Topics in metric fixed point theory. (English) Zbl 0708.47031

Cambridge Studies in Advanced Mathematics, 28. Cambridge: Cambridge University Press. viii, 244 p. £30.00; $ 49.50 (1990).
This book is devoted to some aspects of fixed point theory. To the author’s mind the results considered and described in detail in the book are always couched in at least a metric framework, usually in a Banach spaces setting, and the methods typically involve both the topological and the geometric structure of the space in conjunction with metric constraints on the behavior of the mappings.
The book contains some short chapters: 1. Preliminaries; 2. Banach’s contractions principle; 3. Nonexpansive mappings: introduction; 4. The basic fixed point theorems for nonexpansive mappings; 5. Scaling the convexity of the unit ball; 6. The modulus of convexity and normal structure; 7. Normal structure and smoothness; 8. Conditions involving compactness; 9. Sequential approximation techniques for nonexpansive mappings; 10. Weak sequential approximations; 11. Properties of fixed point sets and minimal sets; 12. Special properties of Hilbert spaces; 13. Applications to accretivity; 14. Ultrafilter methods; 15. Set-valued mappings; 16. Uniformly Lipschitzian mappings; 17. Rotative mappings; 18. The theorems of Brouwer and Schauder; 19. Lipschitzian mappings; 20. Minimal displacement; 21. The retraction problem; it contains an extensive bibliography.
Undoubtedly, the book should be of interested to graduate students seeking a field of interest, to mathematicians interested in learning about the subject and to specialists.
Reviewer: P.Zabreiko


47H10 Fixed-point theorems
47H09 Contraction-type mappings, nonexpansive mappings, \(A\)-proper mappings, etc.
47H04 Set-valued operators
47-02 Research exposition (monographs, survey articles) pertaining to operator theory