On the convolution equation with positive kernel expressed via an alternating measure

Abstract

We consider the integral convolution equation on the half-line or on a finite interval with kernel

$$K(x - t) = \int_a^b {e^{ - \left| {x - t} \right|s} d\sigma (s)} $$

with an alternating measure dσ under the conditions

$$K(x) > 0, \int_a^b {\frac{1}{s}\left| {d\sigma (s)} \right| < + \infty } , \int_{ - \infty }^\infty {K(x)dx = 2} \int_a^b {\frac{1}{s}d\sigma (s) \leqslant 1} .$$

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.