Cauchy problem for a class of integro-differential equations

We consider the Cauchy problem for integro-differential equations containing Wiener-Hopf operators. We define the characteristic polynomial of the problem. We show that if all roots of the characteristic polynomial have negative imaginary parts and the corresponding strong moment problem is solvable, then the problem is equivalent to a Wiener-Hopf integral equation. We consider an example of an application of the result to a problem related to nonlocal wave interaction.