The oscillation of solutions of Volterra integral and integro-differential equations with highly oscillatory kernels

Abstract

We study the oscillatory structures of solutions of Volterra integral and integro-differential equations (VIEs, VIDEs) with highly oscillatory kernels. Based on the structured oscillatory spaces introduced in Wang and Xu \cite {OPG}, we first analyze the degree of oscillation of the solution of VIEs associated with the oscillatory kernels belonging to a certain structured oscillatory space by using the resolvent representation of the solution. According to a decomposition of the oscillatory integrals in the complex plane, we prove that the Volterra integral operator reduces the oscillatory order of the functions in the structured oscillatory spaces corresponding to the oscillatory structure of the kernel. The analogous oscillatory structure of solutions of VIDEs is then analyzed by representing the solution of the VIDEs by the differential resolvent kernel and by exploiting the relationship between the VIDEs and the equivalent VIE. We conclude that the solutions of the VIEs and VIDEs preserve the oscillatory components of the kernel.