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  "body": "\n## Summary\n\nA nonlinear integral equation is an [[integral equation|R3SG1SN7JQ]] that is not a [[linear integral equation|AYZCKG34QK]].\nTypical forms are the Urysohn equation $\\varphi(x) = f(x) + \\int_a^b H(x, s, \\varphi(s)) \\, ds$ and the Hammerstein equation $\\varphi(x) = f(x) + \\int_a^b K(x, s) g(s, \\varphi(s)) \\, ds$.\nThey arise from nonlinear differential equations, such as initial value problems for $y' = F(x, y)$.\nThe main tools are fixed point theorems, such as the [[Banach fixed point theorem|0VK6VEH345]].\n\n## Explanations\n\n### L1 human\n\nIn a nonlinear equation the unknown function appears in a curved way, for example squared or inside a sine.\nThen solutions cannot be added, and there may be no solution, one solution or several.\nFor example, $\\varphi(x) = 1 + \\int_0^x \\varphi(s)^2 \\, ds$ is the integral form of $y' = y^2$, $y(0) = 1$.\nIts solution $\\varphi(x) = 1/(1 - x)$ exists only for $x \u003c 1$.\n",
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