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Wong’s oscillation theorem for second-order delay differential equations

Let H(t) := ∫1/(r(s)z²(s)) (∫z(k)f(k)dk) ds, where z is a positive solution of (r(t)x')' + q(t)x = 0, t ≥ a, satisfying ∫1 / (r(s)z²(s)) ds < ∞. It is well known that, see [J. S. W. Wong, J. Math. Anal. and Appl. — 1999. — 231. — P. 235 – 240], if limt→∞ H(t) = − lim t→∞ H...

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Main Authors:	Özbekler, A., Zafer, A.
Format:	Article
Language:	English
Published:	Інститут математики НАН України 2016
Series:	Нелінійні коливання
Online Access:	http://dspace.nbuv.gov.ua/handle/123456789/177242
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spelling	irk-123456789-1772422021-02-14T01:26:09Z Wong’s oscillation theorem for second-order delay differential equations Özbekler, A. Zafer, A. Let H(t) := ∫1/(r(s)z²(s)) (∫z(k)f(k)dk) ds, where z is a positive solution of (r(t)x')' + q(t)x = 0, t ≥ a, satisfying ∫1 / (r(s)z²(s)) ds < ∞. It is well known that, see [J. S. W. Wong, J. Math. Anal. and Appl. — 1999. — 231. — P. 235 – 240], if limt→∞ H(t) = − lim t→∞ H(t) = ∞, then every solution of (r(t)x') + q(t)x = f(t) is oscillatory. In this paper we extend Wong’s result to delay differential equations of the form (r(t)x' (t))' + q(t)x(τ(t)) = f(t). It is observed that the oscillation behavior may be altered due to presence of the delay. Extensions to Emden – Fowler type delay differential equations are also discussed. Нехай H(t) := ∫1/(r(s)z²(s)) (∫z(k)f(k)dk) ds, де z — додатний розв’язок рiвняння (r(t)x')' + q(t)x = 0, t ≥ a, що задовольняє умову ∫1 / (r(s)z²(s)) ds < ∞. Вiдомо (див. [J. S. W. Wong, J. Math. Anal. and Appl. — 1999. — 231. — P. 235 – 240]), що якщо limt→∞ H(t) = − lim t→∞ H(t) = ∞, то кожен розв’язок рiвняння (r(t)x') + q(t)x = f(t) є осцилюючим. У цiй статтi результат Вонга поширено на диференцiальнi рiвняння з запiзненням вигляду (r(t)x' (t))' + q(t)x(τ(t)) = f(t). Встановлено, що осциляцiйна поведiнка може змiнюватись за рахунок запiзнення. Також розглянуто узагальнення рiвнянь типу Емдена – Фаулера. 2016 Article Wong’s oscillation theorem for second-order delay differential equations / A. Özbekler and A. Zafer // Нелінійні коливання. — 2016. — Т. 19, № 1. — С. 93-100 — Бібліогр.: 15 назв. — англ. 1562-3076 http://dspace.nbuv.gov.ua/handle/123456789/177242 517.9 en Нелінійні коливання Інститут математики НАН України
institution	Digital Library of Periodicals of National Academy of Sciences of Ukraine
collection	DSpace DC
language	English
description	Let H(t) := ∫1/(r(s)z²(s)) (∫z(k)f(k)dk) ds, where z is a positive solution of (r(t)x')' + q(t)x = 0, t ≥ a, satisfying ∫1 / (r(s)z²(s)) ds < ∞. It is well known that, see [J. S. W. Wong, J. Math. Anal. and Appl. — 1999. — 231. — P. 235 – 240], if limt→∞ H(t) = − lim t→∞ H(t) = ∞, then every solution of (r(t)x') + q(t)x = f(t) is oscillatory. In this paper we extend Wong’s result to delay differential equations of the form (r(t)x' (t))' + q(t)x(τ(t)) = f(t). It is observed that the oscillation behavior may be altered due to presence of the delay. Extensions to Emden – Fowler type delay differential equations are also discussed.
format	Article
author	Özbekler, A. Zafer, A.
spellingShingle	Özbekler, A. Zafer, A. Wong’s oscillation theorem for second-order delay differential equations Нелінійні коливання
author_facet	Özbekler, A. Zafer, A.
author_sort	Özbekler, A.
title	Wong’s oscillation theorem for second-order delay differential equations
title_short	Wong’s oscillation theorem for second-order delay differential equations
title_full	Wong’s oscillation theorem for second-order delay differential equations
title_fullStr	Wong’s oscillation theorem for second-order delay differential equations
title_full_unstemmed	Wong’s oscillation theorem for second-order delay differential equations
title_sort	wong’s oscillation theorem for second-order delay differential equations
publisher	Інститут математики НАН України
publishDate	2016
url	http://dspace.nbuv.gov.ua/handle/123456789/177242
citation_txt	Wong’s oscillation theorem for second-order delay differential equations / A. Özbekler and A. Zafer // Нелінійні коливання. — 2016. — Т. 19, № 1. — С. 93-100 — Бібліогр.: 15 назв. — англ.
series	Нелінійні коливання
work_keys_str_mv	AT ozbeklera wongsoscillationtheoremforsecondorderdelaydifferentialequations AT zafera wongsoscillationtheoremforsecondorderdelaydifferentialequations
first_indexed	2023-10-18T22:43:07Z
last_indexed	2023-10-18T22:43:07Z
_version_	1796156251529281536

Wong’s oscillation theorem for second-order delay differential equations

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