Multivalued differential equations.

*(English)*Zbl 0760.34002
De Gruyter Series in Nonlinear Analysis and Applications 1. Berlin: Walter de Gruyter (ISBN 3-11-013212-5/hbk). xi, 260 p. (1992).

Since 1984 when the now classical book on differential inclusions by J.-P. Aubin and A. Cellina [Differential Inclusions (Grundlehren der mathematischen Wissenschaften 264, Springer, Berlin) (1984; Zbl 0538.34007)] was published, many new results and/or improvements of the older results in this field and in the field of set-valued analysis have been obtained. This book reflects most of these achievements and brings to the reader an up-to-date comprehensive treatment of differential inclusions (called here multivalued differential equations) and related topics.

The book consists of five chapters and an appendix. The first chapter covers basic concepts of set-valued analysis like upper and lower semicontinuity of set-valued maps, measurability of set-valued maps, selection theorems etc. Chapters 2 and 3 are concerned with differential inclusions in finite dimensions while the rest of the book is devoted to the infinite dimensional case.

Chapter two gives existence theorems for differential inclusions. It is assumed that the right-hand side may be defined only on a closed set. Solutions to such differential inclusions are also called in literature viable solutions since they must satisfy given constraints. Existence results are given for both upper and lower semicontinuous differential inclusions, including also the Carathéodory case.

Chapter 3 contains results concerning topological properties of the solution set for differential inclusions. For example, connectedness of the solution set is proved for both upper and lower semicontinuous case. Moreover, existence results for maximal and minimal solutions of differential inclusions with respect to the partial ordering given by a cone are given.

Chapter 4 is devoted to existence results for differential inclusions in Banach spaces. The existence theorems are divided into two categories. The first one contains those which are based on some compactness condition (expressed through a measure of noncompactness), the second contains those which are based on a Baire category approach.

Chapter 5 contains some qualitative results on differential inclusions. It covers fixed points theorems, boundary value problems, existence of periodic solutions and stability and asymptotic behavior of solutions of differential inclusions.

In an appendix, four related topics, i.e. discontinuous differential equations, implicit differential equations, functional differential equations and perturbations of dissipative differential inclusions are briefly treated.

It was the aim of the author to include simple and short proofs (although in many cases more sophisticated than the “standard” proofs found elsewhere). Many problems (some of them highly non-trivial) were included in the extensive list of exercises which conclude each paragraph of the book. Each paragraph contains also concluding remarks where references to the existing literature are given together with the author’s comments. Readability of the book is enhanced by several useful examples and/or counterexamples. Some applications to control theory and discontinuous differential equations (for example “dry friction”) are given.

This book, together with other recently published books by J.-P. Aubin and H. Frankowska [Set-valued analysis (Systems and Control 2, Birkhäuser, Boston) (1990; Zbl 0713.49021)], by J.-P. Aubin [Viability theory (Systems and Control, Birkhäuser, Boston) (1991)] and by A. A. Tolstonogov [Differential Inclusions in Banach Spaces (in Russian) (Nauka, Novisibirsk) (1986; Zbl 0689.34014)] provides the most complete overview on the current state-of-the-art in the field of differential inclusions and set-valued analysis.

The book consists of five chapters and an appendix. The first chapter covers basic concepts of set-valued analysis like upper and lower semicontinuity of set-valued maps, measurability of set-valued maps, selection theorems etc. Chapters 2 and 3 are concerned with differential inclusions in finite dimensions while the rest of the book is devoted to the infinite dimensional case.

Chapter two gives existence theorems for differential inclusions. It is assumed that the right-hand side may be defined only on a closed set. Solutions to such differential inclusions are also called in literature viable solutions since they must satisfy given constraints. Existence results are given for both upper and lower semicontinuous differential inclusions, including also the Carathéodory case.

Chapter 3 contains results concerning topological properties of the solution set for differential inclusions. For example, connectedness of the solution set is proved for both upper and lower semicontinuous case. Moreover, existence results for maximal and minimal solutions of differential inclusions with respect to the partial ordering given by a cone are given.

Chapter 4 is devoted to existence results for differential inclusions in Banach spaces. The existence theorems are divided into two categories. The first one contains those which are based on some compactness condition (expressed through a measure of noncompactness), the second contains those which are based on a Baire category approach.

Chapter 5 contains some qualitative results on differential inclusions. It covers fixed points theorems, boundary value problems, existence of periodic solutions and stability and asymptotic behavior of solutions of differential inclusions.

In an appendix, four related topics, i.e. discontinuous differential equations, implicit differential equations, functional differential equations and perturbations of dissipative differential inclusions are briefly treated.

It was the aim of the author to include simple and short proofs (although in many cases more sophisticated than the “standard” proofs found elsewhere). Many problems (some of them highly non-trivial) were included in the extensive list of exercises which conclude each paragraph of the book. Each paragraph contains also concluding remarks where references to the existing literature are given together with the author’s comments. Readability of the book is enhanced by several useful examples and/or counterexamples. Some applications to control theory and discontinuous differential equations (for example “dry friction”) are given.

This book, together with other recently published books by J.-P. Aubin and H. Frankowska [Set-valued analysis (Systems and Control 2, Birkhäuser, Boston) (1990; Zbl 0713.49021)], by J.-P. Aubin [Viability theory (Systems and Control, Birkhäuser, Boston) (1991)] and by A. A. Tolstonogov [Differential Inclusions in Banach Spaces (in Russian) (Nauka, Novisibirsk) (1986; Zbl 0689.34014)] provides the most complete overview on the current state-of-the-art in the field of differential inclusions and set-valued analysis.

Reviewer: V.Křivan (České Budějovice)

##### MSC:

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

39-02 | Research exposition (monographs, survey articles) pertaining to difference and functional equations |

34A60 | Ordinary differential inclusions |

34A40 | Differential inequalities involving functions of a single real variable |

34B15 | Nonlinear boundary value problems for ordinary differential equations |

34C25 | Periodic solutions to ordinary differential equations |

34D20 | Stability of solutions to ordinary differential equations |

34G20 | Nonlinear differential equations in abstract spaces |

47H10 | Fixed-point theorems |

93B99 | Controllability, observability, and system structure |

93C15 | Control/observation systems governed by ordinary differential equations |

26E25 | Set-valued functions |

28B20 | Set-valued set functions and measures; integration of set-valued functions; measurable selections |