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 P x = p admits a T-periodic solution over a T - periodic time scale T. Writing p ( t) = p 0 ( t)+ p, we prove the existence of a compact interval I ( p 0) such that the problem has a solution if and only if p ∈ I ( p 0) 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 ( p 0) is a neighbourhood of 0 for arbitrary p 0. Well known results for the continuous case are generalized to the time scales context.