Abstract
In this short note we show that delay equations can be reformulated as abstract weak*-integral equations (AIE) involving dual semigroups, even in the case of infinite delay and/or when the solution takes values in a non-reflexive Banach space. The advantage is that for such (AIE) the standard local stability and bifurcation results are already available, see [8]. Our motivation derives from models of physiologically structured populations, as explained in more detail in [12].
To the memory of Günter Lumer, a source of inspiration to both of us.
The work of M.G. was supported by the Academy of Finland.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
References
A. Bátkai, S. Piazzera, Semigroups for Delay Equations. Research Notes in Mathematics vol. 10, A.K. Peters, Wellesley, MA, 2005.
P.L. Butzer and H. Berens, Semi-Groups of Operators and Approximation. Springer-Verlag, Berlin, Heidelberg, New York, 1967.
Ph. Clément, O. Diekmann, M. Gyllenberg, H.J.A.M. Heijmans, and H. Thieme, Perturbation theory for dual semigroups I: The sun-reflexive case. Math. Ann. 277 (1987), 709–725.
Ph. Clément, O. Diekmann, M. Gyllenberg, H.J.A.M. Heijmans, and H. Thieme, Perturbation theory for dual semigroups II: Time dependent perturbations in the sun-reflexive case. Proc. Roy. Soc. Edinburgh A 109 (1988), 145–172.
Ph. Clément, O. Diekmann, M. Gyllenberg, H.J.A.M. Heijmans, and H. Thieme, Perturbation theory for dual semigroups III: Nonlinear Lipschitz continuous perturbations in the sun-reflexive case. In: Volterra Integro-Differential Equations in Banach Spaces and Applications, (G. da Prato and M. Iannelli, eds.) Pitman, Boston, (1989), pp. 67–89.
Ph. Clément, O. Diekmann, M. Gyllenberg, H.J.A.M. Heijmans, and H. Thieme, Perturbation theory for dual semigroups IV: The intertwining formula and the canonical pairing. In Trends in semigroup theory and applications. Ph. Clément, S. Invernizzi, E. Mitidieri, I.I. Vrabie (Eds.), Marcel Dekker, (1989), pp. 95–116.
O. Diekmann, M. Gyllenberg, H.R. Thieme, Perturbation theory for dual semigroups. V. Variation of constants formulas. In Semigroup Theory and Evolution Equations, Ph. Clément, E. Mitidieri, and B. de Pagter (Eds.), Marcel Dekker, (1991) 107–123.
O. Diekmann, S.A. van Gils, S.M. Verduyn Lunel and H.-O. Walther, Delay Equations. Functional-, Complex-, and Nonlinear Analysis. Springer-Verlag, New York 1995.
O. Diekmann, M. Gyllenberg, H. Huang, M. Kirkilionis, J.A.J. Metz and H.R. Thieme, On the formulation and analysis of general deterministic structured population models II: Nonlinear theory, J. Math. Biol. 43 (2001), 157–189.
O. Diekmann, M. Gyllenberg and J.A.J. Metz, Steady-state analysis of structured population models, Theor. Pop. Biol. 65 (2003), 309–338.
O. Diekmann, M. Gyllenberg, and J.A.J. Metz, Physiologically Structured Population Models: Towards a General Mathematical Theory. In: Mathematics for Ecology and Environmental Sciences, Y. Takeuchi, Y. Iwasa, K. Sato (Eds.), Springer-Verlag, Berlin-Heidelberg (2007) pp. 5–20.
O. Diekmann, Ph. Getto, M. Gyllenberg, Stability and bifurcation analysis of Volterra functional equations in the light of suns and stars. SIAM Journal on Mathematical Analysis, in the press.
J. Diestel and J.J. Uhl, Vector measures, Math. Surveys, Vol. 15, AMS, Providence, RI, 1977.
K.-J. Engel, R. Nagel, One-Parameter Semigroups for Linear Evolution Equations. Springer-Verlag, New York-Heidelberg-Berlin, 2000.
G. Greiner and J.M.A.M. van Neerven, Adjoints of semigroups acting on vector-valued function spaces, Israel Journal of Mathematics, 77 (1992), 305–333.
Y. Hino, S. Murakami, T. Naito, Functional Differential Equations with Infinite Delay. Springer-Verlag, Heidelberg 1991.
M.A. Krasnosel’skii, P.P. Zabreiko, E.I. Pustyl’nik, P. E. Sobolevskii, Integral operators in spaces of summable functions. Translated from the Russian by T. Ando. Monographs and Textbooks on Mechanics of Solids and Fluids, Mechanics: Analysis. Noordhoff International Publishing, Leiden, 1976.
J. van Neerven, The adjoint of a semigroup of linear operators. Lecture Notes in Mathematics, 1529. Springer-Verlag, Berlin, 1992.
J. Prüß, Stability analysis for equilibria in age-specific population dynamics, Nonl. Anal. 7 (1983), 1291–1313.
G.F. Webb, Nonlinear Age-Dependent Population Dynamics, Marcel Dekker, New York, 1985.
J. Wu, Theory and applications of partial functional-differential equations, Applied Mathematical Sciences, 119 Springer-Verlag, New York, 1996.
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2007 Birkhäuser Verlag Basel/Switzerland
About this chapter
Cite this chapter
Diekmann, O., Gyllenberg, M. (2007). Abstract Delay Equations Inspired by Population Dynamics. In: Amann, H., Arendt, W., Hieber, M., Neubrander, F.M., Nicaise, S., von Below, J. (eds) Functional Analysis and Evolution Equations. Birkhäuser Basel. https://doi.org/10.1007/978-3-7643-7794-6_12
Download citation
DOI: https://doi.org/10.1007/978-3-7643-7794-6_12
Publisher Name: Birkhäuser Basel
Print ISBN: 978-3-7643-7793-9
Online ISBN: 978-3-7643-7794-6
eBook Packages: Mathematics and StatisticsMathematics and Statistics (R0)