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**Convex analysis & approximation of fixed points.
(Totsu kaiseki to fudōten kinji.)**
*(Japanese)*
Zbl 1089.49500

Sūrikaiseki Shirizu 2. Yokohama: Yokohama Publishers (ISBN 4-946552-03-0). iv, 280 pp. (2000).

This book is organized into 27 sections. Each section presents a topic in convex analysis and approximation of fixed points. These sections are distributed among six chapters. Chapter 1 is an introduction to functional analysis. Here metric spaces and topological spaces, Banach spaces and Hilbert spaces, and topological linear spaces are introduced as preparation for other material in this book. Chapter 2 describes convex analysis on the real line in which basic properties of lower semicontinuous and convex functions are discussed. In this chapter convexity in Banach spaces, minimax theorems and a generalized Hahn-Banach theorem are discussed. In Chapter 3 convexity and differentiability of the norm as well as the duality mapping in Banach spaces are discussed. These basic properties are employed to discuss multimappings from a Banach space to another Banach space. Also, accretive operators and monotone operators are considered. Chapter 4 studies subdifferentials of convex functions. The relations between the conjugate function and subdifferentiability are discussed. The Rockafellar theorem is stated: the subdifferential of a proper lower semicontinuous convex function is a maximal monotone mapping. In Chapter 5, approximation of fixed points of monoexpansive mappings defined on Hilbert spaces and Banach spaces is discussed. The last chapter (6) contains applications of convex analysis and fixed point approximations for nonexpansive mappings. Approximate solutions of control problems and minimization problems are shown. A Kuhn-Tucker type theorem as well as a corresponding duality theory are presented.

This book includes many interesting and important theories in convex analysis and approximation of fixed points. It could be a useful reference for mathematics researchers who are interested in this field. Moreover, it could also be helpful for researchers in physics, engineering, operations research, and mathematical economics.

This book includes many interesting and important theories in convex analysis and approximation of fixed points. It could be a useful reference for mathematics researchers who are interested in this field. Moreover, it could also be helpful for researchers in physics, engineering, operations research, and mathematical economics.

Reviewer: Hang-Chin Lai (MR1875068)

### MSC:

49-02 | Research exposition (monographs, survey articles) pertaining to calculus of variations and optimal control |

41A65 | Abstract approximation theory (approximation in normed linear spaces and other abstract spaces) |

46B20 | Geometry and structure of normed linear spaces |

47H10 | Fixed-point theorems |

49J53 | Set-valued and variational analysis |

90C25 | Convex programming |