Volterra equation of the second kind

Summary

A Volterra equation of the second kind is an equation $x(t) = f(t) + \lambda \int_a^t K(t, s) \, x(s) \, ds$, in which the unknown appears both outside and inside the integral. It is the Volterra integral equation case of an integral equation of the second kind. Unlike the Fredholm case, it has exactly one continuous solution for every value of $\lambda$ when the kernel and the function $f$ are continuous. Initial value problems for linear ordinary differential equations lead to equations of this kind.