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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| Дата: | 2016 |
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| Автори: | , |
| Формат: | Стаття |
| Мова: | English |
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Інститут математики НАН України
2016
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| Назва видання: | Нелінійні коливання |
| Онлайн доступ: | http://dspace.nbuv.gov.ua/handle/123456789/177242 |
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| Назва журналу: | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| Цитувати: | 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. |
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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 |
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2023-10-18T22:43:07Z |
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1796156251529281536 |