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**Theory of impulsive differential equations.**
*(English)*
Zbl 0719.34002

Series in Modern Applied Mathematics, 6. Singapore etc.: World Scientific. x, 273 p. $ 58.00/hbk (1989).

The monograph summarizes results obtained by the authors and their colleagues and by the Soviet groups of Myshkis, Samoilenko, Perestyuk (there are 76 references, 57 belonging to these groups). Although there are illustrative examples (some of them interesting) there are no applications. With an extended use of inequalities and comparison arguments, most of the book is devoted to stability.

Chapter 1 introduces the impulsive evolution processes, presents preliminary results and examples. In Chapter 2 variation of parameters formulae, upper and lower solutions, monotone iterative techniques, simple stability criteria are considered. Chapter 3 is devoted to the study of stability by means of discontinuous Lyapunov functions and impulsive differential inequalities. In Chapter 4 different other aspects of impulsive systems are discussed.

Chapter 1 introduces the impulsive evolution processes, presents preliminary results and examples. In Chapter 2 variation of parameters formulae, upper and lower solutions, monotone iterative techniques, simple stability criteria are considered. Chapter 3 is devoted to the study of stability by means of discontinuous Lyapunov functions and impulsive differential inequalities. In Chapter 4 different other aspects of impulsive systems are discussed.

Reviewer: A.Halanay (Bucureşti)

### MSC:

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

34A37 | Ordinary differential equations with impulses |

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

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

34A45 | Theoretical approximation of solutions to ordinary differential equations |

34B05 | Linear boundary value problems for ordinary differential equations |

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

34D05 | Asymptotic properties of solutions to ordinary differential equations |

34D20 | Stability of solutions to ordinary differential equations |

34E20 | Singular perturbations, turning point theory, WKB methods for ordinary differential equations |

34K05 | General theory of functional-differential equations |

34K10 | Boundary value problems for functional-differential equations |

34K20 | Stability theory of functional-differential equations |