{
  "aliases": null,
  "backlinks": null,
  "body": "\n## Statement\n\nLet $J$ be an interval, let $U$ be an open subset of $\\mathbb{R}^n$ and let $K \\ge 0$. Let $f : J \\times U \\to \\mathbb{R}^n$ satisfy $\\|f(t, x) - f(t, y)\\| \\le K \\|x - y\\|$ for all $t \\in J$ and $x, y \\in U$.\nLet $a \u003c b$ with $[a, b] \\subseteq J$, and let $\\delta, \\varepsilon_1, \\varepsilon_2 \\ge 0$.\nLet $\\gamma_1, \\gamma_2 : [a, b] \\to U$ be continuous on $[a, b]$ and differentiable on $(a, b)$, with $\\|\\gamma_i'(t) - f(t, \\gamma_i(t))\\| \\le \\varepsilon_i$ for $i = 1, 2$ and $t \\in (a, b)$.\nIf $\\|\\gamma_1(a) - \\gamma_2(a)\\| \\le \\delta$, then for each $t \\in [a, b]$,\n$$\\|\\gamma_1(t) - \\gamma_2(t)\\| \\le G_{\\delta, K, \\varepsilon_1 + \\varepsilon_2}(t - a),$$\nwhere $G$ is the [[Grönwall bound|0FRX8JD7QH]].\n\n## Remarks\n\nA curve with $\\|\\gamma'(t) - f(t, \\gamma(t))\\| \\le \\varepsilon$ is an $\\varepsilon$-approximate solution.\nFor $\\varepsilon_1 = \\varepsilon_2 = 0$, the result is [[continuous dependence on initial data|0G3DBXRRQJ]].\nThe proof applies the derivative form of [[Grönwall's inequality|07EJHQYR9A]] with the additional term $\\varepsilon_1 + \\varepsilon_2$ to $\\|\\gamma_1 - \\gamma_2\\|$.\n\n## Explanations\n\n### L1 human\n\nA curve that almost solves the equation stays close to a true solution.\nThe error has two sources: the difference of initial values and the defects $\\varepsilon_i$.\nBoth can grow at most exponentially in time.\n",
  "commit": "6ac761d40018146541484a9e8b9eafd6ecc303dc",
  "front": {
    "kind": "statement",
    "msc": [
      "34A12",
      "34A45"
    ],
    "name": "Estimate for approximate solutions",
    "role": "theorem",
    "sources": [
      {
        "locator": "theorem dist_le_of_approx_trajectories_ODE_of_mem",
        "note": "Mathlib at the commit that lean/lake-manifest.json pins. The theorem states the estimate with gronwallBound for two approximate solutions that stay in a set where the field is K-Lipschitz.",
        "url": "https://github.com/leanprover-community/mathlib4/blob/56a005855ea17e0b0958f73ec801ae140b0a1679/Mathlib/Analysis/ODE/Gronwall.lean#L162"
      },
      {
        "locator": "lead section",
        "url": "https://en.wikipedia.org/wiki/Gr%C3%B6nwall%27s_inequality"
      }
    ],
    "tag": "0JTPMCCZT2",
    "uses": [
      "0FRX8JD7QH",
      "055VVSN00V",
      "07EJHQYR9A"
    ]
  },
  "kind": "statement",
  "link": "https://math2.tamnd.com/node/0JTPMCCZT2/",
  "links": [
    {
      "field": "uses",
      "target": "0FRX8JD7QH"
    },
    {
      "field": "uses",
      "target": "055VVSN00V"
    },
    {
      "field": "uses",
      "target": "07EJHQYR9A"
    }
  ],
  "markdown": "https://math2.tamnd.com/node/0JTPMCCZT2.md",
  "msc": [
    "34A12",
    "34A45"
  ],
  "name": "Estimate for approximate solutions",
  "prov": null,
  "role": "theorem",
  "source": "https://github.com/tamnd/math2/blob/main/nodes/0JT/0JTPMCCZT2.md",
  "status": null,
  "tag": "0JTPMCCZT2",
  "uri": "m2://0JTPMCCZT2",
  "url": "https://math2.tamnd.com/wiki/estimate-for-approximate-solutions/",
  "writing": {
    "tag": "0JTPMCCZT2",
    "kind": "statement",
    "level": "W0",
    "next": [
      {
        "what": "an inventory row",
        "note": "the row of the inventory that the node accounts for",
        "have": 0,
        "want": 1
      }
    ]
  }
}