Elementary Fixed Point Theorems

One of the most well-known theorems in topology is Brouwer’s Fixed Point Theorem – that every continuous function from the closed disk to itself has a fixed point. Of course, understanding when a function has the property of having a fixed point has long been of interest more generally, and this book showcases some of that work. The emphasis is on introducing readers to a collection of interesting theorems and applications with the hope of enticing the reader to further study.

 

The chapters are relatively independent. For this reason, this would be a nice book to hand to an advanced undergraduate or a beginning graduate student to supplement coursework or to find a topic for a project. Most of the theorems included here are not found in standard beginning analysis or topology texts, but could be (e.g., Caristi’s fixed point theorem, Tarski’s theorem on fixed points of lattices). That is to say that a first real analysis course is sufficient preparation to fully engage in this material. A first point-set topology course would also be sufficient preparation, but the flavor of the presentation leans towards analysis. 

 

While topology has its roots in analysis, and the necessary prerequisites of both subjects are set forth in the first chapter, the flavor of the text is more analytic than topological. As an example, in the chapter devoted to the aforementioned Brouwer theorem, the proof of the theorem relies on the Weierstrass Approximation. As a further example, the chapter “Applications of the Contraction Principle” revolves primarily around differential equations, offering several theorems about the existence of solutions to certain initial value problems and boundary value problems of ordinary differential equations. It is interesting to note, however, that this chapter also contains a proof of the central limit theorem which is a classical theorem in probability.

 

Each chapter ends with an extensive list of references, which is wonderful, but there is no index in the book, which is not. This omission makes the book a bit challenging to use, especially as it pertains to definitions. Much of the author’s commentary comes in the form of numbered remarks, which makes these easy to track. His introductions to each chapter are minimal, some consisting only of a sentence that captures the chapter title. The better ones very succinctly and expertly provide a bit of history.

 

Overall, the author succeeds in providing an intriguing collection of theorems beyond those widely known from basic study. The text would have benefitted from a careful editor, but this does not detract from its overall usefulness.

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Michele Intermont is an Associate Professor at Kalamazoo College.