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**Methods of nonlinear analysis. Applications to differential equations.**
*(English)*
Zbl 1176.35002

Birkhäuser Advanced Texts. Basler Lehrbücher. Basel: Birkhäuser (ISBN 978-3-7643-8146-2/hbk; 978-3-7643-8147-9/ebook). xii, 568 p. (2007).

This monograph contains old and new basic results from a significant part of the modern Nonlinear Analysis. The authors are well-known experts in the field and their approach in this book is very elegant. All results are presented in an elementary way. Only a basic knowledge of basic functional analysis, topology, and analysis is assumed. The book is well written and contains a wealth of material. The authors make a concerted effort to simplify proofs taken from many sources. That is why the researchers will readily find the information they seek, while students can develop their skills by filling in details of proofs, as well as by using the problem sets that end each chapter. The book is carefully written, the style is informal, and the arguments are clear. It is a delightful read.

Here are the chapter headings followed by brief comments.

(1) Preliminaries. This chapter is rather brief since it is supposed that the reader already has some knowledge of linear algebra and linear functional analysis.

(2) Properties of Linear and Nonlinear Operators. There are pointed out some fundamental properties of linear and nonlinear operators in Banach spaces: uniform boundedness principle,open mapping theorem, Hahn-Banach theorem, separation theorem, Eberlein-Smulyan theorem, contraction principle.

(3) Abstract Integral and Differential Calculus. The authors recall the basic facts on the integration of vector functions and the differential calculus in normed linear spaces. It is also enounced with proof the Newton method which offers a very effective tool for solving nonlinear equations.

(4) Local Properties of Differentiable Mappings. Important topics developed in this chapter include the inverse function and the implicit function theorems together with the rank theorem and the notion of differentiable manifold. Results such as the Lyapunov-Schmidt reduction theorem and the Morse theorem are used to prove the local bifurcation theorem of Crandall and Rabinowitz.

(5) Topological and Monotonicity Methods. The authors focus on the Brouwer and Schauder fixed point theorems, the Sard theorem and the analytic approach of the topological degree of a mapping and the method of monotone iterations based on the notion of sub- and super-solutions.

(6) Variational Methods. There are introduced basic principles such as the Lagrange multipliers method, the mountain pass and the saddle point theorems or Courant-Fischer and Courant-Weinstein principles. The abstract results are illustrated by numerous examples involving boundary value problems for ordinary differential equations.

(7) Boundary Value Problems for Partial Differential Equations. This chapter deals with several applications of the preceding methods to boundary value problems for elementary nonlinear partial differential equations. There are widely discussed the notions of classical and weak solutions for many problems arising in concrete applications.

This book is recommended to researchers in Applied Mathematics who study various linear and nonlinear phenomena but it may be also useful and comprehensive for a broader community of mathematicians, physicists, and engineers. To conclude, this volume provides the reader with an in-depth introduction to a rich and rapidly developing research area that has already produced remarkable results.

Here are the chapter headings followed by brief comments.

(1) Preliminaries. This chapter is rather brief since it is supposed that the reader already has some knowledge of linear algebra and linear functional analysis.

(2) Properties of Linear and Nonlinear Operators. There are pointed out some fundamental properties of linear and nonlinear operators in Banach spaces: uniform boundedness principle,open mapping theorem, Hahn-Banach theorem, separation theorem, Eberlein-Smulyan theorem, contraction principle.

(3) Abstract Integral and Differential Calculus. The authors recall the basic facts on the integration of vector functions and the differential calculus in normed linear spaces. It is also enounced with proof the Newton method which offers a very effective tool for solving nonlinear equations.

(4) Local Properties of Differentiable Mappings. Important topics developed in this chapter include the inverse function and the implicit function theorems together with the rank theorem and the notion of differentiable manifold. Results such as the Lyapunov-Schmidt reduction theorem and the Morse theorem are used to prove the local bifurcation theorem of Crandall and Rabinowitz.

(5) Topological and Monotonicity Methods. The authors focus on the Brouwer and Schauder fixed point theorems, the Sard theorem and the analytic approach of the topological degree of a mapping and the method of monotone iterations based on the notion of sub- and super-solutions.

(6) Variational Methods. There are introduced basic principles such as the Lagrange multipliers method, the mountain pass and the saddle point theorems or Courant-Fischer and Courant-Weinstein principles. The abstract results are illustrated by numerous examples involving boundary value problems for ordinary differential equations.

(7) Boundary Value Problems for Partial Differential Equations. This chapter deals with several applications of the preceding methods to boundary value problems for elementary nonlinear partial differential equations. There are widely discussed the notions of classical and weak solutions for many problems arising in concrete applications.

This book is recommended to researchers in Applied Mathematics who study various linear and nonlinear phenomena but it may be also useful and comprehensive for a broader community of mathematicians, physicists, and engineers. To conclude, this volume provides the reader with an in-depth introduction to a rich and rapidly developing research area that has already produced remarkable results.

Reviewer: Vicenţiu D. Rădulescu (Craiova)

### MSC:

35-02 | Research exposition (monographs, survey articles) pertaining to partial differential equations |

49-02 | Research exposition (monographs, survey articles) pertaining to calculus of variations and optimal control |

58-02 | Research exposition (monographs, survey articles) pertaining to global analysis |

35Jxx | Elliptic equations and elliptic systems |

47Hxx | Nonlinear operators and their properties |

49Jxx | Existence theories in calculus of variations and optimal control |

49Kxx | Optimality conditions |

58Exx | Variational problems in infinite-dimensional spaces |

58Cxx | Calculus on manifolds; nonlinear operators |