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Nonstandard methods in fixed point theory. (English) Zbl 0713.47050
Universitext. New York etc.: Springer-Verlag. ix, 139 p. DM 58.00 (1990).
This book is devoted to some nonconstructive methods in the fixed point theory for nonexpansive mappings of subsets in Banach spaces. The aim of this book is to give a unified account of the major new developments inspired by B. Maurey’s application in 1980 of Banach space ultraproducts. The text begins with careful review of the concepts of Schauder bases, filters, ultrafilters, limits over ultrafilters and nets (chapters 0 and 1). In Chapter 2 both the set-theoretical and Banach space ultraproduct constructions are presented in detail and also the properties of finite representability and Banach-Saks and the description of the classical spaces of Tsirelson and James are given. The final main chapter contents the classical approach to the theory including a discussion of normal structure sets and then after translating these results into ultraproduct language several new fixed point theorems including the B. Maurey results and recent theorems of W. A. Kirk. This book is a good complementary to the recent book by K. Göbel and W. A. Kirk, Topics in metric fixed point theory.
Reviewer: P.Zabreiko

47S20 Nonstandard operator theory
47H10 Fixed-point theorems
47H09 Contraction-type mappings, nonexpansive mappings, \(A\)-proper mappings, etc.
47-01 Introductory exposition (textbooks, tutorial papers, etc.) pertaining to operator theory