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Perturbation theory for dual semigroups II. Time-dependent perturbations in the sun-reflexive case

Published online by Cambridge University Press:  14 November 2011

Ph. Clément ,
O. Diekmann ,
M. Gyllenberg ,
H. J. A. M. Heijmans  and
H. R. Thieme
Ph. Clément
Affiliation:
Delft University of Technology, Department of Mathematics and Informatics, Julianalaan 132, Postbus 356, 2600 AJ Delft, The Netherlands
O. Diekmann
Affiliation:
Centre for Mathematics and Computer Science, P.O. Box 4079, 1009 AB Amsterdam, The Netherlands Institute of Theoretical Biology, University of Leiden, Groenhovenstraat 5, 2311 BT Leiden, The Netherlands
M. Gyllenberg
Affiliation:
Helsinki University of Technology, Department of Mathematics and Systems Analysis, SF-02150 Espoo, Finland
H. J. A. M. Heijmans
Affiliation:
Centre for Mathematics and Computer Science, P.O. Box 4079, 1009 AB Amsterdam, The Netherlands
H. R. Thieme
Affiliation:
Sonderforschungsbereich 123, Universitat Heidelberg, Im Neuenheimer Feld 294, D-6900 Heidelberg, Bundesrepublik Deutschland
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Synopsis

We consider time-dependent perturbations of generators of strongly continuous semigroups on a Banach space. The perturbations map the Banach space into a bigger space, which is the second dual of the original space in a specific semigroup sense. Using the theory of dual semigroups we show that the solutions of a generalised variation-of-constants formuladefine an evolutionary system. We investigate continuity and differentiability propertiesof this evolutionary system and its dual system and examine in what sense the perturbed generator and its adjoint generate these evolutionary systems. It is shown that the results apply naturally to retarded functional differential equations and age structured population dynamics.

Type
Research Article
Information
Proceedings of the Royal Society of Edinburgh Section A: Mathematics , Volume 109 , Issue 1-2 , 1988 , pp. 145 - 172
Copyright
Copyright © Royal Society of Edinburgh 1988

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