{
  "id": "1811.06275",
  "version": "v1",
  "published": "2018-11-15T10:24:09.000Z",
  "updated": "2018-11-15T10:24:09.000Z",
  "title": "Some Class of Linear Operators Involved in Functional Equations",
  "authors": [
    "Janusz Morawiec",
    "Thomas Zürcher"
  ],
  "categories": [
    "math.CA"
  ],
  "abstract": "Fix $N\\in\\mathbb N$ and assume that for every $n\\in\\{1,\\ldots, N\\}$ the functions $f_n\\colon[0,1]\\to[0,1]$ and $g_n\\colon[0,1]\\to\\mathbb R$ are Lebesgue measurable, $f_n$ is almost everywhere approximately differentiable with $|g_n(x)|<|f'_n(x)|$ for almost all $x\\in [0,1]$, there exists $K\\in\\mathbb N$ such that the set $\\{x\\in [0,1]:\\mathrm{card}{f_n^{-1}(x)}>K\\}$ is of Lebesgue measure zero, $f_n$ satisfy Luzin's condition N, and the set $f_n^{-1}(A)$ is of Lebesgue measure zero for every set $A\\subset\\mathbb R$ of Lebesgue measure zero. We show that the formula $Ph=\\sum_{n=1}^{N}g_n\\!\\cdot\\!(h\\circ f_n)$ defines a linear and continuous operator $P\\colon L^1([0,1])\\to L^1([0,1])$, and then we obtain results on the existence and uniqueness of solutions $\\varphi\\in L^1([0,1])$ of the equation $\\varphi=P\\varphi+g$ with a given $g\\in L^1([0,1])$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2018-11-15T10:24:09.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "47A50",
      "26A24",
      "39B12",
      "47B38"
    ],
    "keywords": [
      "lebesgue measure zero",
      "linear operators",
      "functional equations",
      "satisfy luzins condition",
      "continuous operator"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}