On the asymptotics of solutions of the second-order differential equations with rapidly varying nonlinearities
type: Item
instance of: scholarly article
title: On the asymptotics of solutions of the second-order differential equations with rapidly varying nonlinearities
zbMATH Open document ID: 1490.34054
author: A. G. Chernikova
author: Vyacheslav Mikhaĭlovich Evtukhov
published in: Ukrainian Mathematical Journal
publication date: 2021-12-07T00:00:00Z
review text: In the article it is considered the following differential equation \[ y''=\alpha_0 p(t)\varphi(y), \tag{1} \] where \(\alpha_{0}\in\{-1;1\}\), \(p:[a,\omega[~\to~]0,+\infty[\) \((-\infty<a<\omega \leq +\infty)\) is a continuous function, \(\varphi_0:\Delta_{Y_0}~\to~]0,+\infty[\) -- is a twice continuously differentiable on \(\Delta_{Y_0}\) function for which \[ \varphi'_0(y)\neq0 \text{ as } y \in~\Delta_{Y_0}, \ \ \lim_{\substack{y \rightarrow Y_0 \\ y \in \Delta_{Y_0}}}\varphi_0(y)\in\{0,+\infty\},\quad\lim_{\substack{y \rightarrow Y_0 \\ y \in\, \Delta_{Y_0}}} \frac{\varphi_0(y)\varphi_0''(y)}{\left(\varphi_0'( y)\right)^2}=1.\tag{2} \] Here \(Y_0\in\{0, \pm \infty \}\), \(\Delta_{Y_0}\) -- is some one-sided neighborhood of the \(Y_0\). From conditions (2) it follows that the functions \(\varphi_0\) and its derivatives of the first order are rapidly varying functions as \(y \rightarrow Y_0\). For \(Y_0=+\infty\) and \(Z_0=+\infty\), the function \(\varphi_0\) and its first-order derivative belong to the class of functions \(\Gamma\) introduced by Hahn (see, for example, [\textit{N. H. Bingham} et al., Regular variation. Cambridge University Press, Cambridge (1987; Zbl 0617.26001)]). By a change of variables, the authors extend the class \(\Gamma\) to a class \(\Gamma_0(Y_0)\) and formulate the properties of functions of such classes. It is clear that the function \(\varphi_0\) and its first-order derivative belong to the class of functions \(\Gamma_0(Y_0)\) as \(y \rightarrow Y_0\). The authors establish necessary and sufficient conditions for the existence of \(P_{\omega}(Y_0,\lambda_0)\)-solutions of equation (1) in a special case when \(\lambda_0 \in \mathbb{R}\backslash \{0,1\}\), as well as the asymptotic representations of these solutions and their first derivatives as \(t \uparrow \omega\) using the a priori asymptotic properties of these solutions. The main results are given in the Theorems 2.1, 2.2. In Theorem 2.1, necessary conditions for the existenc…
reviewed by: Olga Chepok
Mathematics Subject Classification ID: 34D05
Mathematics Subject Classification ID: 37C60
Mathematics Subject Classification ID: 26A12
zbMATH DE Number: 7439583
MaRDI profile type: Publication
Wikidata QID: Q115379916
full work available at URL: s11253-021-01944-9
OpenAlex ID: W3213949610
cites work: Regular variation and differential equations
cites work: Asymptotic behavior of the solutions of ordinary second-order differential equations with rapidly varying nonlinearities
cites work: Q4726487
cites work: Regularly varying functions
cites work: Conditions for the existence of solutions of real nonautonomous systems of quasilinear differential equations vanishing at a singular point
cites work: Q4692282
DOI: 10.1007/S11253-021-01944-9
description: scientific article; zbMATH DE number 7439583