Handbook of Metric Fixed Point Theory

1. Front Matter

   Pages i-xiii

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2. Contraction Mappings and Extensions

   • W. A. Kirk

   Pages 1-34

3. Examples of Fixed Point Free Mappings

   • Brailey Sims

   Pages 35-48

4. Classical Theory of Nonexpansive Mappings

   • Kazimierz Goebel, W. A. Kirk

   Pages 49-91

5. Geometrical Background of Metric Fixed Point Theory

   • Stanisław Prus

   Pages 93-132

6. Some Moduli and Constants Related to Metric Fixed Point Theory

   • Enrique Llorens Fuster

   Pages 133-175

7. Ultra-Methods in Metric Fixed Point Theory

   • M. A. Khamsi, B. Sims

   Pages 177-199

8. Stability of the Fixed Point Property for Nonexpansive Mappings

   • Jesús Garcia-Falset, Antonio Jiménez-Melado, Enrique Llorens-Fuster

   Pages 201-238

9. Metric Fixed Point Results Concerning Measures of Noncompactness

   • T. Domínguez, M. A. Japón, G. López

   Pages 239-268

10. Renormings of ℓ¹ and C ₀ and Fixed Point Properties

    • P. N. Dowling, C. J. Lennard, B. Turett

    Pages 269-297

11. Nonexpansive Mappings: Boundary/Inwardness Conditions and Local Theory

    • W. A. Kirk, C. H. Morales

    Pages 299-321

12. Rotative Mappings and Mappings with Constant Displacement

    • Wieslawa Kaczor, Małgorzata Koter-Mórgowska

    Pages 323-337

13. Geometric Properties Related to Fixed Point Theory in Some Banach Function Lattices

    • S. Chen, Y. Cui, H. Hudzik, B. Sims

    Pages 339-389

14. Introduction to Hyperconvex Spaces

    • R. Espínola, M. A. Khamsi

    Pages 391-435

15. Fixed Points of Holomorphic Mappings: A Metric Approach

    • Tadeusz Kuczumow, Simeon Reich, David Shoikhet

    Pages 437-515

16. Fixed Point and Non-Linear Ergodic Theorems for Semigroups of Non-Linear Mappings

    • Anthony To-Ming Lau, Wataru Takahashi

    Pages 517-555

17. Generic Aspects of Metric Fixed Point Theory

    • Simeon Reich, Alexander J. Zaslavski

    Pages 557-575

18. Metric Environment of the Topological Fixed Point Theorems

    • Kazimierz Goebel

    Pages 577-611

19. Order-Theoretic Aspects of Metric Fixed Point Theory

    • Jacek Jachymski

    Pages 613-641

20. Fixed Point and Related Theorems for Set-Valued Mappings

    • George Xian-Zhi Yuan

    Pages 643-690

Metric fixed point theory encompasses the branch of fixed point theory which metric conditions on the underlying space and/or on the mappings play a fundamental role. In some sense the theory is a far-reaching outgrowth of Banach's contraction mapping principle. A natural extension of the study of contractions is the limiting case when the Lipschitz constant is allowed to equal one. Such mappings are called nonexpansive. Nonexpansive mappings arise in a variety of natural ways, for example in the study of holomorphic mappings and hyperconvex metric spaces.
Because most of the spaces studied in analysis share many algebraic and topological properties as well as metric properties, there is no clear line separating metric fixed point theory from the topological or set-theoretic branch of the theory. Also, because of its metric underpinnings, metric fixed point theory has provided the motivation for the study of many geometric properties of Banach spaces. The contents of this Handbook reflect all of these facts.
The purpose of the Handbook is to provide a primary resource for anyone interested in fixed point theory with a metric flavor. The goal is to provide information for those wishing to find results that might apply to their own work and for those wishing to obtain a deeper understanding of the theory. The book should be of interest to a wide range of researchers in mathematical analysis as well as to those whose primary interest is the study of fixed point theory and the underlying spaces. The level of exposition is directed to a wide audience, including students and established researchers.

• banach spaces
• compactness
• fixed point theory
• mathematical analysis
• metric space
• stability

`...a wide, up to date and complete presentation of the subject. ...the book appears uniform in style and in the organization of the chapters unncecessary repetitions are practically absent. The large number of authors certainly helped make it possible to treat all the topics presented with the scope and detail that only specialists in the specific subfields can guarantee. Each chapter begins with either a clear and complete exposition of the basic results in its subject area or with the necessary background to get to basic results, then it enters into more specialized results and ends with an exhaustive list of references. ...valuable and agreeable to a large group of mathematicians. For researchers ... a comprehensive reference ... useful for a broad knowledge and understanding of metric fixed point theory. ...editors have succeeded admirably in producing a handbook, not just a collection of papers...'
Australian Mathematical Society GAZETTE, 29:2 (2002)

• Department of Mathematics, The University of Iowa, Iowa City, USA

  William A. Kirk

• School of Mathematical and Physical Sciences, The University of Newcastle, Newcastle, Australia

  Brailey Sims

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