Asymptotic behavior of solutions of half-linear differential equations and generalized Karamata functions
type: Item
instance of: scholarly article
title: Asymptotic behavior of solutions of half-linear differential equations and generalized Karamata functions
zbMATH Open document ID: 1476.34115
author: Kusano Takaŝi
author: Jelena V. Manojlović
published in: Georgian Mathematical Journal
publication date: 2021-08-05T00:00:00Z
review text: In this article, the authors consider the following second-order half-linear differential equation: \[(p(t)|x'|^\alpha \mathrm{sgn} x')'+q(t)|x|^\alpha \mathrm{sgn} x=0,\tag{1}\] where \(\alpha>0, a>0\), \(q\colon[a;+\infty)\rightarrow \mathbb R\) is a continuous function which may take both positive and negative values in any neighborhood of infinity, and \(p\colon[a;+\infty)\rightarrow(0;+\infty)\) is a continuous function satisfying one of the conditions \[ \int\limits_a^{\infty}\frac{ds}{p(s)^{\frac{1}{\alpha}}}=\infty, \qquad \int\limits_a^{\infty}\frac{ds}{p(s)^{\frac{1}{\alpha}}}<\infty. \tag{2}\] The authors, referring to previous works, argue that equation (1) is nonoscillatory if and only if the generalized Riccati differential equation \[u'=\frac{\alpha}{p(t)^{\frac{1}{\alpha}}}|u|^{1+\frac{1}{\alpha}}+q(t)=0 \tag{3}\] has a solution defined in some neighborhood of infinity. For both cases (2), there were obtained conditions for the existence of so called generalized regularly varying solutions of the equation (1) and its asymptotic representations. Examples of specific equations are given for which the conditions of the theorems are verified and the corresponding asymptotics is written also. In my opinion, the results presented in the article are very important and actual.
reviewed by: Olga Chepok
Mathematics Subject Classification ID: 34D05
Mathematics Subject Classification ID: 34C11
Mathematics Subject Classification ID: 26A12
zbMATH DE Number: 7379142
zbMATH Keywords: half-linear differential equations
zbMATH Keywords: asymptotic behavior
zbMATH Keywords: positive solutions
zbMATH Keywords: regularly varying solutions
zbMATH Keywords: generalized Karamata functions
MaRDI profile type: Publication
full work available at URL: gmj-2020-2070
OpenAlex ID: W3056435565
cites work: Q4726487
cites work: Q5466246
cites work: Q4750126
cites work: Self-adjoint differential equations and generalized Karamata functions
cites work: Nonoscillation theory for second order half-linear differential equations in the framework of regular variation
cites work: Nonoscillatory half-linear differential equations and generalized Karamata functions
cites work: Precise asymptotic behavior of regularly varying solutions of second order half-linear differential equations
cites work: Regularly varying functions
cites work: De Haan type increasing solutions of half-linear differential equations
cites work: Methods in half-linear asymptotic theory
DOI: 10.1515/GMJ-2020-2070
description: scientific article; zbMATH DE number 7379142