##
**Handbook of multivalued analysis. Volume I: Theory.**
*(English)*
Zbl 0887.47001

Mathematics and its Applications (Dordrecht). 419. Dordrecht: Kluwer Academic Publishers. xv, 964 p. (1997).

The book is designed as an advanced treatise and a Reference Book which presents a systematical exposition of a new branch in modern analysis. The authors have tried to gather all the basic ideas and methods concerning multivalued (set-valued) functions and to present numerious results on their analysis along with the most important applications in analysis, dynamical systems, optimization theory, mathematical economics and so on.

Volume 1 consists of 8 chapters, devoted to different parts of the theory of multivalued functions, an appendix with basic definitions and results on topology, measure theory, and functional analysis, references, symbols, and index. Each chapter begins with a small introductory section which explains the contents of the chapter and the relations between the chapter and the other parts of the book, further, the basic sections are given; the last section (“Remarks”) is a survey of the literature.

Chapter 1 “Continuity of Multifunctions” deals with the topological theory. Here one can find the detail description of different (Hausdorff, Vietoris, Wijsman, Fell, Mosco, Attouch-Wets) topologies in hyperspaces (spaces of subspaces). Further, the authors expose different continuity concepts: Vietoris, Hausdorff, and one based on Property \(Q\). The basic part of the chapter presents the basic results concerning the continuity property: extension theorems, continuous selections, approximations and parametrization of multifunctions and so on.

Chapter 2 “Measurable Multifunctions” deals with measure-theoretic aspects of multifunctions. Here it presents the theory of measurable multifunctions and their measurable selections, results on extreme points of multifunctions, integration theory for multifunction theory of Carathéodory multifunctions and integral functionals generated by them.

Chapter 3 “Montone and Accretive Operators” presents numerious results on multivalued monotone and maximal monotone operators, surjectivity and perturbation results related to them, subdifferential operators, monotone-like and accretive operators, nonlinear semigroups; some interesting examples and applications are also presented in this chapter.

Chapter 4 “Degree Theory for Multifunctions” is devoted to the systematic exposition of the degree and homotopic theory for different classes of multivalued vector fields: completely continuous, condensing, with basic monotone part and other monotone-like, semicondensing and so on. At the end of the chapter the coincidence degree and fixed point index is considered.

Chapter 5 “Fixed Points” is a nice survey on fixed point results for multivalued operators. Here one can find metric fixed point theorems, relations between topological fixed points and minimax problems, asymptotic fixed point theorems, results on the structure and stability of fixed point sets, and random fixed point theorems.

Chapter 6 “Concave Multifunctions and Tangent Cones” is devoted to special classes of multifunctions and problems related to their differential (“smooth”) properties. The first part of the chapter presents with sufficient completeness the theory of concave, superlinear, and sublinear functions (in particular, the spectral properties of such operators are discussed in detail); the rest part of the chapter deals with the tangent and normal cones of multifunctions, different approaches to the problem of differentiability of multifunctions and some similar aspects.

Chapter 7 “Convergence of Multifunctions” gives a survey on different modes of multifunction convergence (uniform, almost uniform, pointwise, almost everywhere, in measure, epigraphical, Mosco convergence and so on). It contains different results on compactness and completeness properties of different spaces (classes) of multifunctions, theorems on the pass under integral sign, important and interesting examples and so on.

Chapter 8 “Set-Valued Random Processes and Multifunctions” is devoted to some stochastic aspects of multifunction theory. Multivalued semimartingales, laws of large numbers, multimeasures, set-valued Radon-Nikodým derivatives and similar problems are discussed in this chapter. It is not possible to call here all mathematicians whose reasoning and results are used and described in this volume; maybe, it is sufficient to remark that the list of references contains more than a thousand of items.

