Theory of set differential equations in metric spaces.

*(English)*Zbl 1156.34003
Cambridge: Cambridge Scientific Publishers (ISBN 1-904868-46-0/hbk). ix, 204 p. (2006).

This monograph is the first important contribution to a field having a very strong potential: the theory of set differential equations in metric spaces.

It is the merit of the authors of the book to observe that many results in differential calculus, differential equations and set-valued analysis are proved in the absence of a linear structure. It follows a very interesting approach with a lot of powerful research directions.

The monograph is structured in five chapters. References and an index conclude the book. Let us point out the titles of the chapters: I. Preliminaries; II. Basic Theory; III. Stability Theory; IV. Connection to Fuzzy Differential Equations; V. Miscellaneous Topics (just mention several sections: Impulsive Set Differential Equations, Monotone Iteration Technique; Set Differential Equations with Delay; Set Difference Equations; Set Differential Equations with Causal Operators, etc.)

The most important features of this nice monograph are mentioned by the authors in the preface of the book: 6mm

It is the merit of the authors of the book to observe that many results in differential calculus, differential equations and set-valued analysis are proved in the absence of a linear structure. It follows a very interesting approach with a lot of powerful research directions.

The monograph is structured in five chapters. References and an index conclude the book. Let us point out the titles of the chapters: I. Preliminaries; II. Basic Theory; III. Stability Theory; IV. Connection to Fuzzy Differential Equations; V. Miscellaneous Topics (just mention several sections: Impulsive Set Differential Equations, Monotone Iteration Technique; Set Differential Equations with Delay; Set Difference Equations; Set Differential Equations with Causal Operators, etc.)

The most important features of this nice monograph are mentioned by the authors in the preface of the book: 6mm

- “1.
- It is the first book that attempts to describe the theory of set differential equations as an independent discipline;
- 2.
- It incorporates the recent general theory of set differential equations, discusses the interconnections between set differential equations and fuzzy differential equations and uses both smooth and nonsmooth analysis for investigation;
- 3.
- It exhibits several new areas of study by providing the initial apparatus fur further advancement;
- 4.
- It is a timely introduction to a subject that follows the present trend of studying analysis and differential equations in metric spaces.”

Reviewer: Adrian Petruşel (Cluj-Napoca)

##### MSC:

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

49J53 | Set-valued and variational analysis |

47H04 | Set-valued operators |

34G25 | Evolution inclusions |