Perturbations of the half-linear Euler differential equation
type: Item
instance of: scholarly article
title: Perturbations of the half-linear Euler differential equation
zbMATH Open document ID: 0958.34029
DOI: 10.1007/BF03322512
author: Yanyan Li
published in: Results in Mathematics
publication date: 2001-03-28T00:00:00Z
review text: The authors investigate oscillation/nonoscillation properties of the perturbed half-linear Euler differential equation \[ (x'{}^{n*})'+\frac{\gamma_0}{t^{n+1}}[n+2(n+1)\delta(t)]x^{n*}=0, \tag{*} \] where the function \(\delta(t)\) is piecewise continuous on \((t_0,\infty)\), \(t_0\geq 0\), \(n>0\) is a fixed real number and \(u^{n*}=|u|^n \text{sgn} u\). The number \(\gamma_0=\frac{n^n}{(n+1)^{n+1}}\) is a critical constant, which means that in case \(\delta(t)\equiv 0\) the half-linear Euler differential equation is for \(\gamma\leq \gamma_0\) nonoscillatory while for \(\gamma> \gamma_0\) is oscillatory. Using a Riccati technique and a transformation of the independent variable, there are proved the following main results: Suppose that there exists the finite limit \[ \lim_{T\to\infty} \int_{t_0}^T \delta(t)\frac{dt}{t} \] such that \(\int_{t}^\infty \delta(s)\frac{ds}{s}\geq 0\) for \(t>t_0\). (a) If \(n>1\) and the linear equation \[ z''+\delta(e^s)z=0 \tag{**} \] is nonoscillatory then equation (*) is also nonoscillatory. (b) If \(0<n<1\) and equation (*) is nonoscillatory then the linear equation (**) is also nonoscillatory. In addition, the authors establish an asymptotic form of the solution to (*) provided that the solutions to (**) satisfy two integral inequalities.
Mathematics Subject Classification ID: 34C10
zbMATH DE Number: 1448542
zbMATH Keywords: half-linear differential equations
zbMATH Keywords: principal solutions
zbMATH Keywords: Riccati equations
Wikidata QID: Q115390782
reviewed by: Zuzana Došlá
MaRDI profile type: Publication
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full work available at URL: https://doi.org/10.1007/bf03322512
OpenAlex ID: W2001617949
description: scientific article; zbMATH DE number 1448542