##
**Oscillation theory of delay differential equations: with applications.**
*(English)*
Zbl 0780.34048

Oxford Mathematical Monographs. Oxford: Clarendon Press. xii, 368 p. (1991).

Here is a good book in the area of functional differential equations, which will serve as a reference book and a working tool for researchers as well as for the training of students. I will briefly try to justify my appreciation.

In twelve chapters, the book gives the current state-of-the-art in oscillation theory of differential equations with deviating arguments, and not only delay differential equations (as the title leads us to think). Eight chapters are focused on the study of a particular class of equations: linear scalar delay differential equations, autonomous (Chapter 2), and non-autonomous (Chapter 3); systems of delay differential equations (Chapter 5); neutral differential equations (Chapter 6); delay difference equations (Chapter 7); delay and advanced differential equations with piecewise constant deviating arguments (Chapter 8); integrodifferential equations (and equations with unbounded delays) (Chapter 9); higher order delay and neutral differential equations (Chapter 10). The other ones deal with applications, except for Chapter 1, which provides the basic tools of analysis and functional analysis in use throughout the book and Chapter 12 (see below). Applications include notably a food-limited population model, a red blood cell production model and the logistic delay (neutral) equation. Problems considered range from oscillation properties for nonlinear equations to stability and attractivity properties (Chapter 11) in as much as they can be deduced from the study of oscillations. Chapter 12 completes the whole study in an interesting way by presenting results on the nature of oscillations: slow, periodic, as well as rapid oscillations are considered. The central question in each of the eight chapters is to find conditions (best, necessary and sufficient) for all solutions of a given equation to oscillate. A correlated question is: when can we say that a given equation has nonoscillatory solutions? For each class of equations, the authors give the most recent results known. These results were obtained during the eighties. The two authors contributed largely in a great part of these developments: I count about 115 papers in the reference list where one and/or other of the authors is involved. Topics studied by both cover all classes of equations considered in the book. Amongst some of their own results, I will just mention as examples: a sufficient condition for oscillation of nonautonomous equations (Theorem 2.6.1) [Chuanxi and Ladas (1990)] a comparison result for solutions of differential inequalities (Theorem 6.5.1) [Györi (1987)]; a characterization for oscillation of delay differential equations with piecewise constant arguments (Theorem 8.2.1) [Györi and Ladas (1989)]. This is not to say that the book is just a juxtaposition of research works by these authors. First of all, it contains most of the results on the question (as far as I know) obtained during the 80’s. Secondly, it can be seen that all these results have been rewritten, several new proofs have been provided. Things have been done in such a way that the whole book has a unity (in the presentation of chapters) despite the fact that each class of equations entails specific techniques.

As a final remark about the presentation, the book is self-contained, moreover each chapter itself is nearly self-contained. Cross-references throughout the book are scarce and restricted to basic results. Each chapter ends with (historical) notes where credit for results presented is duely paid, and a section on open problems and conjectures.

It is tempting to compare this book with an earlier book on the same subject by G. S. Ladde, V. Lakshmikantham and B. G. Zhang [Oscillation theory of differential equations with deviating arguments. New York (1987; Zbl 0622.34071)]. Comparison brings to light the appearance during the mid 80’s of new trends in the field of oscillation theory: oscillations of difference equations, equations with piecewise constant arguments, oscillations independent of the delay,...

In general, it seems that interest moved from oscillation theory considered as a branch of the classical oscillation theory, (which developed as a subfield of boundary value problems), to the study of oscillations in functional differential equations as a field in itself.

As a result, some of the questions treated in [Ladde, Lakshmikantham and Zhang (log. cit.)] and which may have had recent advances are not considered in this book.

In twelve chapters, the book gives the current state-of-the-art in oscillation theory of differential equations with deviating arguments, and not only delay differential equations (as the title leads us to think). Eight chapters are focused on the study of a particular class of equations: linear scalar delay differential equations, autonomous (Chapter 2), and non-autonomous (Chapter 3); systems of delay differential equations (Chapter 5); neutral differential equations (Chapter 6); delay difference equations (Chapter 7); delay and advanced differential equations with piecewise constant deviating arguments (Chapter 8); integrodifferential equations (and equations with unbounded delays) (Chapter 9); higher order delay and neutral differential equations (Chapter 10). The other ones deal with applications, except for Chapter 1, which provides the basic tools of analysis and functional analysis in use throughout the book and Chapter 12 (see below). Applications include notably a food-limited population model, a red blood cell production model and the logistic delay (neutral) equation. Problems considered range from oscillation properties for nonlinear equations to stability and attractivity properties (Chapter 11) in as much as they can be deduced from the study of oscillations. Chapter 12 completes the whole study in an interesting way by presenting results on the nature of oscillations: slow, periodic, as well as rapid oscillations are considered. The central question in each of the eight chapters is to find conditions (best, necessary and sufficient) for all solutions of a given equation to oscillate. A correlated question is: when can we say that a given equation has nonoscillatory solutions? For each class of equations, the authors give the most recent results known. These results were obtained during the eighties. The two authors contributed largely in a great part of these developments: I count about 115 papers in the reference list where one and/or other of the authors is involved. Topics studied by both cover all classes of equations considered in the book. Amongst some of their own results, I will just mention as examples: a sufficient condition for oscillation of nonautonomous equations (Theorem 2.6.1) [Chuanxi and Ladas (1990)] a comparison result for solutions of differential inequalities (Theorem 6.5.1) [Györi (1987)]; a characterization for oscillation of delay differential equations with piecewise constant arguments (Theorem 8.2.1) [Györi and Ladas (1989)]. This is not to say that the book is just a juxtaposition of research works by these authors. First of all, it contains most of the results on the question (as far as I know) obtained during the 80’s. Secondly, it can be seen that all these results have been rewritten, several new proofs have been provided. Things have been done in such a way that the whole book has a unity (in the presentation of chapters) despite the fact that each class of equations entails specific techniques.

As a final remark about the presentation, the book is self-contained, moreover each chapter itself is nearly self-contained. Cross-references throughout the book are scarce and restricted to basic results. Each chapter ends with (historical) notes where credit for results presented is duely paid, and a section on open problems and conjectures.

It is tempting to compare this book with an earlier book on the same subject by G. S. Ladde, V. Lakshmikantham and B. G. Zhang [Oscillation theory of differential equations with deviating arguments. New York (1987; Zbl 0622.34071)]. Comparison brings to light the appearance during the mid 80’s of new trends in the field of oscillation theory: oscillations of difference equations, equations with piecewise constant arguments, oscillations independent of the delay,...

In general, it seems that interest moved from oscillation theory considered as a branch of the classical oscillation theory, (which developed as a subfield of boundary value problems), to the study of oscillations in functional differential equations as a field in itself.

As a result, some of the questions treated in [Ladde, Lakshmikantham and Zhang (log. cit.)] and which may have had recent advances are not considered in this book.

Reviewer: O.Arino (Pau)

### MSC:

34K99 | Functional-differential equations (including equations with delayed, advanced or state-dependent argument) |

34K40 | Neutral functional-differential equations |

34C10 | Oscillation theory, zeros, disconjugacy and comparison theory for ordinary differential equations |

34-02 | Research exposition (monographs, survey articles) pertaining to ordinary differential equations |