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  "body": "\n## Definition\n\nLet $\\Omega$ be a measurable subset of $\\R^m$, for example a closed interval $[a, b]$.\nLet $K \\colon \\Omega \\times \\Omega \\times \\R \\to \\R$ and $f \\colon \\Omega \\to \\R$ be given functions.\nA *Urysohn integral equation* for an unknown function $\\varphi \\colon \\Omega \\to \\R$ is\n$$\\varphi(x) = f(x) + \\int_\\Omega K\\bigl(x, t, \\varphi(t)\\bigr)\\, dt, \\quad x \\in \\Omega.$$\nThe function $K$ is the *Urysohn kernel*, and the map $\\varphi \\mapsto \\int_\\Omega K(x, t, \\varphi(t))\\, dt$ is the *Urysohn operator*.\n\n## Remarks\n\nIn the classical setting $\\Omega = [a, b]$, $K$ and $f$ are [[continuous|0X3WGX95FE]], and $\\varphi$ is sought among continuous functions.\nIf $K(x, t, u) = k(x, t)\\, u$, the equation is a linear Fredholm equation of the second kind.\nIf $K(x, t, u) = k(x, t)\\, g(t, u)$, it is a Hammerstein equation.\nSome authors also call $\\int_\\Omega K(x, t, \\varphi(t))\\, dt = f(x)$ a Urysohn equation of the first kind.\n\nExample: on $[0, 1]$, take $f(x) = x$ and $K(x, t, u) = \\sin\\bigl(x(u - t)\\bigr)$.\nThen $\\varphi(x) = x$ is a solution: the integrand is $\\sin 0 = 0$, so the right side is $x$.\n\nNon-example: $\\varphi(x)^3 = f(x) + \\int_\\Omega k(x, t)\\, \\varphi(t)\\, dt$ is a nonlinear integral equation, but not a Urysohn equation, since the unknown appears outside the integral through $\\varphi(x)^3$ and not alone.\n",
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