{
  "id": "2005.12851",
  "version": "v1",
  "published": "2020-05-21T17:29:52.000Z",
  "updated": "2020-05-21T17:29:52.000Z",
  "title": "On the Solvability of the Periodically Forced Relativistic Pendulum Equation on Time Scales",
  "authors": [
    "Pablo Amster",
    "Mariel P. Kuna",
    "Dionicio D. Santos"
  ],
  "comment": "11 pages, 2 figures",
  "categories": [
    "math.DS"
  ],
  "abstract": "We study some properties of the range of the relativistic pendulum operator $\\mathcal P$, that is, the set of possible continuous $T$-periodic forcing terms $p$ for which the equation $\\mathcal P x=p$ admits a $T$-periodic solution over a $T$-periodic time scale $\\mathbb T$. Writing $p(t)=p_0(t)+\\bar p$, we prove the existence of a compact interval $\\mathcal I(p_0)$ such that the problem has a solution if and only if $\\bar p\\in \\mathcal I(p_0)$ and at least two different solutions when $\\bar p$ is an interior point. Furthermore, we give sufficient conditions for nondegeneracy; specifically, we prove that if $T$ is small then $\\mathcal I(p_0)$ is a neighbourhood of $0$ for arbitrary $p_0$. Well known results for the continuous case are generalized to the time scales context.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2020-05-21T17:29:52.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "34N05",
      "34C25",
      "47H11"
    ],
    "keywords": [
      "periodically forced relativistic pendulum equation",
      "solvability",
      "relativistic pendulum operator",
      "periodic time scale",
      "time scales context"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 11,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}