{
  "id": "2002.01503",
  "version": "v1",
  "published": "2020-02-04T19:19:02.000Z",
  "updated": "2020-02-04T19:19:02.000Z",
  "title": "Solution estimates for linear differential equations with delay",
  "authors": [
    "Leonid Berezansky",
    "Elena Braverman"
  ],
  "comment": "15 pages, 3 figures. Accepted to \"Applied Mathematics and Computation\"",
  "doi": "10.1016/j.amc.2019.124962",
  "categories": [
    "math.DS",
    "math.AP"
  ],
  "abstract": "In this paper, we give explicit exponential estimates $\\displaystyle |x(t)|\\leq M e^{ -\\gamma (t-t_0) }$, where $t\\geq t_0$, $M>0$, for solutions of a linear scalar delay differential equation $$ \\dot{x}(t)+\\sum_{k=1}^m b_k(t)x(h_k(t))=f(t),~~ t\\geq t_0,~ x(t)=\\phi(t),~t\\leq t_0. $$ We consider two different cases: when $\\gamma>0$ (corresponding to exponential stability) and the case of $\\gamma <0$ when the solution is, generally, growing. In the first case, together with the exponential estimate, we also obtain an exponential stability test, in the second case we get estimation for solution growth. Here both the coefficients and the delays are measurable, not necessarily continuous.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2020-02-04T19:19:02.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "34K20",
      "34K25",
      "34K06"
    ],
    "keywords": [
      "linear differential equations",
      "solution estimates",
      "linear scalar delay differential equation",
      "explicit exponential estimates",
      "exponential stability test"
    ],
    "tags": [
      "journal article"
    ],
    "publication": {
      "publisher": "Elsevier"
    },
    "note": {
      "typesetting": "TeX",
      "pages": 15,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}