{
  "id": "1709.03634",
  "version": "v1",
  "published": "2017-09-12T00:47:19.000Z",
  "updated": "2017-09-12T00:47:19.000Z",
  "title": "A Gronwall inequality and the Cauchy-type problem by means of $ψ$-Hilfer operator",
  "authors": [
    "J. Vanterler da Costa Sousa",
    "E. Capelas de Oliveira"
  ],
  "comment": "19 pages",
  "categories": [
    "math.CA"
  ],
  "abstract": "In this paper, we propose a generalized Gronwall inequality through the fractional integral with respect to another function. The Cauchy-type problem for a nonlinear differential equation involving the $\\psi$-Hilfer fractional derivative and the existence and uniqueness of solutions are discussed. Finally, through generalized Gronwall inequality, we prove the continuous dependence of data on the Cauchy-type problem.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2017-09-12T00:47:19.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "26A33",
      "36A08",
      "34A12",
      "34A40"
    ],
    "keywords": [
      "cauchy-type problem",
      "hilfer operator",
      "generalized gronwall inequality",
      "nonlinear differential equation",
      "fractional integral"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 19,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}