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From the preface: “Metric fixed point theory is a rather loose knit branch of fixed point theory concerning methods and results that involve properties of an essentially isometric nature. That is, the class of mappings and domains satisfying the properties need not be preserved under the move to an equivalent metric. It is this fragility that singles metric fixed point theory out from the more general topological theory, although, as many of the entries in this Handbook serve to illustrate, the divide between the two is often a vague one.”
“The focus of the book is on the major developments of the theory. Some information is put forth in detail, making clear the underlying ideas and the threads that link them. At the same time many peripheral results and extensions leading to the current state of knowledge often appear without proof and the reader is directed to primary sources elsewhere for further information. Because so many authors have been involved in this project, no effort has been made to attain a wholly uniform style or to completely avoid duplication. At the same time an attempt has been made to keep duplication to a minimum, especially where major topics are concerned.
The book begins with an overview of metric contraction principles. Then, to help delineate the theory of nonexpansive mappings and alert the reader to its subtleties, this is followed by a short chapter devoted to examples of fixed point free mappings. A survey of the classical theory of nonexpansive mappings is taken up next. After this, various topics are discussed more or less randomly, including the underlying geometric foundations of the theory in Banach spaces, ultrapower methods, stability of the fixed point property, asymptotic renorming techniques, hyperconvex spaces, holomorphic mappings, generic properties of the theory, nonlinear ergodic theory, rotative mappings, pseudocontractive mappings and local theory, the nonexpansive theory in Banach function lattices, the topological theory in a metric environment, order-theoretic aspects of the theory and set-valued mappings.”
The chapter headings and authors are: 1. Contraction mappings and extensions (W. A. Kirk); 2. Examples of fixed point free mappings (B. Sims); 3. Classical theory of nonexpansive mappings (K. Goebel and W. A. Kirk); 4. Geometrical background of metric fixed point theory (S. Prus); 5. Some moduli and constants related to metric fixed point theory (E. L. Fuster); 6. Ultra-methods in metric fixed point theory (M. A. Khamsi and B. Sims); 7. Stability of the fixed point property for nonexpansive mappings (J. Garcia-Falset, A. Jiménez-Melado, and E. Llorens-Fuster); 8. Metric fixed point results concerning measures of noncompactness (T. Dominguez, M. A. Japón, and G. López); 9. Renormings of \(\ell^1\) and \(c_0\) and fixed point properties (P. N. Dowling, C. J. Lennard, and B. Turett); 10. Nonexpansive mappings: boundary/inwardness conditions and local theory (W. A. Kirk and C. H. Morales); 11. Rotative mappings and mappings with constant displacement (W. Kaczor and M. Koter-Mórgowska); 12. Geometric properties related to fixed point theory in some Banach function lattices (S. Chen, Y. Cui, H. Hudzik, and B. Sims); 13. Introduction to hyperconvex spaces (R. Espinola and M. A. Khamsi); 14. Fixed points of holomorphic mappings: a metric approach (T. Kuczumow, S. Reich, and D. Shoikhet); 15. Fixed point and nonlinear ergodic theorems for semigroups of nonlinear mappings (A. Lau and W. Takahashi); 16. Generic aspects of metric fixed point theory (S. Reich and A. J. Zaslawski); 17. Metric environment of the topological fixed point theorems (K. Goebel); 18. Order-theoretic aspects of metric fixed point theory (J. Jachymski); 19. Fixed point and related theorems for set-valued mappings (G. Yuan).
The articles of this volume will be reviewed individually.
Indexed articles:
Kirk, W. A., Contraction mappings and extensions, 1-34 [Zbl 1019.54001]
Sims, Brailey, Examples of fixed point free mappings, 35-48 [Zbl 1026.47036]
Goebel, Kazimierz; Kirk, W. A., Classical theory of nonexpansive mappings., 49-91 [Zbl 1035.47033]
Prus, Stanisław, Geometrical background of metric fixed point theory, 93-132 [Zbl 1018.46010]
Llorens Fuster, Enrique, Some moduli and constants related to metric fixed point theory, 133-175 [Zbl 1021.47030]
Khamsi, M. A.; Sims, B., Ultra-methods in metric fixed point theory, 177-199 [Zbl 1017.47044]
Garcia-Falset, Jesús; Jiménez-Melado, Antonio; Llorens-Fuster, Enrique, Stability of the fixed point property for nonexpansive mappings., 201-238 [Zbl 1083.46502]
Domínguez, T.; Japón, M. A.; López, G., Metric fixed point results concerning measures of noncompactness, 239-268 [Zbl 1022.47039]
Dowling, P. N.; Lennard, C. J.; Turett, B., Renormings of \(\ell^1\) and \(c_0\) and fixed point properties, 269-297 [Zbl 1026.47037]
Kirk, W. A.; Morales, C. H., Nonexpansive mappings: Boundary/inwardness conditions and local theory, 299-321 [Zbl 1022.47034]
Kaczor, Wiesława; Koter-Mórgowska, Małgorzata, Rotative mappings and mappings with constant displacement, 323-337 [Zbl 1032.47031]
Chen, S.; Cui, Y.; Hudzik, H.; Sims, B., Geometric properties related to fixed point theory in some Banach function lattices, 339-389 [Zbl 1013.46015]
Espínola, R.; Khamsi, M. A., Introduction to hyperconvex spaces, 391-435 [Zbl 1029.47002]
Kuczumow, Tadeusz; Reich, Simeon; Shoikhet, David, Fixed points of holomorphic mappings: A metric approach, 437-515 [Zbl 1019.47041]
Lau, Anthony To-Ming; Takahashi, Wataru, Fixed point and nonlinear ergodic theorems for semigroups of nonlinear mappings., 517-555 [Zbl 1037.47041]
Reich, Simeon; Zaslavski, Alexander J., Generic aspects of metric fixed point theory, 557-575 [Zbl 1016.54021]
Goebel, Kazimierz, Metric environment of the topological fixed point theorems, 577-611 [Zbl 1017.47043]
Jachymski, Jacek, Order-theoretic aspects of metric fixed point theory, 613-641 [Zbl 1027.54065]
Yuan, George Xian-Zhi, Fixed point and related theorems for set-valued mappings, 643-690 [Zbl 1019.47043]