Ordinary differential equations in \(R^ n\). Problems and methods. Transl. from the Italian by A. LoBello.

*(English)*Zbl 0535.34001
Applied Mathematical Sciences, 39. New York etc.: Springer-Verlag. XII, 385 p. DM 96.00; $ 37.70 (1984).

The present book is an excellent rapid introduction to the various problems and methods of the theory of ordinary differential equations.

Besides the classical material this book contains also some unusual questions among which we mention: maximal and minimal solutions, some special methods for the study of stability of nonlinear differential systems (asymptotic equivalence, Olech’s method, the method of the logarithmic norm, invariant sets) as well as some applications in biology, chemistry and automatic control theory.

So the book is useful not only for engineers and physicists but also for mathematicians who can find here motivation for further advanced studies. A great number of interesting exercises completes the theoretical material.

The text is divided into five chapters as follows: Chapter I, Existence and uniqueness for the initial value problem under the hypotheses of Lipschitz; Chapter II, Linear systems; Chapter III, Existence and uniqueness for the Cauchy problem under the condition of continuity; Chapter IV, Boundary value problems; Chapter V, Questions of stability.

Each chapter, except Chapter II, also contains some bibliographical notes which give us a panoramic view related to the subjects and methods discussed in that chapter. In conclusion, the present book is a very welcome addition to the mathematical literature.

Besides the classical material this book contains also some unusual questions among which we mention: maximal and minimal solutions, some special methods for the study of stability of nonlinear differential systems (asymptotic equivalence, Olech’s method, the method of the logarithmic norm, invariant sets) as well as some applications in biology, chemistry and automatic control theory.

So the book is useful not only for engineers and physicists but also for mathematicians who can find here motivation for further advanced studies. A great number of interesting exercises completes the theoretical material.

The text is divided into five chapters as follows: Chapter I, Existence and uniqueness for the initial value problem under the hypotheses of Lipschitz; Chapter II, Linear systems; Chapter III, Existence and uniqueness for the Cauchy problem under the condition of continuity; Chapter IV, Boundary value problems; Chapter V, Questions of stability.

Each chapter, except Chapter II, also contains some bibliographical notes which give us a panoramic view related to the subjects and methods discussed in that chapter. In conclusion, the present book is a very welcome addition to the mathematical literature.

Reviewer: N.Luca

##### MSC:

34-01 | Introductory exposition (textbooks, tutorial papers, etc.) pertaining to ordinary differential equations |

34A34 | Nonlinear ordinary differential equations and systems, general theory |

34D20 | Stability of solutions to ordinary differential equations |

34E99 | Asymptotic theory for ordinary differential equations |

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

34A30 | Linear ordinary differential equations and systems, general |

34Bxx | Boundary value problems for ordinary differential equations |