Periodic solutions of relativistic Liénard-type equations
In this paper, we prove that the relativistic Liénard-type equation \begin{equation*} \begin{split} \frac{d}{dt}\left(\frac{\dot{x}\left\vert \dot{x} \right\vert ^{p-2}}{\big( 1-\left\vert \dot{x}\right\vert ^{p}\big) ^{\frac{p-1}{p}}}\right) +f\left( x\right) \dot{x} +g\left( x\right) =0 \text{,}\q...
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| Format: | Article |
| Language: | English |
| Published: |
University of Szeged
2020-06-01
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| Series: | Electronic Journal of Qualitative Theory of Differential Equations |
| Subjects: | |
| Online Access: | http://www.math.u-szeged.hu/ejqtde/periodica.html?periodica=1¶mtipus_ertek=publication¶m_ertek=8047 |
| _version_ | 1797830535769227264 |
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| author | Mustafa Aktas |
| author_facet | Mustafa Aktas |
| author_sort | Mustafa Aktas |
| collection | DOAJ |
| description | In this paper, we prove that the relativistic Liénard-type equation
\begin{equation*}
\begin{split}
\frac{d}{dt}\left(\frac{\dot{x}\left\vert \dot{x}
\right\vert ^{p-2}}{\big( 1-\left\vert \dot{x}\right\vert
^{p}\big) ^{\frac{p-1}{p}}}\right) +f\left( x\right) \dot{x}
+g\left( x\right) =0
\text{,}\qquad p>1,
\end{split}
\end{equation*}
and its special case, relativistic Van der Pol-type equation, have a periodic solution. Our results are inspired by the results obtained by Mawhin and Villari [Nonlinear Anal. 160(2017), 16–24] and extend their results to this more general case. |
| first_indexed | 2024-04-09T13:37:40Z |
| format | Article |
| id | doaj.art-c8f3de57ae9d4f9390c3611110a532f0 |
| institution | Directory Open Access Journal |
| issn | 1417-3875 |
| language | English |
| last_indexed | 2024-04-09T13:37:40Z |
| publishDate | 2020-06-01 |
| publisher | University of Szeged |
| record_format | Article |
| series | Electronic Journal of Qualitative Theory of Differential Equations |
| spelling | doaj.art-c8f3de57ae9d4f9390c3611110a532f02023-05-09T07:53:10ZengUniversity of SzegedElectronic Journal of Qualitative Theory of Differential Equations1417-38752020-06-0120203811210.14232/ejqtde.2020.1.388047Periodic solutions of relativistic Liénard-type equationsMustafa Aktas0Gazi University, Ankara, TurkeyIn this paper, we prove that the relativistic Liénard-type equation \begin{equation*} \begin{split} \frac{d}{dt}\left(\frac{\dot{x}\left\vert \dot{x} \right\vert ^{p-2}}{\big( 1-\left\vert \dot{x}\right\vert ^{p}\big) ^{\frac{p-1}{p}}}\right) +f\left( x\right) \dot{x} +g\left( x\right) =0 \text{,}\qquad p>1, \end{split} \end{equation*} and its special case, relativistic Van der Pol-type equation, have a periodic solution. Our results are inspired by the results obtained by Mawhin and Villari [Nonlinear Anal. 160(2017), 16–24] and extend their results to this more general case.http://www.math.u-szeged.hu/ejqtde/periodica.html?periodica=1¶mtipus_ertek=publication¶m_ertek=8047closed orbitsperiodic solutionslimit cyclesrelativistic liénard-type equations |
| spellingShingle | Mustafa Aktas Periodic solutions of relativistic Liénard-type equations Electronic Journal of Qualitative Theory of Differential Equations closed orbits periodic solutions limit cycles relativistic liénard-type equations |
| title | Periodic solutions of relativistic Liénard-type equations |
| title_full | Periodic solutions of relativistic Liénard-type equations |
| title_fullStr | Periodic solutions of relativistic Liénard-type equations |
| title_full_unstemmed | Periodic solutions of relativistic Liénard-type equations |
| title_short | Periodic solutions of relativistic Liénard-type equations |
| title_sort | periodic solutions of relativistic lienard type equations |
| topic | closed orbits periodic solutions limit cycles relativistic liénard-type equations |
| url | http://www.math.u-szeged.hu/ejqtde/periodica.html?periodica=1¶mtipus_ertek=publication¶m_ertek=8047 |
| work_keys_str_mv | AT mustafaaktas periodicsolutionsofrelativisticlienardtypeequations |