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  "body": "\n## Definition\n\nLet $a \u003c b$.\nA *nonlinear integral equation* on $[a, b]$ is an [[integral equation|R3SG1SN7JQ]] for an unknown function $\\varphi \\colon [a, b] \\to \\C$ that is not a [[linear integral equation|AYZCKG34QK]].\nThe most common form is the *Urysohn equation*\n$$\\varphi(x) = f(x) + \\int_{I_x} H\\left(x, s, \\varphi(s)\\right) ds \\quad \\text{for all } x \\in [a, b].$$\nHere $I_x$ is $[a, b]$ or $[a, x]$, $f$ is given, and $H \\colon [a, b] \\times [a, b] \\times \\C \\to \\C$ is a given function that is not affine in its third variable.\n\n## Remarks\n\nExample: $\\varphi(x) = x + \\int_0^1 \\varphi(s)^2 \\, ds$ on $[0, 1]$ is nonlinear, with $H(x, s, u) = u^2$.\nA solution has the form $\\varphi(x) = x + c$, where $c = \\int_0^1 (s + c)^2 \\, ds = 1/3 + c + c^2$.\nThis gives $c^2 = -1/3$, so the equation has the two complex solutions $\\varphi(x) = x \\pm i/\\sqrt{3}$ and no real one.\n\nNon-example: $\\varphi(x) = x + \\int_0^1 (x s \\, \\varphi(s) + s^2) \\, ds$ is linear, since $H(x, s, u) = x s u + s^2$ is affine in $u$.\n\nWhen $H(x, s, u) = K(x, s) \\, g(s, u)$ the equation is called a *Hammerstein equation*.\n",
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