Integral equation of the second kind

Summary

An integral equation of the second kind is a linear integral equation of the form $\varphi(x) = f(x) + \lambda \int K(x, s) \varphi(s) \, ds$, where the unknown also appears outside the integral. In operator form it reads $(I - \lambda K) \varphi = f$. For reasonable kernels this equation is well behaved: Fredholm theory shows that it is uniquely solvable for all $f$ unless $\lambda$ is a characteristic value. For small $\abs{\lambda}$ the solution is given by the Liouville–Neumann series.