On the solvability of the periodically forced relativistic pendulum equation on time scales

Abstract

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 $\mathbb T=\mathbb R$.

Keywords

• relativistic pendulum
• periodic solutions
• time scales
• degenerate equations