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**Existence theory for nonlinear ordinary differential equations.**
*(English)*
Zbl 1077.34505

Mathematics and its Applications (Dordrecht) 398. Dordrecht: Kluwer Academic Publishers (ISBN 0-7923-4511-8/hbk). 196 p. (1997).

This monograph is devoted to various questions dealing with the existence of some solutions for ordinary differential equations. It reflects mostly the interests of its author and, in this respect, should be considered more as a research monograph than as a textbook, although some of the results are quite elementary.

The main emphasis is upon the use of topological methods, essentially fixed-point and degree theory, in proving existence theorems, and the corresponding tools are described in Chapter 2 (Chapter 1 is an introduction). The third chapter deals with the initial value problem for a first-order system, with special emphasis upon scalar equations, whose periodic solutions are considered in Chapter 4. Scalar second-order differential equations with various boundary conditions are studied in Chapters 5 and 6, with results in the spirit of the method of upper and lower solutions. Positone and semi-positive boundary value problems are the object of the following two chapters. In Chapters 9 to 12, the author deals with boundary value problems for second-order differential equations which are singular either in the dependent or in the independent variable, for which he proves existence results which are in the line of the method of upper and lower solutions or satisfy nonuniform non-resonance conditions. Chapters 13 and 14 are devoted to various types of solutions of second-order differential equations on non-compact intervals. The book ends with two small chapters dealing with impulsive first-order differential equations and with differential equations in Banach spaces.

Each chapter is followed by a short list of references, and the book ends with an index of topics. This monograph will help the reader in getting acquainted with some aspects of the modern theory of boundary value problems for nonlinear ordinary differential equations.

The main emphasis is upon the use of topological methods, essentially fixed-point and degree theory, in proving existence theorems, and the corresponding tools are described in Chapter 2 (Chapter 1 is an introduction). The third chapter deals with the initial value problem for a first-order system, with special emphasis upon scalar equations, whose periodic solutions are considered in Chapter 4. Scalar second-order differential equations with various boundary conditions are studied in Chapters 5 and 6, with results in the spirit of the method of upper and lower solutions. Positone and semi-positive boundary value problems are the object of the following two chapters. In Chapters 9 to 12, the author deals with boundary value problems for second-order differential equations which are singular either in the dependent or in the independent variable, for which he proves existence results which are in the line of the method of upper and lower solutions or satisfy nonuniform non-resonance conditions. Chapters 13 and 14 are devoted to various types of solutions of second-order differential equations on non-compact intervals. The book ends with two small chapters dealing with impulsive first-order differential equations and with differential equations in Banach spaces.

Each chapter is followed by a short list of references, and the book ends with an index of topics. This monograph will help the reader in getting acquainted with some aspects of the modern theory of boundary value problems for nonlinear ordinary differential equations.

Reviewer: Jean Mawhin (MR1449397)

### MSC:

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

34G20 | Nonlinear differential equations in abstract spaces |

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

47H20 | Semigroups of nonlinear operators |

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