Convolution equation with a completely monotonic kernel on the half-line

The Wiener-Hopf integral equation
\begin {equation} f(x)=g(x)+\int _0^\infty K(x-t) f(t)\,dt,\qquad (I-K)f=g \tag{{1}}\end {equation}
and the related problems of factorization are considered for the kernels $\displaystyle K(\pm x)=\int _a^b e^{-xp}\,d\sigma _\pm (p)$, where $\sigma _\pm (p)\uparrow{}$ and $\displaystyle\mu \equiv \sum _\pm \int _a^b \frac 1p\,d\sigma _\pm (p)<+\infty$. If $K$ is even or the symbol $1-\widehat K(s)$ has a positive zero, then the existence of Volterra factorization is proved in the supercritical case $\mu >1$. An extension of this result to the general supercritical case is indicated. The solubility of the corresponding equation (1) is proved for $g \in L_1(0,\infty )$. Several other results in the supercritical case or for $\mu=1$ are obtained. The approach discussed is essentially based on the method of special factorization and on the generalized Ambartsumyan equations.