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 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 Tperiodic 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.