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

TL;DR

This paper investigates the conditions under which the relativistic pendulum equation on time scales admits periodic solutions, establishing the existence of solution intervals and conditions for multiple solutions, extending previous continuous case results.

Contribution

The paper characterizes the range of the relativistic pendulum operator on time scales and provides new existence and multiplicity results for periodic solutions.

Findings

Existence of a nonempty compact interval for the forcing term allowing solutions.

Multiple solutions exist when the average of the forcing term is an interior point of this interval.

Small period T ensures the interval is a neighborhood of zero, generalizing previous continuous case results.

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 $\mathbb T=\mathbb R$.