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

B. N. Enginbarian

Matematičeskie zametki, Tome 81 (2007) no. 5, pp. 693-702

Résumé

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.

Détail

Keywords: integral convolution equation, nonlinear Ambartsumyan equation, alternating measure, Wiener–Hopf operator, nonlinear factorization equation
Mots-clés : Volterra equation.