{
  "aliases": [
    "cauchy-lipschitz-theorem",
    "existence-and-uniqueness-theorem"
  ],
  "backlinks": [
    {
      "field": "uses",
      "target": "00BT70ND50"
    },
    {
      "field": "uses",
      "target": "00K7ASZ1PR"
    },
    {
      "field": "uses",
      "target": "0BFD3A6CRH"
    }
  ],
  "body": "\n## Statement\n\nLet $J = [t_{\\min}, t_{\\max}]$ and $t_0 \\in J$, let $x_0 \\in \\mathbb{R}^n$, and let $f : J \\times \\mathbb{R}^n \\to \\mathbb{R}^n$ satisfy the [[Picard–Lindelöf hypotheses|0J7HF2PF1H]] at $(t_0, x_0)$ with constants $a, L, K$.\nLet $B$ be the closed ball of radius $a$ around $x_0$.\nThen there is exactly one continuous curve $\\alpha : J \\to B$ with $\\alpha(t_0) = x_0$ and with this property. For each $t \\in J$, $\\alpha$ has the derivative $f(t, \\alpha(t))$ at $t$ within $J$.\n\n## Remarks\n\nThe proof uses the [[integral equation form|015KK993C5]] and the fact that [[an iterate of the Picard operator is a contraction|08XDXP86XE]]. The limit of [[Picard iteration|0RJ5NHWYTP]] is the solution.\nA common form: let $f$ be continuous on a neighbourhood of $(t_0, x_0)$ and locally Lipschitz in $x$. Then the [[initial value problem|075VBY4K4X]] has a unique solution on $[t_0 - \\varepsilon, t_0 + \\varepsilon]$ for some $\\varepsilon \u003e 0$.\nThe theorem is local. For $x' = 1 + x^2$, $x(0) = 0$, the solution $\\tan t$ exists only for $|t| \u003c \\pi/2$.\nWithout the Lipschitz condition, solutions exist by the [[Peano existence theorem|098YP831QY]], but they need not be unique. See the [[non-uniqueness example|00K7ASZ1PR]].\n\n## History\n\nThe theorem is named after Émile Picard and Ernst Lindelöf. It is also called the Cauchy–Lipschitz theorem, after Augustin-Louis Cauchy and Rudolf Lipschitz.\n\n## Explanations\n\n### L1 human\n\nIf the vector field is regular enough, the present state determines the whole motion near the start time.\nRegular enough means continuous in time and Lipschitz in the state.\nThen there is one and only one solution through the initial point, at least for a short time.\n",
  "commit": "6ac761d40018146541484a9e8b9eafd6ecc303dc",
  "front": {
    "aliases": [
      "cauchy-lipschitz-theorem",
      "existence-and-uniqueness-theorem"
    ],
    "kind": "statement",
    "msc": [
      "34A12"
    ],
    "name": "Picard–Lindelöf theorem",
    "role": "theorem",
    "same_as": {
      "wikidata": "Q530152",
      "wikipedia": "Picard–Lindelöf theorem"
    },
    "sources": [
      {
        "locator": "Theorem",
        "url": "https://en.wikipedia.org/wiki/Picard%E2%80%93Lindel%C3%B6f_theorem"
      }
    ],
    "tag": "0P6KRD5B7A",
    "uses": [
      "0J7HF2PF1H",
      "075VBY4K4X",
      "08XDXP86XE",
      "015KK993C5"
    ]
  },
  "kind": "statement",
  "link": "https://math2.tamnd.com/node/0P6KRD5B7A/",
  "links": [
    {
      "field": "uses",
      "target": "0J7HF2PF1H"
    },
    {
      "field": "uses",
      "target": "075VBY4K4X"
    },
    {
      "field": "uses",
      "target": "08XDXP86XE"
    },
    {
      "field": "uses",
      "target": "015KK993C5"
    }
  ],
  "markdown": "https://math2.tamnd.com/node/0P6KRD5B7A.md",
  "msc": [
    "34A12"
  ],
  "name": "Picard–Lindelöf theorem",
  "prov": null,
  "role": "theorem",
  "source": "https://github.com/tamnd/math2/blob/main/nodes/0P6/0P6KRD5B7A.md",
  "status": null,
  "tag": "0P6KRD5B7A",
  "uri": "m2://0P6KRD5B7A",
  "url": "https://math2.tamnd.com/wiki/picard-lindelof-theorem/",
  "writing": {
    "tag": "0P6KRD5B7A",
    "kind": "statement",
    "level": "W1",
    "next": [
      {
        "what": "edge about out",
        "note": "the concepts that the statement is about",
        "have": 0,
        "want": 1
      },
      {
        "what": "edge proves in",
        "note": "at least one proof node",
        "have": 0,
        "want": 1
      },
      {
        "what": "edge instance_of or illustrates in",
        "note": "one example that shows the statement at work",
        "have": 0,
        "want": 1
      },
      {
        "what": "explanation L2",
        "have": 0,
        "want": 1
      }
    ]
  }
}