Methods of Nonlinear Analysis: Applications to Differential Equations

Book Preface

In this second edition we have made minor revisions and added new material to
Chapter 8 and partly also to Chapters 2, 3 and 6. We quote from the Preface to
the first edition in order to clarify the original motivation behind publication of
this book: “There are many books and monographs devoted to the methods of nonlinear analysis and their applications. Typically, such a book is either dedicated to a particular topic and treats details which are difficult to understand for a student, or it deals with an application to complicated nonlinear partial differential equations in which a lot of technicalities are involved. In both cases it is very difficult for a student to get oriented in this kind of material and to pick up the ideas underlying the main tools for treating the problems in question. The purpose of this book is to describe the basic methods of nonlinear analysis and to illustrate them on simple examples. Our aim is to motivate each method considered, to explain it in a general form but in the simplest possible abstract framework, and finally, to show its application (typically to boundary value problems for elementary ordinary or partial differential equations). To keep the text free of technical details and make it accessible also to beginners we did not formulate some key assertions and illustrative examples in the most general form.

In order to make the text self-contained, we decided to comment on several notions and statements in footnotes. To place the material from the footnotes in the text could disturb a more advanced reader and make the exposition more complicated. In order to emphasize the role of the statements in our exposition we identify them as Theorem, Proposition, Lemma and Corollary. However, the reader should be aware of the fact that this by no means expresses the importance of the statement within the whole of mathematics. So, several times, we call important theorems Propositions, Lemmas or Corollaries.

Although the book should primarily serve as a textbook for students on the graduate level, it can be a suitable source for scientists and engineers who have need of modern methods of nonlinear analysis.

At this point we would like to include a few words about our good friend, colleague and mentor Svatopluk Fuˇc´ık to whom we dedicate this book. His work in the field of nonlinear analysis is well recognized and although he died in 1979 at the age of 34, he ranks among the most important and gifted Czech mathematicians of the 20th century.