On the Solvability of the Periodically Forced Relativistic Pendulum Equation on Time Scales
| Autor: | Amster, Pablo, Kuna, Mariel P., Santos, Dionicio D. |
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| Rok vydání: | 2020 |
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| Druh dokumentu: | Working Paper |
| Popis: | 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 $\mathbb T=\mathbb R$. Comment: 12 pages, 2 figures |
| Databáze: | arXiv |
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