{
  "aliases": null,
  "backlinks": null,
  "body": "\n## Proof\n\nWe write the solutions as integrals and apply the integral form of Grönwall's inequality. Here $\\|\\cdot\\|$ is the Euclidean norm on $\\mathbb{R}^n$.\n\n1. As [[solutions|04SW7WECS8]] on $[a, b]$, the curves $\\gamma_1, \\gamma_2$ are differentiable, hence continuous, on $[a, b]$. For $t \\in (a, b)$ they satisfy $\\gamma_i'(t) = f(t, \\gamma_i(t))$. Only these facts are used.\n2. For $i = 1, 2$ let $g_i(s) = f(s, \\gamma_i(s))$ for $s \\in [a, b]$. As a composition of continuous maps, $g_i$ is continuous on $[a, b]$.\n3. Fix $t \\in (a, b]$. Apply the [[second fundamental theorem of calculus|0XYAFCN0PT]] to each coordinate of $\\gamma_i$ on $[a, t]$. The derivative extends to the continuous function $g_i$, so $\\gamma_i(t) = \\gamma_i(a) + \\int_a^t g_i(s)\\, ds$, with coordinatewise integrals.\n4. Let $e(t) = \\gamma_1(t) - \\gamma_2(t)$. By step 3 and [[linearity of the integral|080RWVVSB6]], $e(t) = e(a) + \\int_a^t \\left( g_1(s) - g_2(s) \\right) ds$.\n5. Let $v : [a, t] \\to \\mathbb{R}^n$ be continuous and let $w = \\int_a^t v(s)\\, ds$. We show $\\|w\\| \\le \\int_a^t \\|v(s)\\|\\, ds$.\n6. By linearity, $\\|w\\|^2 = \\sum_{j} w_j \\int_a^t v_j(s)\\, ds = \\int_a^t w \\cdot v(s)\\, ds$.\n7. By the [[Cauchy–Schwarz inequality|0RFWW56Y22]], $w \\cdot v(s) \\le \\|w\\|\\, \\|v(s)\\|$. By [[monotonicity of the integral|0MBWQKM8QR]], $\\|w\\|^2 \\le \\|w\\| \\int_a^t \\|v(s)\\|\\, ds$.\n8. If $w = 0$, then $\\|w\\| = 0$, and the integral of the non-negative function $\\|v\\|$ is non-negative. Otherwise divide by $\\|w\\| \u003e 0$.\n9. By the [[Lipschitz condition|055VVSN00V]] in the hypothesis, $\\|g_1(s) - g_2(s)\\| \\le K \\|e(s)\\|$ for $s \\in [a, b]$.\n10. Let $u(s) = \\|e(s)\\|$, a continuous function on $[a, b]$. By step 4, the [[triangle inequality|0FJZMBX0ZZ]], step 5 and step 9, $u(t) \\le u(a) + \\int_a^t K\\, u(s)\\, ds$.\n11. This also holds for $t = a$, where the integral is $0$. So it holds for each $t \\in [a, b]$.\n12. Apply the integral form of [[Grönwall's inequality|07EJHQYR9A]] on $I = [a, b]$ with $\\beta = K \\ge 0$ and the constant, hence non-decreasing, function $\\alpha = u(a)$.\n13. It gives $u(t) \\le u(a) \\exp\\left( \\int_a^t K\\, ds \\right) = u(a)\\, e^{K(t - a)}$. This is $\\|\\gamma_1(t) - \\gamma_2(t)\\| \\le \\|\\gamma_1(a) - \\gamma_2(a)\\|\\, e^{K(t - a)}$ for each $t \\in [a, b]$.\n\n## Idea\n\nThe difference of two solutions changes at a rate bounded by $K$ times its own size, because of the Lipschitz condition. In integral form this is exactly the hypothesis of Grönwall's inequality, which bounds the size by exponential growth with rate $K$.\n",
  "commit": "4a3290d299c8b2cd6493fce2296920ba7f82d942",
  "front": {
    "kind": "proof",
    "msc": [
      "34A12"
    ],
    "name": "Proof of continuous dependence on initial data by Grönwall's inequality",
    "proves": "0G3DBXRRQJ",
    "sources": [
      {
        "locator": "Integral form for continuous functions",
        "url": "https://en.wikipedia.org/wiki/Gr%C3%B6nwall%27s_inequality"
      }
    ],
    "tag": "RRBCGTD2SB",
    "uses": [
      "04SW7WECS8",
      "055VVSN00V",
      "0XYAFCN0PT",
      "080RWVVSB6",
      "0RFWW56Y22",
      "0MBWQKM8QR",
      "0FJZMBX0ZZ",
      "07EJHQYR9A"
    ]
  },
  "kind": "proof",
  "link": "https://math2.tamnd.com/node/RRBCGTD2SB/",
  "links": [
    {
      "field": "proves",
      "target": "0G3DBXRRQJ"
    },
    {
      "field": "uses",
      "target": "04SW7WECS8"
    },
    {
      "field": "uses",
      "target": "055VVSN00V"
    },
    {
      "field": "uses",
      "target": "0XYAFCN0PT"
    },
    {
      "field": "uses",
      "target": "080RWVVSB6"
    },
    {
      "field": "uses",
      "target": "0RFWW56Y22"
    },
    {
      "field": "uses",
      "target": "0MBWQKM8QR"
    },
    {
      "field": "uses",
      "target": "0FJZMBX0ZZ"
    },
    {
      "field": "uses",
      "target": "07EJHQYR9A"
    }
  ],
  "markdown": "https://math2.tamnd.com/node/RRBCGTD2SB.md",
  "msc": [
    "34A12"
  ],
  "name": "Proof of continuous dependence on initial data by Grönwall's inequality",
  "prov": null,
  "role": "",
  "source": "https://github.com/tamnd/math2/blob/main/nodes/RRB/RRBCGTD2SB.md",
  "status": null,
  "tag": "RRBCGTD2SB",
  "uri": "m2://RRBCGTD2SB",
  "url": "https://math2.tamnd.com/wiki/proof-of-continuous-dependence-on-initial-data-by-gronwalls-inequality/",
  "writing": {
    "tag": "RRBCGTD2SB",
    "kind": "proof",
    "level": "W0",
    "next": [
      {
        "what": "an inventory row",
        "note": "the row of the inventory that the node accounts for",
        "have": 0,
        "want": 1
      }
    ]
  }
}