Differential equations with discontinuous right-hand side. (Differentsial’nye uravneniya s razryvnoj pravoj chast’yu).

*(Russian)*Zbl 0571.34001
Moskva: ”Nauka”. Glavnaya Redaktsiya Fiziko-Matematicheskoj Literatury. 224 p. R. 2.90 (1985).

This interesting and important book is devoted to the theory of differential equations with discontinuous right-hand sides. It consists of five chapters. The first one deals with equations of the Carathéodory type and equations containing distributions.

In Chapter 2 the author surveys the various definitions of solutions of the equations whose right-hand sides are discontinuous both with respect to the independent and the phase variables. He also gives the general theory of the Cauchy problem for such equations and for the differential inclusions.

The classical qualitative theory is extended to the differential equations with discontinuous right-hand sides in Chapter 3. Here the trajectories of autonomous systems, the existence conditions for periodic solutions, stability theory and some other items are considered. Chapters 4 and 5 are concerned with systems having piecewise continuous right-hand sides. For such systems the behavior of solutions on surfaces of discontinuity is studied.

In Chapter 2 the author surveys the various definitions of solutions of the equations whose right-hand sides are discontinuous both with respect to the independent and the phase variables. He also gives the general theory of the Cauchy problem for such equations and for the differential inclusions.

The classical qualitative theory is extended to the differential equations with discontinuous right-hand sides in Chapter 3. Here the trajectories of autonomous systems, the existence conditions for periodic solutions, stability theory and some other items are considered. Chapters 4 and 5 are concerned with systems having piecewise continuous right-hand sides. For such systems the behavior of solutions on surfaces of discontinuity is studied.

Reviewer: B.Šehter

##### MSC:

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

34A12 | Initial value problems, existence, uniqueness, continuous dependence and continuation of solutions to ordinary differential equations |

34C05 | Topological structure of integral curves, singular points, limit cycles of ordinary differential equations |

34C25 | Periodic solutions to ordinary differential equations |

34A60 | Ordinary differential inclusions |

34D20 | Stability of solutions to ordinary differential equations |