##
**Geometric properties of Banach spaces and nonlinear iterations.**
*(English)*
Zbl 1167.47002

Lecture Notes in Mathematics 1965. Berlin: Springer (ISBN 978-1-84882-189-7/pbk; 978-1-84882-190-3/ebook). xvii, 326 p. (2009).

This monograph gives an introduction to and overview of the author’s extensive work on fixed point iterations. It consists of three parts. Part 1 (Chapters 1 to 5) is dedicated to the geometric properties of Banach spaces, namely, convexity, smoothness and the duality map.

In Part 2 (Chapters 6 to 14), the author gives many results about fixed points of different classes of mappings. He focuses the main attention on the iterative processes that converge to a fixed point. The celebrated Banach contraction theorem assures that the Picard iteration formula

\[ x_0 \in K, \quad x_{n+1} = T x_n \quad (n \geq 0) \]

provides a sequence which converges to the unique fixed point, if \(K\) is a complete metric space and \(T: K \to K\) is contractive. If \(T\) is nonexpansive, other iteration processes are considered. Under certain conditions, the Mann iteration formula

\[ x_0 \in K, \quad x_{n+1} = (1 - \alpha_n) x_n + \alpha_n T x_n \quad (n \geq 0), \]

where \((\alpha_n)_{n \geq 0} \subset ]0,1[\), \(\lim_{n \rightarrow \infty} \alpha_n = 0\) and \(\sum_{n \geq 0} \alpha_n = \infty\), provides a sequence convergent to a fixed point. Throughout the book, the author considers different classes of operators: contractive, nonexpansive, quasi-nonexpansive, asymptotically regular, uniformly asymptotically regular, etc.

In Part 3 (Chapters 15 to 22), common fixed points for (finite, countable) families of mappings are studied. The final Chapter 23 presents a lot of the same results on set-valued mappings, plus some general comments, examples and open questions.

Each chapter contains a section of exercises and another of historical remarks. The book ends with 561 references, of which 100 are of the author, and a short index. It contains some minor mistakes: for example, the Goldstine theorem is called Goldstein theorem.

Almost all the theorems of this monograph are due to the author and his collaborators.

In Part 2 (Chapters 6 to 14), the author gives many results about fixed points of different classes of mappings. He focuses the main attention on the iterative processes that converge to a fixed point. The celebrated Banach contraction theorem assures that the Picard iteration formula

\[ x_0 \in K, \quad x_{n+1} = T x_n \quad (n \geq 0) \]

provides a sequence which converges to the unique fixed point, if \(K\) is a complete metric space and \(T: K \to K\) is contractive. If \(T\) is nonexpansive, other iteration processes are considered. Under certain conditions, the Mann iteration formula

\[ x_0 \in K, \quad x_{n+1} = (1 - \alpha_n) x_n + \alpha_n T x_n \quad (n \geq 0), \]

where \((\alpha_n)_{n \geq 0} \subset ]0,1[\), \(\lim_{n \rightarrow \infty} \alpha_n = 0\) and \(\sum_{n \geq 0} \alpha_n = \infty\), provides a sequence convergent to a fixed point. Throughout the book, the author considers different classes of operators: contractive, nonexpansive, quasi-nonexpansive, asymptotically regular, uniformly asymptotically regular, etc.

In Part 3 (Chapters 15 to 22), common fixed points for (finite, countable) families of mappings are studied. The final Chapter 23 presents a lot of the same results on set-valued mappings, plus some general comments, examples and open questions.

Each chapter contains a section of exercises and another of historical remarks. The book ends with 561 references, of which 100 are of the author, and a short index. It contains some minor mistakes: for example, the Goldstine theorem is called Goldstein theorem.

Almost all the theorems of this monograph are due to the author and his collaborators.

Reviewer: Antonio Martinón (La Laguna)

### MSC:

47-02 | Research exposition (monographs, survey articles) pertaining to operator theory |

47J25 | Iterative procedures involving nonlinear operators |

47Hxx | Nonlinear operators and their properties |

46-02 | Research exposition (monographs, survey articles) pertaining to functional analysis |

46B20 | Geometry and structure of normed linear spaces |