TY - JOUR
T1 - On Volterra integral operators with highly oscillatory kernels
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
UR - http://www.scopus.com/inward/record.url?scp=84886389039&partnerID=8YFLogxK
U2 - 10.3934/dcds.2014.34.915
DO - 10.3934/dcds.2014.34.915
M3 - Journal article
AN - SCOPUS:84886389039
SN - 1078-0947
VL - 34
SP - 915
EP - 929
JO - Discrete and Continuous Dynamical Systems
JF - Discrete and Continuous Dynamical Systems
IS - 3
ER -