In many cases, existence problems in the theory of differential equations and in the calculus of variations refer to results of nonlinear analysis in abstract infinite-dimensional spaces. For example, the problem of finding stationary points of a functional in the calculus of variations may lead to an abstract saddle point theorem. To prove existence of a solution of a problem in the theory of partial differential equations, fixed points and other topological tools of nonlinear operator theory are often useful. The book shows methods of how to apply abstract nonlinear analysis to specific problems. Thus both nonlinear functional analysis and application sections are well developed.

The list of all the topics would be too long so we will only mention the main themes. The exposition of differential calculus in normed linear spaces discusses the inverse function theorem and the implicit function theorem. Some further topics are specific to finite dimensions; they include the rank theorem, finite-dimensional bifurcation theorems, integration of differential forms on manifolds with Stokes’ theorem and the Brouwer degree. The topological methods in nonlinear analysis are based on infinite-dimensional generalizations of the degree. The Leray-Schauder degree is available for compact perturbations of the identity operator. Another extension of the concept of degree can be used for monotone operators and their generalizations. Applications include fixed point theorems, global bifurcation theorems and existence theorems based on subsolutions and supersolutions.

The exposition of variational methods starts with a classical analysis of extrema on infinite dimensional spaces and direct methods of the calculus of variations. The analysis of stationary points covers bifurcation of potential operators, the mountain pass lemma, the Lusternik-Schnirelmann method and other tools for searching saddle points. Finally, the last chapter is devoted to a systematic treatment of classical and weak solutions of partial differential equations, demonstrating techniques developed in the preceding text. The authors reveal the fruits of their long and rich teaching experience. The presentation is self-contained and covers all the fundamental tools of nonlinear analysis. On the other hand, they also organize advanced excursions, mostly as appendices, to topics of interest and some parts are developed up to recent research level. Consequently, this volume can be used as a reference book or a textbook, especially as an important source of knowledge and inspiration for university students and teachers of all levels.

Reviewer:

jama