{
  "aliases": null,
  "backlinks": [
    {
      "field": "uses",
      "target": "00BT70ND50"
    },
    {
      "field": "uses",
      "target": "015KK993C5"
    },
    {
      "field": "uses",
      "target": "08XDXP86XE"
    },
    {
      "field": "uses",
      "target": "0RJ5NHWYTP"
    }
  ],
  "body": "\n## Definition\n\nLet $J = [t_{\\min}, t_{\\max}]$, let $t_0 \\in J$ and let $x_0 \\in \\mathbb{R}^n$. Let $f : J \\times \\mathbb{R}^n \\to \\mathbb{R}^n$ be a function such that $s \\mapsto f(s, \\gamma(s))$ is integrable on $J$ for each continuous $\\gamma : J \\to \\mathbb{R}^n$.\nThe Picard operator of $f$ at $(t_0, x_0)$ maps a continuous curve $\\gamma : J \\to \\mathbb{R}^n$ to the curve $P\\gamma : J \\to \\mathbb{R}^n$ given by\n$$(P \\gamma)(t) = x_0 + \\int_{t_0}^{t} f\\left(s, \\gamma(s)\\right) ds.$$\n\n## Remarks\n\nAssume the [[Picard–Lindelöf hypotheses|0J7HF2PF1H]] with constants $a, L, K$. Then $P$ maps the continuous curves with values in the closed ball $\\|x - x_0\\| \\le a$ to curves in this ball.\nA fixed point of $P$ is a solution of the [[initial value problem|075VBY4K4X]]. See [[the integral equation form|015KK993C5]].\n[[Picard iteration|0RJ5NHWYTP]] applies $P$ again and again, starting from the constant curve $x_0$.\n\n## Explanations\n\n### L1 human\n\nThe operator takes a guess for the solution and makes a new guess.\nIt integrates the field along the old guess, starting from $x_0$.\nA curve that the operator does not change is a solution.\n",
  "commit": "6ac761d40018146541484a9e8b9eafd6ecc303dc",
  "front": {
    "defines": "07PB7HH1T9",
    "kind": "definition",
    "msc": [
      "34A12",
      "47H10"
    ],
    "name": "Picard operator",
    "sources": [
      {
        "locator": "Proof sketch",
        "url": "https://en.wikipedia.org/wiki/Picard%E2%80%93Lindel%C3%B6f_theorem"
      }
    ],
    "tag": "00WAVB68F9",
    "uses": [
      "0J7HF2PF1H",
      "075VBY4K4X"
    ]
  },
  "kind": "definition",
  "link": "https://math2.tamnd.com/node/00WAVB68F9/",
  "links": [
    {
      "field": "defines",
      "target": "07PB7HH1T9"
    },
    {
      "field": "uses",
      "target": "0J7HF2PF1H"
    },
    {
      "field": "uses",
      "target": "075VBY4K4X"
    }
  ],
  "markdown": "https://math2.tamnd.com/node/00WAVB68F9.md",
  "msc": [
    "34A12",
    "47H10"
  ],
  "name": "Picard operator",
  "prov": null,
  "role": "",
  "source": "https://github.com/tamnd/math2/blob/main/nodes/00W/00WAVB68F9.md",
  "status": null,
  "tag": "00WAVB68F9",
  "uri": "m2://00WAVB68F9",
  "url": "https://math2.tamnd.com/wiki/picard-operator-definition/",
  "writing": {
    "tag": "00WAVB68F9",
    "kind": "definition",
    "level": "W0",
    "next": [
      {
        "what": "an inventory row",
        "note": "the row of the inventory that the node accounts for",
        "have": 0,
        "want": 1
      }
    ]
  }
}