.