Fredholm equation of the second kind

Summary

A Fredholm equation of the second kind is an integral equation $\varphi(x) - \lambda \int_a^b K(x, y) \varphi(y) \, dy = f(x)$ with fixed limits $a$ and $b$. The unknown $\varphi$ appears both outside and under the integral. It is the Fredholm integral equation form of an integral equation of the second kind. Boundary value problems for differential equations lead to it, and its theory is very complete. For all but a discrete set of values of the parameter $\lambda$, it has exactly one solution for every right-hand side.