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**Introduction to various aspects of degree theory in Banach spaces.**
*(English)*
Zbl 0597.47040

Mathematical Surveys and Monographs, No. 23. Providence, R.I.: American Mathematical Society (AMS). VI, 242 p. $ 60.00 (1986).

This book is devoted to some aspects of the theory of degree for different classes of nonlinear mappings in Banach spaces; the choice of results and constructions in this book is determined partly by the fact that the book is written from the point of view of an analyst; partly by the author’s desire to throw light on the theory from various angles, and partly by the author’s taste.

The book contains 9 chapters and 2 applications: function-analytic preliminaries (Banach spaces and different classes of nonlinear mappings, differentiability of mappings, Dugundji’s extension theorem and so on), the Leray-Schauder degree for differentiable maps, the Leray-Schauder degree for not necessarily differentiable maps, the Poincaré-Bohl theorem and some of its applications, the product theorem and some of its consequences (the invariance of the domain and the Jordan-Leray theorem), the finite-dimensional case (simplicial approximations in the degree theory), on spheres (the degree theory for mappings from one sphere \(S_ 1^ n\) into another \(S_ 2^ n)\), some extension and homotopy theorems (Hopf-Krasnoselskii extension theorems and some near results), the Borsuk theorem and some of its consequences; appendix 1 contains the proof of the linear homotopic theorem, appendix 2 gives the proof of the Sard- Smale theorem.

This book will be interesting to all analysts that apply geometrical and topological methods in their investigations; it can be used for acquaintance with some problems in the field.

The book contains 9 chapters and 2 applications: function-analytic preliminaries (Banach spaces and different classes of nonlinear mappings, differentiability of mappings, Dugundji’s extension theorem and so on), the Leray-Schauder degree for differentiable maps, the Leray-Schauder degree for not necessarily differentiable maps, the Poincaré-Bohl theorem and some of its applications, the product theorem and some of its consequences (the invariance of the domain and the Jordan-Leray theorem), the finite-dimensional case (simplicial approximations in the degree theory), on spheres (the degree theory for mappings from one sphere \(S_ 1^ n\) into another \(S_ 2^ n)\), some extension and homotopy theorems (Hopf-Krasnoselskii extension theorems and some near results), the Borsuk theorem and some of its consequences; appendix 1 contains the proof of the linear homotopic theorem, appendix 2 gives the proof of the Sard- Smale theorem.

This book will be interesting to all analysts that apply geometrical and topological methods in their investigations; it can be used for acquaintance with some problems in the field.

Reviewer: P.Zabrejko

### MSC:

47J05 | Equations involving nonlinear operators (general) |

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

46G05 | Derivatives of functions in infinite-dimensional spaces |