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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.

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 |