Nonlinear Analysis and its Application to Differential Equations Edited by M.R. Grossinho, M. Ramos, C. Rebelo, L. Sanchez. Birkhauser, 456 pp., Boston 2000, price Eur 95.- ISBN: 0817641882
Reviewer: Dario Bambusi, Department of Mathematics, Milano, Italy

This book contains a series of papers giving a nice survey of the main themes of present research in nonlinear analysis. It originated from an autumn school held in Lisbon in 1998. The book is separated in two parts. The first one contains six short `tutorials' on some important topics, and the second part contains a series of review papers on research topics.

The wide range of arguments touched by the book is already evident by reading the first part. Here one can find an introduction to the use of lower and upper solutions in ODEs together with a paper devoted to the dynamics of one of the more classic PDEs: the Navier Stokes equation. An important and quite new stream of research is related to the study of twist maps: this is also presented. Other tutorials are devoted to classical topics like variational inequalities, the construction of periodic solutions and the use of geometry for the description of mechanical systems.

The papers in the second part of the book deal with an even wider range of topics, but each paper is written in a style that tries to avoid technicalities presenting some recent developments of nonlinear analysis.