Asymptotic analysis of solutions of ordinary differential equations with distribution coefficients

A. M. Savchuk ; A. A. Shkalikov

Sbornik. Mathematics, Tome 211 (2020) no. 11, pp. 1623-1659

Résumé

Ordinary differential equations of the form $$ \tau(y)- \lambda ^{2m} \varrho(x) y=0, \qquad \tau(y) =\sum_{k,s=0}^m(\tau_{k,s}(x)y^{(m-k)}(x))^{(m-s)}, $$ on the finite interval $x\in[0,1]$ are under consideration. Here the functions $\tau_{0,0}$ and $\varrho$ are absolutely continuous and positive and the coefficients of the differential expression $\tau(y)$ are subject to the conditions $$ \tau_{k,s}^{(-l)}\in L_2[0,1], \qquad 0\le k,s \le m, \quad l=\min\{k,s\}, $$ where $f^{(-k)}$ denotes the $k$th antiderivative of the function $f$ in the sense of distributions. Our purpose is to derive analogues of the classical asymptotic Birkhoff-type representations for the fundamental system of solutions of the above equation with respect to the spectral parameter as $\lambda \to \infty$ in certain sectors of the complex plane $\mathbb C$. We reduce this equation to a system of first-order equations of the form $$ \mathbf y'=\lambda\rho(x)\mathrm B\mathbf y+\mathrm A(x)\mathbf y+\mathrm C(x,\lambda)\mathbf y, $$ where $\rho$ is a positive function, $\mathrm B$ is a matrix with constant elements, the elements of the matrices $\mathrm A(x)$ and $\mathrm C(x,\lambda)$ are integrable functions, and $\|\mathrm C(x,\lambda)\|_{L_1}=o(1)$ as $\lambda \to \infty$. For systems of this kind, we obtain new results concerning the asymptotic representation of the fundamental solution matrix, which we use to make an asymptotic analysis of the above scalar equations of high order. Bibliography: 44 titles.

Détail

Keywords: differential equations with distribution coefficients, asymptotics with respect to the spectral parameter, Birkhoff asymptotics, spectral asymptotics.