{
  "aliases": null,
  "backlinks": null,
  "body": "\n## Statement\n\nLet $t_0 \\in \\R$, $x_0 \\in \\R^n$ and $a, b \u003e 0$, and let $R = [t_0 - a, t_0 + a] \\times \\set{x \\in \\R^n : \\norm{x - x_0} \\le b}$.\nLet $f : R \\to \\R^n$ satisfy these three conditions.\n\n1. For each $x$, the map $t \\mapsto f(t, x)$ is Lebesgue measurable.\n2. For almost every $t$, the map $x \\mapsto f(t, x)$ is continuous.\n3. There is a Lebesgue integrable function $m : [t_0 - a, t_0 + a] \\to [0, \\infty)$ with $\\norm{f(t, x)} \\le m(t)$ for all $(t, x) \\in R$.\n\nThen there are $\\beta \\in (0, a]$ and an absolutely continuous function $x : [t_0 - \\beta, t_0 + \\beta] \\to \\R^n$ with $\\norm{x(t) - x_0} \\le b$ for all $t$, with $x(t_0) = x_0$, and with $x'(t) = f(t, x(t))$ for almost every $t$.\n\n## Remarks\n\nSuch a function $x$ is called a solution in the sense of Carathéodory; it is the same as a continuous solution of the integral equation $x(t) = x_0 + \\int_{t_0}^{t} f(s, x(s))\\, ds$.\nOne can take any $\\beta$ with $\\int_{t_0 - \\beta}^{t_0 + \\beta} m(s)\\, ds \\le b$.\nThe proof follows that of the [[Peano existence theorem|098YP831QY]]: one builds approximate solutions with delayed arguments.\nThe bound by $m$ makes them equicontinuous, and the dominated convergence theorem allows passage to the limit.\nExample: let $f(t, x) = 0$ for $t \u003c 0$ and $f(t, x) = 1$ for $t \\ge 0$, with $x(0) = 0$.\nThe function $x(t) = \\max(t, 0)$ is a Carathéodory solution.\nThere is no solution that is differentiable at every point, because a derivative cannot have a jump discontinuity, by Darboux's theorem.\nSo the theorem covers equations with discontinuous time dependence, where the classical theory gives nothing.\n\n## Explanations\n\n### L1 human\n\nClassical existence theorems ask the right side of $x' = f(t, x)$ to be continuous.\nIn applications the forcing is often switched on and off, so $f$ jumps in time.\nCarathéodory's theorem still gives a solution, if one only asks the equation to hold at almost every time.\nThe solution is then a continuous curve with corners where the forcing jumps.\n\n### L2 human\n\nThe theorem replaces continuity in $t$ by measurability, and boundedness of $f$ by an integrable bound.\nSolutions are absolutely continuous functions, which is the natural class for the fundamental theorem of calculus of the Lebesgue integral.\nThis framework underlies control theory, where controls are measurable functions of time.\nAs for Peano's theorem, there is no uniqueness in general; a Lipschitz condition in $x$ with an integrable Lipschitz constant gives uniqueness, as in the [[Picard–Lindelöf theorem|0P6KRD5B7A]].\n",
  "commit": "5fc425f97f1f6b11da5892c2948ef8c5d8805665",
  "front": {
    "about": [
      "0FXDW992DS"
    ],
    "kind": "statement",
    "msc": [
      "34A12",
      "34A06"
    ],
    "name": "Carathéodory's existence theorem",
    "role": "theorem",
    "same_as": {
      "wikidata": "Q5037752",
      "wikipedia": "Carathéodory's existence theorem"
    },
    "sources": [
      {
        "url": "https://en.wikipedia.org/wiki/Carath%C3%A9odory%27s_existence_theorem"
      }
    ],
    "tag": "4D090WJ84H",
    "uses": [
      "075VBY4K4X",
      "098YP831QY"
    ]
  },
  "kind": "statement",
  "link": "https://math2.tamnd.com/node/4D090WJ84H/",
  "links": [
    {
      "field": "about",
      "target": "0FXDW992DS"
    },
    {
      "field": "uses",
      "target": "075VBY4K4X"
    },
    {
      "field": "uses",
      "target": "098YP831QY"
    }
  ],
  "markdown": "https://math2.tamnd.com/node/4D090WJ84H.md",
  "msc": [
    "34A12",
    "34A06"
  ],
  "name": "Carathéodory's existence theorem",
  "prov": null,
  "role": "theorem",
  "source": "https://github.com/tamnd/math2/blob/main/nodes/4D0/4D090WJ84H.md",
  "status": null,
  "tag": "4D090WJ84H",
  "uri": "m2://4D090WJ84H",
  "url": "https://math2.tamnd.com/wiki/caratheodorys-existence-theorem/",
  "writing": {
    "tag": "4D090WJ84H",
    "kind": "statement",
    "level": "W0",
    "next": [
      {
        "what": "an inventory row",
        "note": "the row of the inventory that the node accounts for",
        "have": 0,
        "want": 1
      },
      {
        "what": "a source with a locator",
        "have": 0,
        "want": 1
      }
    ]
  }
}