Measures of noncompactness in metric fixed point theory. (English) Zbl 0885.47021

Ayerbe Toledano, J. M.; Domínguez Benavides, Tomas; López Acedo, Genaro

Operator Theory: Advances and Applications. 99. Basel: Birkhäuser. vi, 211 p. (1997).

This well-written text is mainly addressed to graduate students and all who wish to learn and work in the field of measures of noncompactness and metric fixed point theory.
The text is divided into ten chapters. Chapters 1, 2 and 3 are devoted to discussion of the basic problems on the field: the fixed point theorems of Brouwer and Schauder, measure of noncompactness, and minimal sets for a measure of noncompactness. In Chapters 4, 5, 6 and 7, the authors consider geometric notions which play an important role in metric fixed point theory – convexity and smoothness, nearly uniform convexity and nearly uniform smoothness, normal structure and its absence. They apply the geometric and compactness conditions to metric fixed point theory. Chapters 8 and 9 are devoted to uniformly Lipschitzian and asymptotically regular mappings. The final Chapter 10 is devoted to studying the packing rates and \(\phi\)-contractiveness constants.
Finally, this is an interesting, useful and informative work, which presents the beauty of the subject. The book can be warmly recommended to any student of analysis.

Reviewer: J.Górnicki (Rzeszów)

MSC:

47H10	Fixed-point theorems
47H09	Contraction-type mappings, nonexpansive mappings, \(A\)-proper mappings, etc.
54-02	Research exposition (monographs, survey articles) pertaining to general topology
46-02	Research exposition (monographs, survey articles) pertaining to functional analysis
55M20	Fixed points and coincidences in algebraic topology
47-02	Research exposition (monographs, survey articles) pertaining to operator theory
54H25	Fixed-point and coincidence theorems (topological aspects)
46A50	Compactness in topological linear spaces; angelic spaces, etc.
46B20	Geometry and structure of normed linear spaces
46B25	Classical Banach spaces in the general theory

Keywords:

measures of noncompactness; metric fixed point theory; fixed point theorems of Brouwer and Schauder; minimal sets; convexity; smoothness; nearly uniform convexity; nearly uniform smoothness; normal structure; asymptotically regular mappings; packing rates; \(\phi\)-contractiveness constants