Urysohn integral equation

Summary

A Urysohn integral equation is a nonlinear integral equation in which the unknown function enters the integrand through a general function of three variables: $\varphi(x) = f(x) + \int_\Omega K(x, t, \varphi(t))\, dt$. It is the most general form of a nonlinear integral equation of Fredholm type in one unknown. It contains the Hammerstein equation as a special case. Existence results for it usually come from fixed point theorems, such as the Schauder fixed-point theorem, applied to the operator on the right side. The name is also written Uryson.