An infinite sequence of nonoscillation theorems for second-order nonlinear differential equations of Euler type
type: Item
instance of: scholarly article
title: An infinite sequence of nonoscillation theorems for second-order nonlinear differential equations of Euler type
zbMATH Open document ID: 1013.34029
DOI: 10.1016/S0362-546X(01)00768-4
author: Jitsuro Sugie
author: Naoto Yamaoka
published in: Nonlinear Analysis. Theory, Methods \& Applications. Series A: Theory and Methods
publication date: 2002-08-25T00:00:00Z
review text: The purpose of this paper is to improve nonoscillation criteria for the nonlinear differential equation \[ t^2 x''+ g(x)= 0,\quad t> 0,\tag{\(*\)} \] where \(g(x)\) satisfies a suitable smoothness condition for the uniqueness of solutions to the initial value problem and the signum condition \[ xg(x)> 0\quad\text{if }x\neq 0. \] Equation \((*)\) is said to be oscillatory (resp. nonoscillatory) in case all nontrivial solutions are oscillatory (resp. nonoscillatory). A nontrivial solution \(x(t)\) to \((*)\) is said to be oscillatory if there exists a sequence \(\{t_n\}\) tending to \(\infty\) such that \(x(t_n)= 0\). Otherwise, it is said to be nonoscillatory. The Riemann-Weber version of the Euler differential equation and positive orbits to a Liénard system are considered.
reviewed by: Vasile Marinca
Mathematics Subject Classification ID: 34C10
zbMATH DE Number: 1788018
zbMATH Keywords: oscillation
zbMATH Keywords: nonlinear Euler differential equations
zbMATH Keywords: Liénard system
zbMATH Keywords: phase plane analyses
MaRDI profile type: Publication
cites work: Non-Oscillation Theorems
cites work: Q5584775
cites work: Classification of global phase portraits of a system of Liénard type
cites work: Nonlinear oscillations of second order differential equations of Euler type
cites work: Oscillation criteria for second order nonlinear differential equations of Euler type
cites work: Oscillation theorems for second-order nonlinear differential equations of Euler type
description: scientific article; zbMATH DE number 1788018