## 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.