On the Solvability of the Periodically Forced Relativistic Pendulum Equation on Time Scales - Amster, Pablo Gustavo (2020)
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.