On the Solvability of the Periodically Forced Relativistic Pendulum Equation on Time Scales

Pablo Amster, Mariel P. Kuna, Dionicio D. Santos

Published 2020-05-21Version 1

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 $p(t)=p_0(t)+\bar p$, we prove the existence of a compact interval $\mathcal I(p_0)$ such that the problem has a solution if and only if $\bar p\in \mathcal I(p_0)$ and at least two different solutions when $\bar p$ 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 $p_0$. Well known results for the continuous case are generalized to the time scales context.