Nonlinear integral equation

Summary

A nonlinear integral equation is an integral equation that is not a linear integral equation. Typical forms are the Urysohn equation $\varphi(x) = f(x) + \int_a^b H(x, s, \varphi(s)) \, ds$ and the Hammerstein equation $\varphi(x) = f(x) + \int_a^b K(x, s) g(s, \varphi(s)) \, ds$. They arise from nonlinear differential equations, such as initial value problems for $y' = F(x, y)$. The main tools are fixed point theorems, such as the Banach fixed point theorem.