Multiple solutions of boundary value problems on time scales for a φ-Laplacian operator

Amster, Pablo; Kuna, Mariel Paula; Santos, Dionicio Pastor Dallos

doi:10.7494/opmath.2020.40.4.405

Cited by 2 publications

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“…In the past decades, periodic problems involving the relativistic forced pendulum differential equation for the continuous case T = R were studied by many authors, see [3,4,8,13,17,18]. In particular, the works [3,18] are concerned with the so-called solvability set, that is, the set I(p 0 ) of values of s for which (1) has at least one T -periodic solution. We remark that problem (1) is 2πperiodic and, consequently, if x is a T -periodic solution then x + 2kπ is also a T -periodic solution for all k ∈ Z.…”

Section: Introductionmentioning

confidence: 99%

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

Amster¹,

Kuna²,

Santos³

2020

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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.

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