*(English)*Zbl 0641.47066

The book of Jean-Pierre Aubin and Ivar Ekeland, two well-known names in the area of nonlinear analysis, is intended as a (high-level) introduction to this domain with the declarated goal to develop the theory having in the mind some current applications.

Here are its Contents: a Preface, (1) Background Notes, (2) Smooth Analysis, (3) Set-Valued Maps, (4) Convex Analysis and Optimization, (5) a General Variational Principle, (6) Solving Inclusions, (7) Non-smooth Analysis, (8) Hamiltonian Systems, Comments, Bibliography, Author Index, and Subject index.

All is written in a medium-size volume of 518 pages. It is worthy to note that the theory is developed not as a subsidiary of linear analysis but via its own methods which are specifical to it and lead to more simple and direct proofs. The prerequisites are the elementary theory of Hilbert spaces since most interesting results are those which hold in Euclidian space. The two major themes of the book are

(a) the resolution of the equations $f\left(x\right)=0$ in finite or infinite dimensional spaces and

(b) the resolution of $\partial V\left(x\right)=0$ where $\partial $ is the generalized gradient.

In order to give the reader a deeper insight into the very nature of the very realm of nonlinearity, the authors first develop the theory for smooth maps and then for nonsmooth ones. Although the usual field of applications of nonlinear analysis is mathematical economics, particularly convex optimization, other domains of interests are covered in the book; we remark the chapter devoted to Hamiltonian dynamical systems. Throughout the book not only existence-type theorems are proved but iterative procedures for obtaining the solution are discussed which gives a more practical value to the book. In conclusion, we think the book will soon become an excellent reference book and a valuable “instrument de travail” for mathematicians and research engineers working in the area of applied nonlinear analysis.

##### MSC:

47J05 | Equations involving nonlinear operators (general) |

47-01 | Textbooks (operator theory) |

47H10 | Fixed point theorems for nonlinear operators on topological linear spaces |

46G05 | Derivatives, etc. (functional analysis) |

58C05 | Real-valued functions on manifolds |

58C25 | Differentiable maps on manifolds (global analysis) |

49J52 | Nonsmooth analysis (other weak concepts of optimality) |

35J20 | Second order elliptic equations, variational methods |

58E30 | Variational principles on infinite-dimensional spaces |