Semigroups and delay equations, spring 2014

Last modified by mirka@helsinki_fi on 2024/03/27 10:22

Semigroups and delay equations, spring 2014


Mats Gyllenberg


10 sp.


Advanced studies


Basic knowledge of ordinary differential equations and linear partial differential equations, elements of functional analysis (including the Hahn-Banach, Banach-Steinhaus and closed graph theorems), elements of measure and integration (including the Riesz representation theorems) and a passion for pure mathematics.


The basic theory of one-parameter semigroups of linear operators on Banach spaces. Strongly continuous semigroups, adjoint semigroups. The abstract Cauchy problem. Perturbation theory. Spectral Analysis. Applications to nonlinear delay equations. The principle of linearised (in)stability.

The lectures will be based on material that has been published in books and journal articles (see bibliography below).


Weeks 3-9 and 11-18, Tuesday 14-16 in room D123 and Wednesday 14-16 in room C122. Two hours of exercise classes per week.

Easter Holiday 17.-23.4.


Final exam.



P.L. Butzer, H. Berens, Semi-Groups of Operators and Approximation, Springer-Verlag, Berlin, Heidelberg, New York, 1967.

Clément, Ph.; Heijmans, H. J. A. M.; Angenent, S.; van Duijn, C. J.; de Pagter, B. One-parameter semigroups. CWI Monographs, 5. North-Holland Publishing Co., Amsterdam, 1987. x+312 pp. ISBN: 0-444-70284-9

O. Diekmann, S.A. van Gils, S.M. Verduyn Lunel, H.-O. Walther, Delay Equations. Functional-, Complex-, and Nonlinear Analysis, Springer-Verlag, New York, 1995.

Engel, Klaus-Jochen; Nagel, Rainer One-parameter semigroups for linear evolution equations. Graduate Texts in Mathematics, 194. Springer-Verlag, New York, 2000. xxii+586 pp. ISBN: 0-387-98463-1

Goldstein, Jerome A. Semigroups of linear operators and applications. Oxford Mathematical Monographs. The Clarendon Press, Oxford University Press, New York, 1985. x+245 pp. ISBN: 0-19-503540-2

A. Pazy, Semigroups of Linear Operators and Applications to Partial Differential Equations, Springer-Verlag, New York, 1983.


Clément, Ph., Diekmann, O., Gyllenberg, M., Heijmans, H.J.A.M., Thieme,H.R.: Perturbation theory for dual semigroups. I. The sun-reflexive case, Math. Ann. 277 (1987), 709-725.

Clément, Ph., Diekmann, O., Gyllenberg, M., Heijmans, H.J.A.M., Thieme,H.R.: Perturbation theory for dual semigroups III. Nonlinear Lipschitz continuous perturbations in the sun reflexive case. in Proceedings of the Conference on Volterra integro-differential equations in Banach spaces and applications, Trento 1987, G. Da Prato and M. Iannelli (Eds.), Pitman research Notes in Mathematics Series 190 (1989), 67-89.

Diekmann, O., Getto, Ph. and Gyllenberg, M.: Stability and bifurcation analysis of Volterra functional equations in the light of suns and stars, SIAM Journal on Mathematical Analysis, 39 (2007) 1023-1069.

Diekmann, O. and Gyllenberg, M.: Abstract delay equations inspired by population dynamics, In Functional Analysis and Evolution Equations, H. Amann, W. Arendt, M. Hieber, F. Neubrander, S. Nicaise, J. von Below (Eds.), Birkhäuser (2007) pp. 187-200.

Diekmann, O. and Gyllenberg, M.: The second half - with a quarter of a century delay, Mathematical Modelling of Natural Phenomena 3 (2008) 36-48.

Diekmann, O. and Gyllenberg, M.: Equations with infinite delay: blending the abstract and the concrete, Journal of Differential Equations, 252 (2012) 819-851.



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Anssi Mirka

Exercises 1        Solutions 1.

Exercises 2        Solutions 2

Exercises 3        Solutions 3

Exercises 4        Solutions 4

Exercises 5        Solutions 5.

Exercises 6        Solutions 6.

Exercises 7        Solutions 7

Exercises 8a      Solutions 8

Exercises 8b

Exercises 8c

Exercises 9a       Solutions 9

Exercises 9b