This book deals with the semigroup approach to delay equations in Banach spaces, building the theory from scratch. In the first part, semigroup theory and spectral theory for semigroups are presented. These results are required for the rest of the book. The second part deals with the well-posedness of problems containing a leading undelayed term and a delayed term that is in some sense smaller and can therefore be handled as a small perturbation. The third part is concerned with stability. The authors investigate asymptotic behavior via the spectral mapping theorem and critical spectrum, show how things become easier when positivity is involved, and discuss the influence of small perturbations and the size of the delay on stability. Part IV is devoted to second order equations with and without delay in Hilbert spaces and to delay equations where the delay terms contain derivatives of the same order as the leading undelayed terms.

The book describes the semigroup approach to delay equations in a well-arranged and understandable way. Moreover, it contains many references for different approaches to delay equations. Theoretical results are always followed by illustrative examples. Therefore, the book will be useful for graduate students as well as for researchers in evolution equations.

Reviewer:

tb