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Nonlinear functional analysis and its applications. I: Fixed-point theorems. Transl. from the German by Peter R. Wadsack. (English) Zbl 0583.47050
New York etc.: Springer-Verlag. XXI, 897 p. DM 298.00 (1986).
The five volumes (100 chapters) of this treatise are devoted to: I) Fixed-point theorems, II) Monotone operators, III) Variational methods and optimization (see next review), IV-V) Applications in mathematical physics, and are the result of the author’s wide experience as a researcher and educator involved in the solvability of operator equations. The first three volumes are a translated extension of the German original [(1976; Zbl 0326.47053 and 1980; Zbl 0434.47042)]. Although the titles of the chapters are changed according to the new enlarged content, the succession of the basic topics is almost unaltered. That is why, after some general considerations on the entire work, we will point out only certain augmented sections which we feel to be more significant in contemporary nonlinar analysis. As distinct from the lecture note schemes of the German version, every chapter in the English translation has a flowing, coherent form and contains nice comments, overviews, and perspectives on the strategy and implementations of the considered procedures, and is concluded with complementary problems. Moreover, at the end of each volume there is a comprehensive and up-to- date bibliography.
The work is clearly written and organized so that each chapter can be independently approached. The survey of the basic features of most topics of nonlinear functional analysis makes Zeidler’s treatise, more than any other recent books concerning this subject, a deskbook for a broad range of researchers as well as a guide for many graduate courses.
The first volume (chapters 1-17) describes what fixed-point theory actually means. In the part that deals with general principles and applications, more emphasis is placed on iterative methods, generalizations of the implicit function theorem, super- and subsolutions, the nonlinear Krein-Rutman theorem, multiple solutions, the Ljapunov-Schmidt branching equations, the generic and Hopf bifurcation, minimax theorems, Michael’s selection theorem and differential inclusions, the Bourbaki-Kneser theorem. In the context of the degree theory one also studies the fixed-point index and eigenvalue principle, Leray’s product rule and homeomorphisms, continuation principles in the topological bifurcation theory.
Reviewer: D.Pascali

MSC:
47Hxx Nonlinear operators and their properties
47-02 Research exposition (monographs, survey articles) pertaining to operator theory