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SMOOTH SOLUTIONS OF VOLTERRA EQUATIONS VIA SEMIGROUPS

Part of: Groups and semigroups of linear operators, their generalizations and applications

Published online by Cambridge University Press:  01 October 2008

TOMÁŠ BÁRTA
TOMÁŠ BÁRTA*
Affiliation:
Faculty of Mathematics and Physics, Department of Mathematical Analysis, Charles University, Prague, Sokolovska 83,180 00 Prague 8, Czech Republic (email: barta@karlin.mff.cuni.cz)
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Abstract

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In this paper we introduce a class of left shift semigroups that are differentiable. With the help of perturbation theory for differentiable semigroups we show that solutions of an integrodifferential equation can be infinitely differentiable if the convolution kernel is sufficiently smooth and regular.

Keywords

differentiable semigroupbounded perturbationshift semigroupintegro-differential equation

MSC classification

Secondary: 47D06: One-parameter semigroups and linear evolution equations
Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 78 , Issue 2 , October 2008 , pp. 249 - 260
Copyright
Copyright © 2008 Australian Mathematical Society

References

[1]Barta, T., ‘Analytic solutions for integrodifferential equations via semigroups’, Semigroup Forum 76 (2008), 142–148.CrossRefGoogle Scholar
[2]Barta, T., ‘On R-sectorial derivatives on Bergman spaces’, Bull. Aust. Math. Soc. 77 (2008), 305–313.CrossRefGoogle Scholar
[3]Batty, C. J. K., Differentiability of Perturbed Semigroups and Delay Semigroups, Banach Center Publications, 75 (Polish Academy of Science, Warsaw, 2007).CrossRefGoogle Scholar
[4]Desch, W. and Schappacher, W., ‘On relatively bounded perturbations of linear C 0 semigroups’, Ann. Scuola Norm. Sup. Pisa Cl. Sci. 11 (1984), 327–341.Google Scholar
[5]Doytchinov, B., Hrusa, W. J. and Watson, S. J., ‘On perturbations of differentiable semigroups’, Semigroup Forum 54 (1997), 100–111.CrossRefGoogle Scholar
[6]Engel, K. J. and Nagel, R., One-parameter Semigroups for Linear Evolution Equations (Springer, New York, 2000).Google Scholar
[7]Iley, P., ‘Perturbations of differentiable semigroups’, J. Evol. Eq. 7 (2007), 765–781.CrossRefGoogle Scholar
[8]Nagel, R., ‘Towards a ‘matrix theory’ for unbounded operator matrices’, Math. Z. 201 (1989), 57–68.CrossRefGoogle Scholar
[9]Pazy, A., ‘On the differentiability and compactness of semi-groups of linear operators’, J. Math. Mech. 17 (1968), 1131–1141.Google Scholar
[10]Pruss, J., Evolutionary integral equations and applications (Birkhauser, Basel, 1993).CrossRefGoogle Scholar
[11]Renardy, M., ‘On the stability of differentiability of semigroups’, Semigroup Forum 51 (1995), 343–346.CrossRefGoogle Scholar