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**Volterra integral and differential equations.
2nd ed.**
*(English)*
Zbl 1075.45001

Mathematics in Science and Engineering 202. Amsterdam: Elsevier (ISBN 0-444-51786-3/hbk). x, 353 p. (2005).

[For the first edition (1983) see Zbl 0515.45001.]

This is the second edition of the author’s book on Volterra integral and differential equations. The main approach is to consider those Volterra equations that can be treated as perturbations of ordinary differential equations and there is a very strong emphasis on Lyapunov functions.

Chapter 1 gives a gentle introduction to the subject and discusses integral equations reducible to ordinary differential equations. In chapter 2 linear equations are studied and various stability concepts introduced. In chapter 3 the basic existence, uniqueness and continuation results are established and chapter 4 treats the history and motivating examples of the theory. In chapters 5 and 6 Lyapunov’s method is used to derive results about stability, instability and boundedness of the solutions. Chapter 7 which is to a large part new in this edition considers the theory of resolvents and chapter 8 is a treatment of functional differential equations.

This is the second edition of the author’s book on Volterra integral and differential equations. The main approach is to consider those Volterra equations that can be treated as perturbations of ordinary differential equations and there is a very strong emphasis on Lyapunov functions.

Chapter 1 gives a gentle introduction to the subject and discusses integral equations reducible to ordinary differential equations. In chapter 2 linear equations are studied and various stability concepts introduced. In chapter 3 the basic existence, uniqueness and continuation results are established and chapter 4 treats the history and motivating examples of the theory. In chapters 5 and 6 Lyapunov’s method is used to derive results about stability, instability and boundedness of the solutions. Chapter 7 which is to a large part new in this edition considers the theory of resolvents and chapter 8 is a treatment of functional differential equations.

Reviewer: Gustaf Gripenberg (Hut)

### MSC:

45D05 | Volterra integral equations |

45J05 | Integro-ordinary differential equations |

34K05 | General theory of functional-differential equations |

34D20 | Stability of solutions to ordinary differential equations |

45M05 | Asymptotics of solutions to integral equations |

45M10 | Stability theory for integral equations |

45M15 | Periodic solutions of integral equations |

45-02 | Research exposition (monographs, survey articles) pertaining to integral equations |

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

45G10 | Other nonlinear integral equations |

45-03 | History of integral equations |

92D25 | Population dynamics (general) |