The Leading Term of the Asymptotics of Solutions of Linear Differential Equations with First-Order Distribution Coefficients

N. N. Konechnaja ; K. A. Mirzoev

Matematičeskie zametki, Tome 106 (2019) no. 1, pp. 74-83

Résumé

Let $a_1,a_2,\dots,a_n$, and $\lambda$ be complex numbers, and let $p_1,p_2,\dots,p_n$ be measurable complex-valued functions on $\mathbb R_+$ ($:=[0,+\infty)$) such that $$ |p_1|+(1+|p_2-p_1|)\sum_{j=2}^n|p_j| \in L^1_{\mathrm{loc}}(\mathbb R_+). $$ A construction is proposed which makes it possible to well define the differential equation $$ y^{(n)}+(a_1+p_1(x))y^{(n-1)} +(a_2+p'_2(x)) y^{(n-2)}+\dotsb +(a_n+p'_n(x))y=\lambda y $$ under this condition, where all derivatives are understood in the sense of distributions. This construction is used to show that the leading term of the asymptotics as $x\to +\infty$ of a fundamental system of solutions of this equation and of their derivatives can be determined, as in the classical case, from the roots of the polynomial $$ Q(z)=z^n+a_1 z^{n-1}+\dotsb+a_n-\lambda, $$ provided that the functions $p_1,p_2,\dots,p_n$ satisfy certain conditions of integral decay at infinity. The case where $a_1=\dotsb=a_n=\lambda=0$ is considered separately and in more detail.

Détail

Keywords: differential equations with distribution coefficients, quasiderivatives, quasidifferential expression, leading term of the asymptotics of solutions of differential equations.