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Topological fixed point theory of multivalued mappings. (English) Zbl 0937.55001
Mathematics and its Applications (Dordrecht). 495. Dordrecht: Kluwer Academic Publishers. ix, 399 p. (1999).
A multivalued function $$\phi$$ assigns to each point $$x$$ of the domain set $$X$$ a subset $$\phi(x)$$ of the range set $$Y$$. Although topology is primarily concerned with single-valued functions, multivalued functions arise naturally in topological contexts. For instance, for any single-valued map $$f: X\to Y$$ there is a multivalued function $$\phi$$ with domain $$Y$$ defined by $$\phi(y) =f^{-1}(y)$$. With regard to fixed point theory, the fixed point equation $$f(x) = x$$ for a self-map of a space $$X$$ readily generalizes to the inclusion $$x\in\phi(x)$$ for a multivalued function that takes points of a space $$X$$ to subsets of the same space. In 1946, when the Lefschetz Fixed Point Theorem was barely 20 years old, Eilenberg and Montgomery extended that fundamental result to a useful class of multivalued functions. The 500 or so books and papers listed as references in this book quite properly makes no attempt to be a comprehensive bibliography on multivalued functions but instead focuses on relevant parts of the more recent literature. Nevertheless, the length of that list suggests how extensive the literature of multivalued functions has become.
The author is widely recognized as an authority on the fixed point theory of multivalued functions. Not only is he author of more than 30 of the works referenced, but a significant proportion of the rest is the work of his colleagues and students. The author’s concerns as a teacher as well as researcher are quite evident. The first two chapters, which occupy more than 100 out of a total of 400 pages, give the reader a carefully designed introduction to the topics that a person with a reasonable foundation in topology will need to read the rest of the book. The first chapter begins with some background from retract theory and spaces of subsets, followed by a number of topics in homology theory, concluding with the Lefschetz theorem and some of its extensions. This chapter is no mere listing of definitions and facts but includes many proofs and, when a proof is not furnished, the reader is directed to the appropriate literature. Multivalued functions make their appearance in the second chapter. No background concerning such functions is assumed of the readers and many examples, as well as proofs of just about all the results, are included. After introducing the various notions of continuity in the multivalued function setting, the chapter is devoted to an extensive discussion of selections, that is, a single-valued function $$f$$ that selects one point $$f(x)$$ out of the set $$\phi(x)$$ for each $$x$$ in the domain of $$\phi$$.
The next two chapters, which present methods for establishing the existence of fixed points of multivalued functions in a variety of settings, are the heart of the book. The first of these chapters collects results based on approximation by a single-valued function and then the use of a limiting process to determine to what extent properties of the approximating function carry over to the original multivalued function. Although approximation methods are useful, more sophisticated methods based on homology theory are considerably more powerful and lead to conditions for the existence of fixed points of multivalued functions under a variety of hypotheses. The chapter on these methods is a very thorough exposition of what is presently known. The final two chapters of the book apply the results of the two chapters on methods to a variety of contexts. The first presents applications primarily to nonlinear functional analysis, but also to dynamical systems and mathematical economics. The final chapter is devoted to the application of the theory to differential inclusions, that is, problems of the form $$x'(t) \in \phi(t,x(t))$$ where $$\phi$$ is a multivalued function and the solution $$x(t)$$ is single-valued.
As the outline above indicates, the overall planning of the book is very good. The style of exposition is spare and formal, with a definition-theorem-proof format, but with helpful examples where they are needed. There are numerous remarks, some to guide the reader through the text by connecting results presently being discussed to material yet to come, many that indicate ways in which results may be generalized and others that clarify difficult technical points. The author has taken considerable care to make the proofs throughout the book as direct and simple as possible. This is a virtue in any area of mathematics, but it is a necessity when dealing with a subject as inherently complicated as this one. Portions of the book would function well as the text for advanced courses, but its greatest value will likely be as a reference to what is presently known about the fixed point theory of multi-valued functions. For that reason, it is unfortunate that the index is rather inadequate (for instance, there is no index entry for the definition of a selection) and there is no index of notation. This latter problem is particularly acute because a reader who wants a specific result has to search backward through the text from its statement to find the precise meaning of the symbols. These problems could be solved if the author or publisher would post a more detailed word index and an index of notation on a web site for the convenience of purchasers of the book. Another, though minor, problem with the book is the too-numerous misprints. Although these do not appear to affect the mathematical content of the book, they are annoying and many of them could have been caught by running the file through a spell checker. This does not seem too much to ask of a publisher that is charging \$ 175 for a 400 page book.

##### MSC:
 55M20 Fixed points and coincidences in algebraic topology 54H25 Fixed-point and coincidence theorems (topological aspects) 47H10 Fixed-point theorems