{
  "aliases": null,
  "backlinks": null,
  "body": "\n## Definition\n\nLet $a \u003c b$, let $\\Delta = \\set{(t, s) : a \\le s \\le t \\le b}$, let $K \\colon \\Delta \\times \\R \\to \\R$ and $f \\colon [a, b] \\to \\R$ be [[continuous|0X3WGX95FE]].\nA *nonlinear Volterra equation of the second kind* for an unknown continuous function $\\varphi$ is\n$$\\varphi(t) = f(t) + \\int_a^t K\\bigl(t, s, \\varphi(s)\\bigr)\\, ds, \\quad a \\le t \\le b.$$\nA *nonlinear Volterra equation of the first kind* is $\\int_a^t K(t, s, \\varphi(s))\\, ds = f(t)$.\nA solution on a subinterval $[a, c]$ with $c \\le b$ is a continuous function on $[a, c]$ that satisfies the equation there.\n\n## Remarks\n\nIf $K(t, s, u) = k(t, s)\\, u$, this is the linear [[Volterra integral equation|DND8CBEVXT]] of the second kind.\nIf $K$ satisfies a [[Lipschitz condition|0AW1E6YTJY]] $\\abs{K(t, s, u) - K(t, s, v)} \\le L \\abs{u - v}$ on all of $\\Delta \\times \\R$, the equation has exactly one continuous solution on $[a, b]$, and the successive approximations converge to it.\nIf the Lipschitz condition holds only for $u, v$ in bounded sets, there is still a unique solution on some interval $[a, c]$ with $c \u003e a$.\n\nExample: $\\varphi(t) = 1 + \\int_0^t \\varphi(s)^2\\, ds$ on $[0, 1)$, with $K(t, s, u) = u^2$.\nThe function $\\varphi(t) = 1/(1 - t)$ solves it, since $1 + \\int_0^t (1 - s)^{-2}\\, ds = 1 + \\frac{1}{1 - t} - 1 = \\frac{1}{1 - t}$.\nIt is the integral form of $y' = y^2$, $y(0) = 1$, and the solution does not extend past $t = 1$.\n\nNon-example: $\\varphi(t) = f(t) + \\int_a^b K(t, s, \\varphi(s))\\, ds$ has the fixed upper limit $b$; it is a Urysohn equation, not a Volterra one.\n",
  "commit": "ff8a40749dc16545e3d2fc637f6b0196e81d0bfa",
  "front": {
    "defines": "F18MBATVVP",
    "kind": "definition",
    "msc": [
      "45D05",
      "45G10"
    ],
    "name": "Nonlinear Volterra equation",
    "sources": [
      {
        "url": "https://en.wikipedia.org/wiki/Volterra_integral_equation"
      }
    ],
    "tag": "1DFG72YV3N",
    "uses": [
      "DND8CBEVXT",
      "BSCYBP6AS6",
      "0X3WGX95FE",
      "0AW1E6YTJY"
    ]
  },
  "kind": "definition",
  "link": "https://tamnd.github.io/math2/node/1DFG72YV3N/",
  "links": [
    {
      "field": "defines",
      "target": "F18MBATVVP"
    },
    {
      "field": "uses",
      "target": "DND8CBEVXT"
    },
    {
      "field": "uses",
      "target": "BSCYBP6AS6"
    },
    {
      "field": "uses",
      "target": "0X3WGX95FE"
    },
    {
      "field": "uses",
      "target": "0AW1E6YTJY"
    }
  ],
  "markdown": "https://tamnd.github.io/math2/node/1DFG72YV3N.md",
  "msc": [
    "45D05",
    "45G10"
  ],
  "name": "Nonlinear Volterra equation",
  "prov": null,
  "role": "",
  "source": "https://github.com/tamnd/math2/blob/main/nodes/1DF/1DFG72YV3N.md",
  "status": null,
  "tag": "1DFG72YV3N",
  "uri": "m2://1DFG72YV3N",
  "url": "https://tamnd.github.io/math2/wiki/nonlinear-volterra-equation-definition/",
  "writing": {
    "tag": "1DFG72YV3N",
    "kind": "definition",
    "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
      }
    ]
  }
}