Note on ``Oscillation criteria for second order linear differential equations''
type: Item
instance of: scholarly article
title: Note on ``Oscillation criteria for second order linear differential equations''
zbMATH Open document ID: 0862.34024
published in: Journal of Mathematical Analysis and Applications
publication date: 1996-12-09T00:00:00Z
review text: \textit{Horng Jaan Li} [J. Math. Anal. Appl. 194, 217-234 (1995; Zbl 0836.34033)] proved some oscillation criteria for the equation \([r(t)u'(t)]' + c(t)u(t) =0\) by using a generalized Riccati transformation due to Yu. He improved oscillation criteria of Yan and Philos and obtained, as a special case, the criterion of Kamenev. One of his assumptions has the form \((C_1)\) \(\int^t_{t_0} a(s)r(s) h^2(t,s) ds < \infty\) for all \(t\geq t_0\) where \(a(t) >0\), \(r(t) >0\) are continuous on \([t_0, \infty)\) and \(h(t,s)\) is continuous on \(D_0= \{(t,s): t>s \geq t_0\}\). By the author of this note assumption \((C_1)\) means that there exists a constant \(K>0\) such that \(\int^t_{t_0} a(s) r(s)h^2(t,s) ds<K\) for all \(t\geq t_0\). Then he shows that Horng Jaan Li's results do not improve the results of Yan and Philos and that we do not obtain the Kamenev criterion as a special case. Moreover, he states theorems and corollaries of Horng Jaan Li in which assumption \((C_1)\) should be omitted. \{Reviewer's remark: The function \(h(t,s)\) in \((C_1)\) may be discontinuous for \(t=s\). Hence assumption \((C_1)\) by Horng Jaan Li only means that \(\int^t_{t_0} a(s) r(s)h^2(t,s)ds\) is a Lebesgue integrable function on any interval \([t_0,T]\), \(T>0\}\).
Mathematics Subject Classification ID: 34C10
Mathematics Subject Classification ID: 34A30
zbMATH DE Number: 944370
zbMATH Keywords: second order equation
zbMATH Keywords: oscillation
zbMATH Keywords: generalized Riccati transformation
author: Yuriy V. Rogovchenko
reviewed by: Svatoslav Staněk
MaRDI profile type: Publication
full work available at URL: jmaa.1996.0396
OpenAlex ID: W1981786626
DOI: 10.1006/JMAA.1996.0396
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description: scientific article; zbMATH DE number 944370