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

PABLO AMSTER; MARIEL PAULA KUNA; DIONICIO PASTOR DALLOS

ELECTRONIC JOURNAL OF QUALITATIVE THEORY OF DIFFERENTIAL EQUATIONS

UNIV SZEGED

Año: 2020

1417-3875

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 theequation $\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.