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Fixed point theory. (English) Zbl 1171.54034
Cluj-Napoca: Cluj University Press (ISBN 978-973-610-810-5/hbk). xx, 509 p. (2008).
This is an awkward book. The authors claim “to present the most important results in the field of fixed point theory”, but actually they just have compiled a huge list of references – some sections contain no result at all (e.g., 24.10 “Fixed point theory in ordered linear spaces”, 24.18 “Classification of fixed points”, 24.19 “Fixed point theory for fuzzy operators”, 24.20 “Fixed point theory in algebraic structures”). In some cases the selection of topics exhibits a chauvinistic bias (e.g., section 24.21 “Fixed point theory in algebraic topology” lists just one result, viz., Deleanu’s theorem without explaining the technical terms in this theorem.) To be sure, Deleanu’s work is certainly interesting, but algebraic topology has made a vast contribution to fixed point theory and selecting just one result is certainly inadequate.
The authors devote much space to the contraction mapping principle and its countless (more or less useful) generalizations, and they provide proofs for many theorems. Whereas the contraction principle is fairly trivial, nonexpansive mappings provide challenging mathematical problems. The authors spend just eight pages on this topic (two of which deal with the geometry of Banach spaces), they don’t prove a single theorem, and highly important contributions (e.g., by M. Edelstein or R. Nussbaum) are not even mentioned. There is also a discussion of multivalued contractions, of common fixed points and of coincidence points. Degree theory is dealt with on nine pages without a single proof. In the chapter on “topological spaces with the fixed point property” the authors provide two proofs of the Brouwer fixed point theorem. For the first one they use the KKM theorem (for the proof of which they refer to the Brouwer theorem!), for the second one they use degree theory (which they treated without proofs). In the section on multivalued operators they don’t even mention the fundamental article by M. J. Powers [Proc. Camb. Philos. Soc. 68, 619–630 (1970; Zbl 0201.25001)], and L. Gorniewicz’s contributions [e.g. in Dissertationes Math., Warszawa 129, 66 p. (1976; Zbl 0324.55002)] are mentioned in the list of references but none of his results is discussed. Further chapters deal with fixed point structures, fixed point structures for multivalued operators, fixed point theory for operators on product spaces, fixed point theory for nonself operators, a generic view on fixed point theory, iterated function systems. The last chapter “other results” is just a ragbag containing a sub-ragbag “Applications of the fixed point theory” which doesn’t contain a single result – it’s just a huge list of references.
The main purpose of the book seems to be the promulgation of the first author’s papers: if this reviewer has counted correctly he appears 136 times in the list of references – a kind of autapotheosis. Whereas this is at most comical the authors’ attitude towards theorems and proofs is annoying: as long as results are fairly trivial they get a chance of obtaining a proof, when things become more difficult results are mentioned without a proof, whereas really interesting results are not mentioned at all. In any case at present there is no need on a new book on fixed point theory since there exists the encyclopedic monograph by [A. Granas and J. Dugundji, Fixed point theory. Springer Monographs in Mathematics. New York, NY: Springer (2003; Zbl 1025.47002)] which covers the topological as well the analytical aspects of the theory with full worked-out proofs. A reader who wants to learn fixed point theory without bothering about algebraic topology is best advised to read the corresponding sections in Deimling’s unparalleled [K. Deimling, Nonlinear functional analysis. Berlin, etc.: Springer-Verlag (1985; Zbl 0559.47040)]. {It is a shame that the publisher removed this excellent book from his stock-list.}

54H25 Fixed-point and coincidence theorems (topological aspects)
54-02 Research exposition (monographs, survey articles) pertaining to general topology
47H10 Fixed-point theorems
47H04 Set-valued operators
47H09 Contraction-type mappings, nonexpansive mappings, \(A\)-proper mappings, etc.