Methods of Nonlinear Analysis: Applications to Differential Equations Pavel Drábek and Jaroslav Milota Series: Birkhäuser Advanced Texts / Basler Lehrbücher Birkhäuser (2007), 568 pp., Price: US$89.95 ISBN 978-3-7643-8146-2.
Reviewer: Mark Groves Department of Mathematical Sciences, Loughborough University, Loughborough, England, U.K.

According to the authors, the purpose of this book is "to describe the basic methods of nonlinear analysis and illustrate them on simple examples," and a glance at the table of contents gives an impression of what they have in mind:

1. Preliminaries (linear algebra, Banach spaces)
2. Properties of linear and nonlinear operators (bounded linear operators, compact operators, contraction-mapping principle)
3. Abstract integral and differential calculus
4. Local properties of differentiable mappings (inverse and implicit function theorems, manifolds)
5. Topological and monotonicity methods (degree theory, monotone operators)
6. Variational methods (direct methods, mountain-pass theory, Lusternik-Schnirelmann category)
7. Boundary value problems for partial differential equations (weak solutions to linear and nonlinear boundary-value problems)

In view of the large number of good introductory texts on applied nonlinear analysis, any new volume has to offer something "different" in order to qualify as a welcome addition to the existing literature in this area. The distinguishing feature of Drábek and Milota is the large number of subjects covered, including many which are not usually found in beginners' texts. In fact, one could describe it as a small encyclopædia rather than as a textbook. Apart from solid discussions of familiar topics such as abstract differential calculus and bifurcation theory, the book contains a thorough treatment of modern aspects of the calculus of variations and degree theory and an excellent introduction to the Lusternik-Schnirelmann category. The theory of monotone operators (which is today sightly out of fashion but still very useful) is also covered in some detail, and there is an entertaining final chapter on applications of the methods developed in the text to existence theories for weak solutions of partial differential equations.

The authors do however score some of their own pedagogical goals. Many important points are relegated to in-text exercises ("the reader is invited to prove that...."), when a proper discussion would have been more appropriate. This is after all supposed to be an introductory text. Furthermore, material which the authors consider to be "more advanced" is placed in small-print appendices at the end of chapters. This material, which accounts for over 10% of the total text, is in reality informative and relevant, and should be promoted to the regular print size.