  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 $\mathbb T=\mathbb
R$.
