Volterra equations with completely monotone kernels

Remarks

Example: $k(t) = t^{-1/2}$. Its derivatives are $k^{(j)}(t) = (-1)^j \frac{1 \cdot 3 \cdots (2j - 1)}{2^j} t^{-1/2 - j}$, so $(-1)^j k^{(j)} \ge 0$, and $\int_0^1 t^{-1/2} \, dt = 2$. So Abel's integral equation is a first-kind equation with a completely monotone kernel. Non-example: $k(t) = t$ is positive but increasing, so $(-1)^1 k'(t) = -1 < 0$ and clause 1 fails. Non-example: $k(t) = 1/t$ is completely monotone but not integrable near $0$, so clause 2 fails. By Bernstein's theorem, clause 1 holds exactly when $k(t) = \int_{[0, \infty)} e^{-st} \, d\mu(s)$ for a positive Borel measure $\mu$ for which the integral converges for $t > 0$. For $k(t) = t^{-1/2}$ the measure is $d\mu(s) = \pi^{-1/2} s^{-1/2} \, ds$.