Hammerstein integral equation

Summary

A 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. In operator form it is $\varphi = f + K N_g \varphi$, where $K$ is a linear integral operator and $N_g$ is the Nemytskii operator of $g$. It is a special case of the Urysohn equation. Its split structure lets one combine linear tools, such as spectral theory of $K$, with monotone operators and fixed point theorems. Boundary value problems $-u'' = g(x, u)$ with Dirichlet conditions become Hammerstein equations with the Green's function as kernel.