Oscillation of second-order forced nonlinear dynamic equations on time scales
In this paper, we discuss the oscillatory behavior of the second-order forced nonlinear dynamic equation \begin{equation*} \left( a(t)x^{\Delta }(t)\right) ^{\Delta }+p(t)f(x^{\sigma })=r(t), \end{equation*} on a time scale ${\mathbb{T}}$ when $a(t)>0$. We establish some sufficient conditions wh...
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| Format: | Article |
| Language: | English |
| Published: |
University of Szeged
2005-11-01
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| Series: | Electronic Journal of Qualitative Theory of Differential Equations |
| Online Access: | http://www.math.u-szeged.hu/ejqtde/periodica.html?periodica=1¶mtipus_ertek=publication¶m_ertek=231 |
| _version_ | 1797830772756840448 |
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| author | Samir Saker |
| author_facet | Samir Saker |
| author_sort | Samir Saker |
| collection | DOAJ |
| description | In this paper, we discuss the oscillatory behavior of the second-order forced nonlinear dynamic equation
\begin{equation*}
\left( a(t)x^{\Delta }(t)\right) ^{\Delta }+p(t)f(x^{\sigma })=r(t),
\end{equation*}
on a time scale ${\mathbb{T}}$ when $a(t)>0$. We establish some sufficient conditions which ensure that every solution oscillates or satisfies $\lim \inf_{t\rightarrow \infty }\left\vert x(t)\right\vert =0.$ Our oscillation results when $r(t)=0$ improve the oscillation results for dynamic equations on time scales that has been established by Erbe and Peterson [Proc. Amer. Math. Soc \ 132 (2004), 735-744], Bohner, Erbe and Peterson [J. Math. Anal. Appl. 301 (2005), 491--507] since our results do not require $\int_{t_{0}}^{\infty }q(t)\Delta t>0$ and $\int_{\pm t_{0}}^{\pm \infty } \frac{du}{f(u)}<\infty .$ Also, as a special case when ${\mathbb{T=R}}$, and $r(t)=0$ our results improve some oscillation results for differential equations. Some examples are given to illustrate the main results. |
| first_indexed | 2024-04-09T13:41:28Z |
| format | Article |
| id | doaj.art-480af5673bb9457da87885b1f5a8d033 |
| institution | Directory Open Access Journal |
| issn | 1417-3875 |
| language | English |
| last_indexed | 2024-04-09T13:41:28Z |
| publishDate | 2005-11-01 |
| publisher | University of Szeged |
| record_format | Article |
| series | Electronic Journal of Qualitative Theory of Differential Equations |
| spelling | doaj.art-480af5673bb9457da87885b1f5a8d0332023-05-09T07:52:57ZengUniversity of SzegedElectronic Journal of Qualitative Theory of Differential Equations1417-38752005-11-0120052311710.14232/ejqtde.2005.1.23231Oscillation of second-order forced nonlinear dynamic equations on time scalesSamir Saker0Mansoura University, Mansoura, EgyptIn this paper, we discuss the oscillatory behavior of the second-order forced nonlinear dynamic equation \begin{equation*} \left( a(t)x^{\Delta }(t)\right) ^{\Delta }+p(t)f(x^{\sigma })=r(t), \end{equation*} on a time scale ${\mathbb{T}}$ when $a(t)>0$. We establish some sufficient conditions which ensure that every solution oscillates or satisfies $\lim \inf_{t\rightarrow \infty }\left\vert x(t)\right\vert =0.$ Our oscillation results when $r(t)=0$ improve the oscillation results for dynamic equations on time scales that has been established by Erbe and Peterson [Proc. Amer. Math. Soc \ 132 (2004), 735-744], Bohner, Erbe and Peterson [J. Math. Anal. Appl. 301 (2005), 491--507] since our results do not require $\int_{t_{0}}^{\infty }q(t)\Delta t>0$ and $\int_{\pm t_{0}}^{\pm \infty } \frac{du}{f(u)}<\infty .$ Also, as a special case when ${\mathbb{T=R}}$, and $r(t)=0$ our results improve some oscillation results for differential equations. Some examples are given to illustrate the main results.http://www.math.u-szeged.hu/ejqtde/periodica.html?periodica=1¶mtipus_ertek=publication¶m_ertek=231 |
| spellingShingle | Samir Saker Oscillation of second-order forced nonlinear dynamic equations on time scales Electronic Journal of Qualitative Theory of Differential Equations |
| title | Oscillation of second-order forced nonlinear dynamic equations on time scales |
| title_full | Oscillation of second-order forced nonlinear dynamic equations on time scales |
| title_fullStr | Oscillation of second-order forced nonlinear dynamic equations on time scales |
| title_full_unstemmed | Oscillation of second-order forced nonlinear dynamic equations on time scales |
| title_short | Oscillation of second-order forced nonlinear dynamic equations on time scales |
| title_sort | oscillation of second order forced nonlinear dynamic equations on time scales |
| url | http://www.math.u-szeged.hu/ejqtde/periodica.html?periodica=1¶mtipus_ertek=publication¶m_ertek=231 |
| work_keys_str_mv | AT samirsaker oscillationofsecondorderforcednonlineardynamicequationsontimescales |