Theorems of Leray-Schauder type and applications.

*(English)*Zbl 1045.47002
Series in Mathematical Analysis and Applications 3. London: Gordon and Breach Science Publishers (ISBN 90-5699-295-3/hbk). ix, 206 p. (2001).

The authors describe the motivation of the book as follows: “In this book we present basic continuation theorems for several classes of nonlinear operators and typical applications to differential equations. Our approach is elementary and does not use degree theory. In addition, in this book we present in a global setting various Leray-Schauder type theorems from nonlinear analysis.”

The book has an overview section and 10 chapters, which treat the following main topics: Theorems of Leray-Schauder type for contractions, Continuation theorems for nonexpansive maps, Theorems of Leray-Schauder type for accretive maps, Continuation theorems involving compactness, Applications to semilinear elliptic problems, Theorems of Leray-Schauder type for coindicences, Theorems of selective continuation, The unified theory, Multiplicity, Local continuation theorems. Part of the material comes from the authors’ own work and some of the results appear here for the first time. The list of references consists of 161 items, all referenced in the text.

The text is essentially self-contained and hence provides a good introduction to nonlinear analysis. The book is well written and organized. The results are clearly stated and the proofs are explained in detail. This is a useful book for all those who are interested in the applications of the continuation method without knowledge of degree theory.

The book has an overview section and 10 chapters, which treat the following main topics: Theorems of Leray-Schauder type for contractions, Continuation theorems for nonexpansive maps, Theorems of Leray-Schauder type for accretive maps, Continuation theorems involving compactness, Applications to semilinear elliptic problems, Theorems of Leray-Schauder type for coindicences, Theorems of selective continuation, The unified theory, Multiplicity, Local continuation theorems. Part of the material comes from the authors’ own work and some of the results appear here for the first time. The list of references consists of 161 items, all referenced in the text.

The text is essentially self-contained and hence provides a good introduction to nonlinear analysis. The book is well written and organized. The results are clearly stated and the proofs are explained in detail. This is a useful book for all those who are interested in the applications of the continuation method without knowledge of degree theory.

Reviewer: Vesa Mustonen (Oulu)

##### MSC:

47-02 | Research exposition (monographs, survey articles) pertaining to operator theory |

47J25 | Iterative procedures involving nonlinear operators |

47N20 | Applications of operator theory to differential and integral equations |

47Hxx | Nonlinear operators and their properties |

35J25 | Boundary value problems for second-order elliptic equations |