On Volterra integral operators with highly oscillatory kernels

Research output: Contribution to journalArticle

5 Citations (Scopus)

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.
LanguageEnglish
Pages915-929
Number of pages15
JournalDiscrete and Continuous Dynamical Systems - Series A
Volume34
Issue number3
DOIs
Publication statusPublished - Mar 2014

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Volterra Operator
Integral Operator
Integral equations
kernel
Oscillation
Volterra Integral Equations
Uncountable
Compact Operator
Integral Equations

Keywords

  • Volterra integral operators
  • highly oscillatory kernels
  • non-compact Volterra operators
  • cordial Volterra operators
  • spectra
  • highly oscillatory solutions

Cite this

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title = "On Volterra integral operators with highly oscillatory kernels",
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.",
keywords = "Volterra integral operators, highly oscillatory kernels, non-compact Volterra operators, cordial Volterra operators, spectra, highly oscillatory solutions",
author = "Hermann Brunner",
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language = "English",
volume = "34",
pages = "915--929",
journal = "Discrete and Continuous Dynamical Systems - Series A",
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}

On Volterra integral operators with highly oscillatory kernels. / Brunner, Hermann.

In: Discrete and Continuous Dynamical Systems - Series A , Vol. 34, No. 3, 03.2014, p. 915-929.

Research output: Contribution to journalArticle

TY - JOUR

T1 - On Volterra integral operators with highly oscillatory kernels

AU - Brunner, Hermann

PY - 2014/3

Y1 - 2014/3

N2 - 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.

AB - 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.

KW - Volterra integral operators

KW - highly oscillatory kernels

KW - non-compact Volterra operators

KW - cordial Volterra operators

KW - spectra

KW - highly oscillatory solutions

U2 - 10.3934/dcds.2014.34.915

DO - 10.3934/dcds.2014.34.915

M3 - Article

VL - 34

SP - 915

EP - 929

JO - Discrete and Continuous Dynamical Systems - Series A

T2 - Discrete and Continuous Dynamical Systems - Series A

JF - Discrete and Continuous Dynamical Systems - Series A

SN - 1078-0947

IS - 3

ER -