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  "body": "\n## Summary\n\nA Hammerstein integral equation has the form $\\varphi(x) = f(x) + \\int_\\Omega k(x, t)\\, g(t, \\varphi(t))\\, dt$: a linear kernel $k$ acts on a pointwise nonlinear function of the unknown.\nIn operator form it is $\\varphi = f + K N_g \\varphi$, where $K$ is a linear integral operator and $N_g$ is the [[Nemytskii operator|W8G3AZ0DNG]] of $g$.\nIt is a special case of the [[Urysohn equation|WZ4Z26XSVC]].\nIts split structure lets one combine linear tools, such as spectral theory of $K$, with monotone operators and fixed point theorems.\nBoundary value problems $-u'' = g(x, u)$ with Dirichlet conditions become Hammerstein equations with the Green's function as kernel.\n\n## Explanations\n\n### L1 human\n\nFirst apply a nonlinear rule to the unknown at each point, for example squaring it.\nThen smear the result out with a fixed weight function and integrate.\nThe Hammerstein equation asks that this give back the unknown, up to a known term.\nFor example, on $[0, 1]$ the function $\\varphi(x) = 4x$ solves $\\varphi(x) = \\int_0^1 x t\\, \\varphi(t)^2\\, dt$, because $\\int_0^1 x t \\cdot 16 t^2\\, dt = 4x$.\n",
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