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

D. P. SANTOS; P. AMSTER; M. P. KUNA

ELECTRONIC JOURNAL OF QUALITATIVE THEORY OF DIFFERENTIAL EQUATIONS

UNIV SZEGED

Año: 2020 vol. 2020 p. 1 - 11

1417-3875

We study some properties of the range of the relativistic pendulum operator P, that is, the set of possible continuous T-periodic forcing terms p for which the equation Px=p admits a T-periodic solution over a T-periodic time scale T. Writing p(t)=p0(t)+p¯¯¯, we prove the existence of a nonempty compact interval I(p0), depending continuously on p0, such that the problem has a solution if and only if p¯¯¯∈I(p0) and at least two different solutions when p¯¯¯ is an interior point. Furthermore, we give sufficient conditions for nondegeneracy; specifically, we prove that if T is small then I(p0) is a neighbourhood of 0 for arbitrary p0. The results in the present paper improve the smallness condition obtained in previous works for the continuous case T=R.