zbMATH — the first resource for mathematics

A fixed-point farrago. (English) Zbl 1352.47002
Universitext. Cham: Springer (ISBN 978-3-319-27976-3/hbk; 978-3-319-27978-7/ebook). xiv, 221 p. (2016).
This is a well-written and essentially self-contained survey on several applications of fixed-point theory. The book starts quite traditionally with applications of Brouwer’s fixed-point theorem and contraction mappings. It seems to be fashionable today to include a section on the Google matrix (cf. [I. Farmakis and M. Moskowitz, Fixed point theorems and their applications. Hackensack, NJ: World Scientific (2013; Zbl 1276.47001)]). The author then gives C. A. Rogers’ version [Am. Math. Mon. 87, 525–527 (1980; Zbl 0447.57020)] of Milnor’s proof of the Brouwer theorem. (Neither the publication year (2010) nor the volume number (67) are correctly stated in the bibliography.) There is an interesting section on game theory covering the minimax theorem and Nash equilibrium. Proceeding to the infinite-dimensional case, the author proves the classical Schauder fixed-point theorem and describes the traditional application of this result to Peano’s theorem on solutions of the initial value problem for ODE’s. There is also a section on Lomonosov’s theorem on hyperinvariant subspaces for compact operators. This is a somewhat extravagant application of Schauder’s fixed point theorem; as a matter of fact, it is well known (cf. [A. J. Michaels, Adv. Math. 25, 56–58 (1977; Zbl 0356.47003)]) that the existence of nontrivial hyperinvariant subspaces for non-zero compact operators in infinite-dimensional Banach spaces can easily be proven without recurrence to any fixed-point theorem.
The last (fourth) part makes this book really interesting. Here, the author proves the Markov-Kakutani theorem and a special version of the Ryll-Nardzewski theorem (for duals of separable Banach spaces) which is sufficient for the proof of the existence of a Haar measure on compact topological groups. Similarly, there is a proof of the Kreĭn-Mil’man theorem for duals of separable Banach spaces (here, in the statement of Lemma 13.5 \(K\) should be \(\overline{\text{conv}}K_0\)). As another application of the Markov-Kakutani theorem, he shows that each abelian group is amenable. For the reviewer, the most interesting item is Chapter 13 on “paradoxical decompositions”, where the author discusses the Banach-Tarski paradox.
The book makes easy reading because the proofs are remarkably well structured. They are broken up into easily digestible parts and the claims are always clearly stated. There are five appendices which help to make the book almost self-contained. The reviewer would nevertheless have welcomed an appendix on axiomatic set theory. When discussing paradoxical decompositions it would be helpful to know which model of set theory the author prefers. The reviewer’s guess is ZFC.
A final remark about the book’s title is in order. It seems funny that the author considers it necessary to explain the meaning of “farrago” by quoting an online dictionary instead of the ODE or Webster’s whereas everybody with a modest knowledge of Latin will know the word anyway. More important, using this farrago (in the original meaning) will result in an unbalanced diet. The author restricts himself completely to analytical methods without even mentioning the entire realm of topological methods. Of course, one cannot deal with any topic, but a book on fixed-point theory should at least mention the Lefschetz fixed point theorem, Borsuk-Ulam type results and equivariant fixed point theory.

47-02 Research exposition (monographs, survey articles) pertaining to operator theory
00A05 Mathematics in general
20F16 Solvable groups, supersolvable groups
47A15 Invariant subspaces of linear operators
47H10 Fixed-point theorems
54H25 Fixed-point and coincidence theorems (topological aspects)
Full Text: DOI