Theory of differential equations with unbounded delay.

*(English)*Zbl 0823.34069
Mathematics and its Applications (Dordrecht). 298. Dordrecht: Kluwer Academic Publishers. xi, 385 p. (1994).

Differential equations with unbounded delay date back to at least about 1920 when Volterra studied various problems from population dynamics and materials with memory. Systematic study of these equations was initiated by the Carnegie-Mellon group in 1960’s, and the axiomatic treatment by Hele, Kato and Schumacher in 1978 has inspired explosive growth of the subject. The survey paper by C. Corduneanu and V. Lakshmikantham [Nonlineaar Anal., Theory Methods Appl. 4, 831-877 (1980; Zbl 0449.34048)] discussed numerous topics and the current status of the subject to that date and the lecture notes of Y. Hino, S. Murakami and T. Naito [Lecture Notes in Mathematics, 1473 (1991; Zbl 0732.34051)] provided a unified theory of equations with infinite delay from the view point of dynamical systems and functional analysis. Applications of equations with infinite delay arises from many fields and can be found in the aforementioned survey paper by Corduneanu and Lakshmikantham and other books such as the one by J. Cushing [Lecture Notes in Biomathematics, 20 (1977; Zbl 0363.92014)] and the one by R. May [Stability and Complexity in Model Ecosystems (1973)].

The monograph under review has several important nice features different from the above literature. First of all, many materials of this book are based on recent results of Chinese mathematicians unknown to other researchers due to language obstacles. Secondly, it provides a classification of equations with unbounded delay so as to bring out the subtle differences between them. Thirdly, it is the first monograph which presents a somehow systematic treatment of neutral functional differential equations with infinite delay. This monograph has also provided some recent developments on oscillations and on stability in terms of two measures.

The book is accessible to any beginning graduate student and even to some bright undergraduates. It should also be helpful for those applied mathematicians, engineers and biologists who have interests in delay problems. This makes the book a valuable addition to the literature in the field of functional differential equations.

Chapter 1 gives a detailed classification of equations with delay. Chapters 2 and 3 study the basic existence-uniqueness theory for neutral equations with unbounded delay and infinite delay. Chapters 4-6 are devoted to various stability problems by Lyapunov’s second method and Razumikhin’s technique for equations with finite delay, unbounded delay and infinite delay. The asymptotic and oscillatory behaviors of solutions are investigated in Chapters 7 and 8. The final chapter discusses the existence of periodic solutions.

It would be nice if the book could select some examples from application problems.

The monograph under review has several important nice features different from the above literature. First of all, many materials of this book are based on recent results of Chinese mathematicians unknown to other researchers due to language obstacles. Secondly, it provides a classification of equations with unbounded delay so as to bring out the subtle differences between them. Thirdly, it is the first monograph which presents a somehow systematic treatment of neutral functional differential equations with infinite delay. This monograph has also provided some recent developments on oscillations and on stability in terms of two measures.

The book is accessible to any beginning graduate student and even to some bright undergraduates. It should also be helpful for those applied mathematicians, engineers and biologists who have interests in delay problems. This makes the book a valuable addition to the literature in the field of functional differential equations.

Chapter 1 gives a detailed classification of equations with delay. Chapters 2 and 3 study the basic existence-uniqueness theory for neutral equations with unbounded delay and infinite delay. Chapters 4-6 are devoted to various stability problems by Lyapunov’s second method and Razumikhin’s technique for equations with finite delay, unbounded delay and infinite delay. The asymptotic and oscillatory behaviors of solutions are investigated in Chapters 7 and 8. The final chapter discusses the existence of periodic solutions.

It would be nice if the book could select some examples from application problems.

Reviewer: J.Wu (North York)

##### MSC:

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

34K05 | General theory of functional-differential equations |

34K20 | Stability theory of functional-differential equations |

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

34K40 | Neutral functional-differential equations |