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 Mat. Zametki, 2019, Volume 106, Issue 1, Pages 74–83 (Mi mz12290)  The Leading Term of the Asymptotics of Solutions of Linear Differential Equations with First-Order Distribution Coefficients

N. N. Konechnajaa, K. A. Mirzoevb

a Northern (Arctic) Federal University named after M. V. Lomonosov, Arkhangelsk
b Lomonosov Moscow State University

Abstract: Let $a_1,a_2,…,a_n$, and $\lambda$ be complex numbers, and let $p_1,p_2,…,p_n$ 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 $p_1,p_2,…,p_n$ 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.

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

 Funding Agency Grant Number Russian Foundation for Basic Research 18-01-00250 Russian Science Foundation 17-11-01215 The work on Lemma 1 and Theorem 1 was supported by the Russian Science Foundation under grant no. 17-11-01215. The work on the corollary and Theorem 2 was supported by the Russian Foundation for Basic Research under grant no. 18-01-00250.

DOI: https://doi.org/10.4213/mzm12290  Full text: PDF file (519 kB) First page: PDF file References: PDF file   HTML file

English version:
Mathematical Notes, 2019, 106:1, 81–88 Bibliographic databases:   UDC: 517.928
Revised: 16.12.2018

Citation: N. N. Konechnaja, K. A. Mirzoev, “The Leading Term of the Asymptotics of Solutions of Linear Differential Equations with First-Order Distribution Coefficients”, Mat. Zametki, 106:1 (2019), 74–83; Math. Notes, 106:1 (2019), 81–88 Citation in format AMSBIB
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• http://mi.mathnet.ru/eng/mz12290
• https://doi.org/10.4213/mzm12290
• http://mi.mathnet.ru/eng/mz/v106/i1/p74

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