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Geometry of Banach spaces, duality mappings and nonlinear problems. (English) Zbl 0712.47043

From Preface: “With this book we intend to familiarize the reader with nonlinear operators related to nonlinear functional and evolution equations; that is, with monotone and accretive operators. Our approach to this subject will be means of duality mappings and to this purpose we develop some convex analysis and a lot of geometry of Banach spaces.”
The book contains the following chapters.
Chapter 1. Subdifferentiality and duality mappings (Generalities on convex functions; the subdifferential and the conjugate of a convex function; smooth Banach spaces; duality mappings in Banach spaces; positive duality mappings. Exercises. Bibliographic comments).
Chapter 2. Characterizations of some classes of Banach spaces by duality mappings (strictly convex Banach spaces; uniformly convex Banach spaces; duality mappings in reflexive Banach spaces; duality mappings in \(L^ p\) spaces; duality mappings in Banach spaces with the properties (h) and \((\pi)_ 1\). Exercises. Bibliographical comments).
Chapter 3. Renorming of Banach spaces (Classical renorming results; Lindenstrauss and Troyanski theorems. Exercises. Bibliographical comments).
Chapter 4. On the topological degree in finite and infinite dimensions (Brouwer’s degree; Browder-Petryshin’s degree for A-proper mappings; P- compact mappings. Exercises. Bibliographical comments).
Chapter 5. Nonlinear monotone mappings (Demicontinuity and hemicontinuity for monotone operators; monotone and maximal monotone mappings; the role of duality mappings in surjectivity and maximality problems; again on subdifferentials of convex functions. Exercises. Bibliographical comments).
Chapter 6. Accretive mappings and semigroups of nonlinear contractions (general properties of maximal accretive mappings; semigroups of nonlinear contractions in uniformly convex Banach spaces; the exponential formula of Crandall-Liggett; the abstract Cauchy problem for accretive mappings; semigroups of nonlinear contractions in Hilbert spaces; the inhomogeneous case. Exercises. Bibliographical comments).
Referee’s remarks:
1. By Šmulian’s theorem [J. Diestel, Geometry of Banach spaces (1975; Zbl 0307.46009), Ch. 1, § 3, Corollary 1] the local uniform smoothness of \(X^*\) implies the reflexivity of X - compare with Prop. 3.14.
2. It follows from the Šmulian’s theorem that implication [X is weakly uniformly convex \(\Rightarrow\) \(X^*\) is locally uniformly smooth] in Theorem 2.13 is false.
Reviewer: M.I.Kadets

MSC:

47H05 Monotone operators and generalizations
47J05 Equations involving nonlinear operators (general)
46B20 Geometry and structure of normed linear spaces
47-02 Research exposition (monographs, survey articles) pertaining to operator theory
47H06 Nonlinear accretive operators, dissipative operators, etc.
47H20 Semigroups of nonlinear operators
46B03 Isomorphic theory (including renorming) of Banach spaces
46-02 Research exposition (monographs, survey articles) pertaining to functional analysis

Citations:

Zbl 0307.46009
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