Many years ago, as a young and overconfident graduate student, I thought, based on the fact that I had taken and enjoyed an undergraduate senior-seminar course in basic functional analysis, that it would be fun to sit in on a course being offered by Louis Nirenberg on nonlinear functional analysis. I remember the first class quite well. Professor Nirenberg began by writing the equation “F(x) = 0” on the blackboard and explaining that we would be interested in solving that equation for nonlinear functions F. *So far so good*, I remember thinking. And then Nirenberg began talking about homotopy equivalence and differential forms, and I began to realize that I might have bitten off a bit more than I could chew. After one or two other lectures I regretfully concluded that I simply didn’t have the appropriate background for this course and that therefore my education in nonlinear analysis would have to be put off for a while.

What with the press of other business that attends a graduate education in mathematics, particularly with a thesis in a completely different area, not to mention the general demands of growing up, “a while” turned out to be about 40 years. However, thanks to the excellent book now under review, I have finally had a chance to get exposed to some of the ideas that were being discussed in that course.

Brown’s text is pitched at a lower level than were Nirenberg’s lectures (which, by the way, were codified in 1974 as a set of lecture notes published by the Courant Institute; a bibliographically updated version of these notes is now published by the AMS) and has a more modest scope. The basic goal of this book is to explain, prove and apply a famous result in bifurcation theory called the Krasnoselski-Rabinowitz theorem. The author chose this as a goal not only because of the beauty of this particular result but also because it helps illustrate a fundamental point that he wishes to make, namely that topological methods are very valuable in the study of nonlinear analysis. (The term “topological” should be interpreted in a broad sense here, so as to include both point-set and algebraic topology.)

The trip up to this particular peak is reasonably gradual and quite scenic. It also contains a detour that those pressed for time may, if they wish, skip. In more detail: the book is divided into four parts, each one containing about half a dozen relatively short chapters. Part I begins with a very nice introductory chapter designed to illustrate the utility and beauty of the topological approach. Specifically, the author considers a standard existence theorem in differential equations and proceeds to sketch two proofs. The first proof successively approximates a solution by a sequence of functions. The second uses a generalization of the Brouwer fixed point theorem (taken on faith at this point, but proved later in the text).

With this as motivation, the rest of Part I addresses the Arzelà-Ascoli theorem and fixed point theory. Starting with the Brouwer fixed point theorem (assumed in the main body of the text but proved in an Appendix, using basic notions of homology theory) the author proves a generalized version of it and then generalizes things still further to Schauder fixed point theory (replacing Euclidean space by more general normed linear spaces). The rest of Part I gives applications of these fixed point theorems.

Part II addresses degree theory. As in part I of the text, the author starts in finite-dimensional spaces with Brouwer degree theory and then generalizes to infinite-dimensional normed spaces to discuss a generalization, the Leray-Schauder degree. Applications to differential equations arising from physical problems are also discussed.

At this point, the reader has a choice of what path to take. (Or, to put it in terms appropriate for the book, the text bifurcates.) Readers who are anxious to get to the Krasnoselski-Rabinowitz theorem may proceed directly to part IV, which proves this theorem (after developing some background material in topology and the spectral theory of compact operators). After the theorem is proved, an application to the theory of Euler buckling is provided. Euler buckling concerns the mathematical modeling of a situation where a load is placed on a vertical column; for a sufficiently large load, the column will buckle. Under some mild physical assumptions, the analysis of this situation leads to a differential equation, which is an example of a Sturm-Liouville problem. So, the chapter on Euler buckling is preceded by two chapters discussing these problems in general.

An alternative route would be to proceed directly to Part III, which is new to this edition of the text. (Or, of course, any reader who proceeded directly to part IV can, time permitting, return to this part of the book.) In this part of the book, the Leray-Schauder theory of Part II is extended and a mathematical theory called the fixed point index is discussed. This part of the book then ends with several chapters discussing applications of this theory to differential equations and fixed point theorems.

The lack of any exercises in this text may complicate its usefulness as a textbook for a course. However, in all other respects the book has a great deal to recommend it, particularly the author’s exceptional writing style: clear, conversational, and a pleasure to read, yet mathematically precise and rigorous. The author talks directly to the readers, but does not talk down to them. Witness, for example, the discussion of the Brouwer fixed point theorem, which the author is about to generalize:

I’m not going to prove Brouwer’s theorem at this point. If you are a fan of algebraic topology, you believe the result is true and can probably remember how to prove it. If you aren’t, I’m betting you’d rather not start off your journey through topology and analysis by scaling a wall of homology theory. So I’ll just assume the theorem. However, if it bothers you to generalize a result that you have not seen proved, by all means pay a visit to Appendix A. There you will find the basic definitions of singular homology theory, a summary of the properties of this theory, and the necessary computations and argument to establish Brouwer’s theorem in a reasonably self-contained manner. The purpose of the appendix is not only to present a proof of the Brouwer theorem but also to exhibit the basic facts of homology theory, because we will need to refer to them later.

This Appendix, in turn, begins by stating that it presents “a good-sized piece of basic algebraic topology, summarized at approximately the speed of sound. It should, therefore, be approached with all the care you would normally use in the vicinity of any object moving at that speed.”

Thanks to this and other appendices that summarize some basic prerequisite material, a large portion of this book should be reasonably understandable even to upper-level undergraduates with a good real analysis course under their belts; certainly a beginning graduate student should find this book quite comprehensible, very informative, and enjoyable as well. The author deserves both congratulations and thanks for making such nontrivial mathematics so readily accessible.

Mark Hunacek ([email protected]) teaches mathematics at Iowa State University.