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An introduction to nonlinear analysis. Theory. (English) Zbl 1040.46001

Dordrecht: Kluwer Academic/Plenum Publishers (ISBN 0-306-47392-5/hbk). xv, 689 p. (2003).
This book provides a self-contained and systematic introduction to the mathematical foundations of nonlinear analysis. The authors have written a splendid text. The book can be used as a textbook. It can also serve as a reference for researchers. The work is organized as follows:
Chapter 1 gives a thorough treatment of topology. Theorems are precisely formulated and all proofs (here and in subsequent chapters as well) are given. Chapter 2, entitled “Elements of measure theory”, contains a detailed elaboration of the subject. Chapter 3 is devoted to Banach spaces. Among others, Sobolev spaces and vector-valued functions and the Bochner integral are presented here. The last two chapters, namely Chapter 4, “Set-valued analysis”, and Chapter 5, “Nonsmooth analysis”, are, in the opinion of the reviewer, major parts of the monograph.
Both of these chapters constitute the foundations of modern nonlinear analysis with a view towards applications in many areas. The authors give a deep insight into the theory of multifunctions and into nonsmooth analysis. (Let us remark that even in the monumental work “Nonlinear functional analysis and its applications”, Volumes 1–5, of E. Zeidler (1986; Zbl 0583.47050, 1990; Zbl 0684.47028, 1990; Zbl 0684.47029, 1985; Zbl 0583.47051, 1988; Zbl 0648.47036) these themes are not included.)
Each chapter contains illustrative examples perfectly adjusted to the context. At the end of each chapter there are valuable comments, both historical and bibliographical; they provide also, without proofs, additional material. The well-chosen numerous (circa 55) exercises at the end of each chapter with complete solutons make the book especially attractive as a text both for students and teachers. To sum up, the work is an important addition to a rather meager set of text/reference books on modern nonlinear analysis.
Because of contents and a lucid and transparent style of presentation it was a pleasure to study and review the book. The second volume of the work is devoted to various applications of modern nonlinear analysis and will be reviewed separately.
Reviewer: S. Burys (Kraków)

MSC:

46-01 Introductory exposition (textbooks, tutorial papers, etc.) pertaining to functional analysis
49-01 Introductory exposition (textbooks, tutorial papers, etc.) pertaining to calculus of variations and optimal control
28B20 Set-valued set functions and measures; integration of set-valued functions; measurable selections
49-02 Research exposition (monographs, survey articles) pertaining to calculus of variations and optimal control
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