Some oscillation problems for a second-order linear delay differential equation
type: Item
instance of: scholarly article
title: Some oscillation problems for a second-order linear delay differential equation
zbMATH Open document ID: 0915.34064
published in: Journal of Mathematical Analysis and Applications
publication date: 1999-01-24T00:00:00Z
review text: The following scalar delay differential equation is considered \[ \ddot x(t)+\sum_{k=1}^m a_k(t) x(g_k(t))=0,\quad g_k(t)\leq t . \] Nonoscillation and properties related to nonoscillation are studied. The main result is that under some natural assumptions for a delay differential equation the following four assertions are equivalent: nonoscillation of solutions to this equation and the corresponding differential inequality, positiveness of the fundamental function, and the existence of a nonnegative solution to a generalized Riccati inequality. The equivalence of the oscillation properties with a differential inequality is applied to obtain new explicit conditions for nonoscillation and oscillation and to prove some known results in a different way. A generalized Riccati inequality is applied to compare oscillation properties of two equations without comparing their solutions. These results can be regarded as a natural generalization of the Sturm comparison theorem for a second-order ordinary differential equation. By applying the positiveness of the fundamental function the positive solutions to two nonoscillatory equations are compared. Conditions on the initial function are given implying the positiveness of the corresponding solution.
Mathematics Subject Classification ID: 34K11
Mathematics Subject Classification ID: 34C15
zbMATH DE Number: 1218589
zbMATH Keywords: scalar linear delay differential equation
zbMATH Keywords: oscillations
zbMATH Keywords: fundamental function
zbMATH Keywords: nonnegative solutions
zbMATH Keywords: Riccati inequality
zbMATH Keywords: Sturm comparison theorem
Wikidata QID: Q115395401
author: Leonid Berezansky
author: E. Braverman
reviewed by: Ivan Ginchev
MaRDI profile type: Publication
full work available at URL: jmaa.1997.5879
OpenAlex ID: W2040803490
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DOI: 10.1006/JMAA.1997.5879
description: scientific article; zbMATH DE number 1218589