Journal of Integral Equations and Applications

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

Hermann Brunner, Yunyun Ma, and Yuesheng Xu

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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.

Article information

Source
J. Integral Equations Applications, Volume 27, Number 4 (2015), 455-487.

Dates
First available in Project Euclid: 8 February 2016

Permanent link to this document
https://projecteuclid.org/euclid.jiea/1454939250

Digital Object Identifier
doi:10.1216/JIE-2015-27-4-455

Mathematical Reviews number (MathSciNet)
MR3457679

Zentralblatt MATH identifier
1381.45009

Subjects
Primary: 45D05: Volterra integral equations [See also 34A12] 45H05: Miscellaneous special kernels [See also 44A15] 45J05: Integro-ordinary differential equations [See also 34K05, 34K30, 47G20]

Keywords
Volterra integral equation Volterra integro-differential equation highly oscillatory kernel oscillatory structured space decomposition of the oscillatory integral oscillation preserving solution

Citation

Brunner, Hermann; Ma, Yunyun; Xu, Yuesheng. The oscillation of solutions of Volterra integral and integro-differential equations with highly oscillatory kernels. J. Integral Equations Applications 27 (2015), no. 4, 455--487. doi:10.1216/JIE-2015-27-4-455. https://projecteuclid.org/euclid.jiea/1454939250


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