Semigroups for Delay Equations

András Bátkai and Susanna Piazzera

A. K. Peters: Wellesley MA, 2005,

Research Notes in Mathematics vol. 10, 

ISBN: 1-56881-243-4.

The book presents in a systematic way how delay equations can be studied in Lp-history spaces. Using recent operator semigroup methods the existence of solutions and the asymptotic behavior is analyzed in detail. Main emphasis is laid on uniform exponential stability and hyperbolicity of the solution semigroup, but other properties such as positivity, almost periodicity or non-uniform exponential stability are also considered. The efficiency of the theoretical results is demonstrated on various examples.

To make the book self-contained, the needed operator semigroup theoretic results are collected in the first two chapters. Then the well-posedness of the delay equations is showed using perturbation theoretic arguments.

The asymptotic behavior of the solutions is first analyzed for regular equations using spectral mapping theorems and characteristic equations. This is extended to the non-regular case using resolvent growth estimates and perturbation arguments. As an important application, the problem of small delays is also considered.

Finally, the theory is extended to wave equations with delay and to parabolic equations with delays in the highest order derivatives.

Some less well-known results on vector valued integration and Sobolev spaces are presented in the appendix.

 An extensive bibliography and a detailed index may help to use the book as a reference tool.


You find the homepage of the publisher here. The official flyer of the book is here (PDF-file). In Europe,  you may also want to consider

See the entry in Zentralblatt Math database (PDF-format).

See the entry in Mathematical Reviews (PDF-format)

A Review in the Newsletter of the European Mathematical Society


Related publications:

Hadd, S., Unbounded Perturbations of C0-Semigroups on Banach Spaces and Applications, Semigroup Forum 70 (2005), 451–465.

Bounit, H., Idrissi, A., Regular bilinear systems, IMA Journal of Mathematical Control and Information 22 (2005), 26–57.

Fuhrman, M., Tessitore, G., Generalized directional gradients, backward stochastic differential equations and mild solutions of semilinear parabolic equations, Appl. Math. Optim. 51 (2005), 279–332.

Sikolya, E., Flows in networks with dynamic ramification nodes, J. Evol. Eqs. 5 (2005), 441-463.

Hadd, S., Idrissi, A., Regular linear systems governed by systems with state, input and output delays, Preprint, Tübinger Berichte zur Funktionalanalysis 13 (2003/2004).

Hadd, S., Idrissi, A., Rhandi, A., The regular linear systems associated to the shift semigroups and applications to control delay systems, Preprint, Tübinger Berichte zur Funktionalanalysis 13 (2003/2004).

Hadd, S., On well-posedness of neutral equations with nonatomic difference operators in Banach spaces, Preprint, Tübinger Berichte zur Funktionalanalysis 14 (2004/2005).

Hadd, S., Rhandi, A., Feedback theory for neutral equations in infinite dimensional state spaces, Preprint, Tübinger Berichte zur Funktionalanalysis 14 (2004/2005).

Hadd, S., Rhandi, A., Schnaubelt, R., Feedback theory for non-autonomous regular linear systems with input and state delays, Preprint, Tübinger Berichte zur Funktionalanalysis 14 (2004/2005).