On Volterra integral operators with highly oscillatory kernels

Hermann Brunner

doi:10.3934/dcds.2014.34.915

On Volterra integral operators with highly oscillatory kernels

Hermann Brunner^*

^*Corresponding author for this work

Department of Mathematics

Research output: Contribution to journal › Article › peer-review

15 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.

Original language	English
Pages (from-to)	915-929
Number of pages	15
Journal	Discrete and Continuous Dynamical Systems
Volume	34
Issue number	3
DOIs	https://doi.org/10.3934/dcds.2014.34.915
Publication status	Published - Mar 2014

Scopus Subject Areas

Analysis
Discrete Mathematics and Combinatorics
Applied Mathematics

User-Defined Keywords

Cordial volterra operators
Highly oscillatory kernels
Highly oscillatory solutions
Non-compact volterra operators
Spectra
Volterra integral operators

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10.3934/dcds.2014.34.915

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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 = "Cordial volterra operators, Highly oscillatory kernels, Highly oscillatory solutions, Non-compact volterra operators, Spectra, Volterra integral operators",

author = "Hermann Brunner",

note = "Funding information: The author{\textquoteright}s work is supported by the Hong Kong Research Grants Council (GRF Grant No. HKBU 200211) and the Natural Sciences and Engineering Research Council of Canada (NSERC Discovery Grant No. 9406).",

year = "2014",

month = mar,

doi = "10.3934/dcds.2014.34.915",

language = "English",

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pages = "915--929",

journal = "Discrete and Continuous Dynamical Systems",

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AU - Brunner, Hermann

N1 - Funding information: The author’s work is supported by the Hong Kong Research Grants Council (GRF Grant No. HKBU 200211) and the Natural Sciences and Engineering Research Council of Canada (NSERC Discovery Grant No. 9406).

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 - Cordial volterra operators

KW - Highly oscillatory kernels

KW - Highly oscillatory solutions

KW - Non-compact volterra operators

KW - Spectra

KW - Volterra integral operators

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DO - 10.3934/dcds.2014.34.915

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VL - 34

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JO - Discrete and Continuous Dynamical Systems

JF - Discrete and Continuous Dynamical Systems

SN - 1078-0947

IS - 3

ER -

On Volterra integral operators with highly oscillatory kernels

Abstract

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