On the Convolution Equation with Positive Kernel Expressed via an Alternating Measure

We consider the integral convolution equation on the half-line or on a finite interval with kernel
$$ K(x-t)=\int_a^be^{-|x-t|s}\,d\sigma(s) $$
with an alternating measure $d\sigma$ under the conditions
$$ K(x)>0, \quad \int_a^b\frac{1}{s}\,|d\sigma(s)|<+\infty, \quad \int_{-\infty}^\infty K(x)\,dx=2\int_a^b\frac{1}{s}\,d\sigma(s)\le1. $$
The solution of the nonlinear Ambartsumyan equation
$$ \varphi(s)=1+\varphi(s)\int_a^b\frac{\varphi(p)}{s+p}\,d\sigma(p), $$
is constructed; it can be effectively used for solving the original convolution equation.