On the solvability of the periodically forced relativistic pendulum equation on time scales
We study some properties of the range of the relativistic pendulum operator $\mathcal P$, that is, the set of possible continuous $T$-periodic forcing terms $p$ for which the equation $\mathcal P x=p$ admits a $T$-periodic solution over a $T$-periodic time scale $\mathbb T$. Writing $p(t)=p_0(t)+\ov...
| Main Authors: | , , |
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| Format: | Article |
| Language: | English |
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University of Szeged
2020-11-01
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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=8493 |
| _version_ | 1797830528496304128 |
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| author | Pablo Amster Mariel Kuna Dionicio Santos |
| author_facet | Pablo Amster Mariel Kuna Dionicio Santos |
| author_sort | Pablo Amster |
| collection | DOAJ |
| description | We study some properties of the range of the relativistic pendulum operator $\mathcal P$, that is, the set of possible continuous $T$-periodic forcing terms $p$ for which the equation $\mathcal P x=p$ admits a $T$-periodic solution over a $T$-periodic time scale $\mathbb T$. Writing $p(t)=p_0(t)+\overline p$, we prove the existence of a nonempty compact interval $\mathcal I(p_0)$, depending continuously on $p_0$, such that the problem has a solution if and only if $\overline p\in \mathcal I(p_0)$ and at least two different solutions when $\overline p$ is an interior point. Furthermore, we give sufficient conditions for nondegeneracy; specifically, we prove that if $T$ is small then $\mathcal I(p_0)$ is a neighbourhood of $0$ for arbitrary $p_0$. The results in the present paper improve the smallness condition obtained in previous works for the continuous case $\mathbb T=\mathbb R$. |
| first_indexed | 2024-04-09T13:37:33Z |
| format | Article |
| id | doaj.art-a6beb260ac914448bfae7316b0222c37 |
| institution | Directory Open Access Journal |
| issn | 1417-3875 |
| language | English |
| last_indexed | 2024-04-09T13:37:33Z |
| publishDate | 2020-11-01 |
| publisher | University of Szeged |
| record_format | Article |
| series | Electronic Journal of Qualitative Theory of Differential Equations |
| spelling | doaj.art-a6beb260ac914448bfae7316b0222c372023-05-09T07:53:10ZengUniversity of SzegedElectronic Journal of Qualitative Theory of Differential Equations1417-38752020-11-0120206211110.14232/ejqtde.2020.1.628493On the solvability of the periodically forced relativistic pendulum equation on time scalesPablo Amster0Mariel Kuna1Dionicio Santos2Universidad de Buenos Aires, Buenos Aires, ArgentinaUniversidad de Buenos Aires, Buenos Aires, ArgentinaUniversidad de Buenos Aires, Buenos Aires, ArgentinaWe study some properties of the range of the relativistic pendulum operator $\mathcal P$, that is, the set of possible continuous $T$-periodic forcing terms $p$ for which the equation $\mathcal P x=p$ admits a $T$-periodic solution over a $T$-periodic time scale $\mathbb T$. Writing $p(t)=p_0(t)+\overline p$, we prove the existence of a nonempty compact interval $\mathcal I(p_0)$, depending continuously on $p_0$, such that the problem has a solution if and only if $\overline p\in \mathcal I(p_0)$ and at least two different solutions when $\overline p$ is an interior point. Furthermore, we give sufficient conditions for nondegeneracy; specifically, we prove that if $T$ is small then $\mathcal I(p_0)$ is a neighbourhood of $0$ for arbitrary $p_0$. The results in the present paper improve the smallness condition obtained in previous works for the continuous case $\mathbb T=\mathbb R$.http://www.math.u-szeged.hu/ejqtde/periodica.html?periodica=1¶mtipus_ertek=publication¶m_ertek=8493relativistic pendulumperiodic solutionstime scalesdegenerate equations |
| spellingShingle | Pablo Amster Mariel Kuna Dionicio Santos On the solvability of the periodically forced relativistic pendulum equation on time scales Electronic Journal of Qualitative Theory of Differential Equations relativistic pendulum periodic solutions time scales degenerate equations |
| title | On the solvability of the periodically forced relativistic pendulum equation on time scales |
| title_full | On the solvability of the periodically forced relativistic pendulum equation on time scales |
| title_fullStr | On the solvability of the periodically forced relativistic pendulum equation on time scales |
| title_full_unstemmed | On the solvability of the periodically forced relativistic pendulum equation on time scales |
| title_short | On the solvability of the periodically forced relativistic pendulum equation on time scales |
| title_sort | on the solvability of the periodically forced relativistic pendulum equation on time scales |
| topic | relativistic pendulum periodic solutions time scales degenerate equations |
| url | http://www.math.u-szeged.hu/ejqtde/periodica.html?periodica=1¶mtipus_ertek=publication¶m_ertek=8493 |
| work_keys_str_mv | AT pabloamster onthesolvabilityoftheperiodicallyforcedrelativisticpendulumequationontimescales AT marielkuna onthesolvabilityoftheperiodicallyforcedrelativisticpendulumequationontimescales AT dioniciosantos onthesolvabilityoftheperiodicallyforcedrelativisticpendulumequationontimescales |