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  "body": "\n## Summary\n\nA Urysohn integral equation is a nonlinear integral equation in which the unknown function enters the integrand through a general function of three variables: $\\varphi(x) = f(x) + \\int_\\Omega K(x, t, \\varphi(t))\\, dt$.\nIt is the most general form of a [[nonlinear integral equation|BSCYBP6AS6]] of Fredholm type in one unknown.\nIt contains the Hammerstein equation as a special case.\nExistence results for it usually come from fixed point theorems, such as the [[Schauder fixed-point theorem|DCPFVVZZ54]], applied to the operator on the right side.\nThe name is also written Uryson.\n\n## Explanations\n\n### L1 human\n\nIn a linear integral equation the unknown is multiplied by a kernel and then integrated.\nIn a Urysohn equation the kernel may depend on the unknown in any way, for example through $\\sin(x\\, \\varphi(t))$.\nThe equation still says that the unknown at a point is a weighted total of its values elsewhere.\nFor instance, $\\varphi(x) = x$ solves $\\varphi(x) = x + \\int_0^1 \\sin\\bigl(x(\\varphi(t) - t)\\bigr)\\, dt$, since the sine is then $0$.\n",
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