On Volterra integral operators with highly oscillatory kernels

Hermann Brunner

doi:10.3934/dcds.2014.34.915

Article Contents

2014, Volume 34, Issue 3: 915-929. Doi: 10.3934/dcds.2014.34.915

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On Volterra integral operators with highly oscillatory kernels

Hermann Brunner^1,

1.
Department of Mathematics, Hong Kong Baptist University, Hong Kong

Received: November 30, 2012

Revised: December 31, 2012

Published: February 2014

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Abstract

We study the high-oscillation properties of solutions to integral equations associated with two classes of Volterra integral operators: compact operators with highly oscillatory kernels that are either smooth or weakly singular, and noncompact cordial Volterra integral operators with highly oscillatory kernels. In the latter case the focus is on the dependence of the (uncountable) spectrum on the oscillation parameter. It is shown that the results derived in this paper merely open a window to a general theory of solutions of highly oscillatory Volterra integral equations, and many questions remain to be answered.

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Mathematics Subject Classification: 45D05, 47G10, 65R20.

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