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 $\mathbb{T}=\mathbb{R}$.

12 pages, 2 figures