Chapman and Hall/CRC
Published December 12, 2013
Reference - 232 Pages - 30 B/W Illustrations
ISBN 9781466595217 - CAT# K20714
Series: Chapman & Hall/CRC Pure and Applied Mathematics
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Classification of Lipschitz Mappings presents a systematic, self-contained treatment of a new classification of Lipschitz mappings and its application in many topics of metric fixed point theory. Suitable for readers interested in metric fixed point theory, differential equations, and dynamical systems, the book only requires a basic background in functional analysis and topology.
The author focuses on a more precise classification of Lipschitzian mappings. The mean Lipschitz condition introduced by Goebel, Japón Pineda, and Sims is relatively easy to check and turns out to satisfy several principles:
The Lipschitz Condition
Nonlinear spectral radius
Uniformly lipschitzian mappings
Basic Facts on Banach Spaces
The operator norm
Dual spaces, reexivity, the weak, and weak* topologies
Mean Lipschitz Condition
Nonexpansive and mean nonexpansive mappings in Banach spaces
On the Lipschitz Constants for Iterates of Mean Lipschitzian Mappings
A bound for Lipschitz constants of iterates
A bound for the constant k∞(T)
Moving averages in Banach spaces
A bound for the constant k0(T)
More about k(Tn), k0(T), and k∞(T)
Subclasses Determined by p-Averages
Basic definitions and observations
A bound for k(Tn), k∞(T), and k0(T)
On the moving p-averages
Classical Banach’s contractions
On characterizations of contractions
On the rate of convergence of iterates
Nonexpansive Mappings in Banach Space
The asymptotic center technique
Minimal invariant sets and normal structure
Uniformly nonsquare, uniformly noncreasy, and reflexive Banach spaces
Remarks on the stability of f.p.p.
The case of ℓ1
Mean Nonexpansive Mappings
Some new results of stability type
Sequential approximation of fixed points
The case of n = 3
On the structure of the fixed points set
Mean Lipschitzian Mappings with k > 1
Losing compactness in Brouwer’s Fixed Point Theorem
Retracting onto balls in Banach spaces
Generalized characteristics of minimal displacement
"The book is well written and contains new interesting results along with some classical ones in metric fixed point theory. The prerequisites are modest—some basic results in topology and functional analysis—so it can be used by advanced undergraduate and graduate students for an introduction to this domain and by researchers as a reference text. Experts in other areas, such as differential equations and dynamical systems, will find it useful as well."
—Studia Universitatis Babes-Bolyai Mathematica, 59, 2014
"Every mathematician knows the importance of Lipschitz maps and, in particular, of the behavior of Lipschitz constants of the iterates. This book is highly recommended to anyone interested in getting insight on new developments in this area. The main part of this volume is devoted to present the basic theory of the so called ‘mean Lipschitz condition,’ a recent extension of the classical Lipschitz property which involves not only the property of the map itself but also of its iterates. In particular, the author present the deep influence that this condition has on the behavior of the sequence of Lipschitz constants for consecutive iterates and on its asymptotic behavior. In addition, it contains a large number of example and various applications in metric fixed point theory. The book is self-contained and addressed to advanced undergraduate and graduate students as well to researchers interested in this topic. Students will find a rich collection of examples ranging from simple to non-trivial, while specialists will be challenged by new interesting open problems."
—Emanuele Casini, Dip. Scienza ed Alta Tecnologia, Insubria University
"The Lipschitz condition is one of the most elegant classical concepts in mathematical analysis. It appears in university courses of differential equations and nonlinear analysis as well as in contemporary research in both pure and applied mathematics. Deep understanding of the properties of Lipschitzian mappings is therefore important for all levels of study in many branches of mathematics. This book by Lukasz Piasecki is a good choice for achieving such an understanding in the framework of mappings in general metric spaces, in particular, Banach spaces. Moreover, it gives new insight into the theory of Lipschitzian mappings via a study of the mean Lipschitz condition. This quite natural modification of the classical Lipschitz condition turns out to be very useful in the problem of estimating Lipschitz constants of the consecutive iterates of a given mapping. The author also presents its various applications to metric fixed-point theory.
The book is written in a very clear and reader-friendly way. The author gives many examples illustrating various aspects of presented results. As emphasized in the introduction, the book is self-contained and only a basic knowledge of functional analysis and topology is required. It can be advised for graduate students, but specialists will also find some interesting ideas and results in it."
—Stanislaw Prus, Marie Curie-Sklodowska University, Lublin, Poland
"… a self-contained, readable and precise course on the subject. It is addressed to advanced undergraduate and graduate students interested in nonlinear analysis. The main stress is put on operator theory, applications to metric fixed point theory and related fields. Besides the presentation of the theory, the true value of the book lies in a collection of cleverly chosen interesting examples. As prerequisites, only a basic knowledge of functional analysis and topology is required.
Students will find here materials for seminar works and presentations. Teachers can select topics for advanced courses on analysis and use the book as a supplementary text. A number of problems open a stream of new directions for research."
—Kazimierz Goebel, Maria Curie-Sklodowska University, Lublin, Poland
"I strongly recommend this book for advanced undergraduate and graduate students because this is the first book devoted to a classification of Lipschitzian mappings … The reader will find in this book a new classification of this kind of mapping as well as many examples and illustrations designed to help the reader understand the definitions, properties, and results. Since the book states new theorems concerning the asymptotic behaviour of some mappings and sequences of real numbers, along with some open problems, I also recommend this book for analysts or mathematicians who are looking for new topics to research."
—Victor Perez-Garcia, University of Veracruz, Mexico
"... self-contained and systematically arranged. Many interesting examples are also presented."
—Satit Saejung (Khon Kaen), Zentralblatt MATH