Volterra equations with completely monotone kernels

Summary

These are Volterra equations of convolution type whose kernel $k$ is completely monotone on $(0, \infty)$: it is infinitely differentiable and its derivatives alternate in sign. By Bernstein's theorem on monotone functions, such a kernel is a superposition $k(t) = \int e^{-st} \, d\mu(s)$ of decaying exponentials with a positive measure $\mu$. The class contains the exponential kernels and the Abel kernels $t^{\alpha - 1}$ with $0 < \alpha < 1$. It is a natural setting for positivity and decay results for solutions. Such kernels model fading memory, for instance in linear viscoelasticity.