One can see that the book under consideration contains an ocean of new material, which earlier one could find only in numerous research papers. All concepts and results of multivalued analysis gathered in this fundamental book make it almost an encyclopedia in field. Of course, it is easy to find some “bad places” and flaws in this book; for example, despite the big bibliography one can discover annoying omissions; some proofs can be simplified or improved, and so on. However defects of such a type cannot play any role for this book. The reviewer thinks that the acquaintance with the book by Shouchuan Hu and Nikolas S. Papageorgiu will be useful for everyone who deals with multifunctions, as researcher so also as a lecturer; it is clear that this book can be used as a reference book in the field; however, any its chapter can be used as a textbook for the corresponding one or two semester course.

Volume 1 consists of 8 chapters, devoted to different parts of the theory of multivalued functions, an appendix with basic definitions and results on topology, measure theory, and functional analysis, references, symbols, and index. Each chapter begins with a small introductory section which explains the contents of the chapter and the relations between the chapter and the other parts of the book, further, the basic sections are given; the last section (“Remarks”) is a survey of the literature.

Chapter 1 “Continuity of Multifunctions” deals with the topological theory. Here one can find the detail description of different (Hausdorff, Vietoris, Wijsman, Fell, Mosco, Attouch-Wets) topologies in hyperspaces (spaces of subspaces). Further, the authors expose different continuity concepts: Vietoris, Hausdorff, and one based on Property \(Q\). The basic part of the chapter presents the basic results concerning the continuity property: extension theorems, continuous selections, approximations and parametrization of multifunctions and so on.

Chapter 2 “Measurable Multifunctions” deals with measure-theoretic aspects of multifunctions. Here it presents the theory of measurable multifunctions and their measurable selections, results on extreme points of multifunctions, integration theory for multifunction theory of Carathéodory multifunctions and integral functionals generated by them.

Chapter 3 “Montone and Accretive Operators” presents numerious results on multivalued monotone and maximal monotone operators, surjectivity and perturbation results related to them, subdifferential operators, monotone-like and accretive operators, nonlinear semigroups; some interesting examples and applications are also presented in this chapter.

Chapter 4 “Degree Theory for Multifunctions” is devoted to the systematic exposition of the degree and homotopic theory for different classes of multivalued vector fields: completely continuous, condensing, with basic monotone part and other monotone-like, semicondensing and so on. At the end of the chapter the coincidence degree and fixed point index is considered.

Chapter 5 “Fixed Points” is a nice survey on fixed point results for multivalued operators. Here one can find metric fixed point theorems, relations between topological fixed points and minimax problems, asymptotic fixed point theorems, results on the structure and stability of fixed point sets, and random fixed point theorems.

Chapter 6 “Concave Multifunctions and Tangent Cones” is devoted to special classes of multifunctions and problems related to their differential (“smooth”) properties. The first part of the chapter presents with sufficient completeness the theory of concave, superlinear, and sublinear functions (in particular, the spectral properties of such operators are discussed in detail); the rest part of the chapter deals with the tangent and normal cones of multifunctions, different approaches to the problem of differentiability of multifunctions and some similar aspects.

Chapter 7 “Convergence of Multifunctions” gives a survey on different modes of multifunction convergence (uniform, almost uniform, pointwise, almost everywhere, in measure, epigraphical, Mosco convergence and so on). It contains different results on compactness and completeness properties of different spaces (classes) of multifunctions, theorems on the pass under integral sign, important and interesting examples and so on.

Chapter 8 “Set-Valued Random Processes and Multifunctions” is devoted to some stochastic aspects of multifunction theory. Multivalued semimartingales, laws of large numbers, multimeasures, set-valued Radon-Nikodým derivatives and similar problems are discussed in this chapter. It is not possible to call here all mathematicians whose reasoning and results are used and described in this volume; maybe, it is sufficient to remark that the list of references contains more than a thousand of items.

One can see that the book under consideration contains an ocean of new material, which earlier one could find only in numerous research papers. All concepts and results of multivalued analysis gathered in this fundamental book make it almost an encyclopedia in field. Of course, it is easy to find some “bad places” and flaws in this book; for example, despite the big bibliography one can discover annoying omissions; some proofs can be simplified or improved, and so on. However defects of such a type cannot play any role for this book. The reviewer thinks that the acquaintance with the book by Shouchuan Hu and Nikolas S. Papageorgiu will be useful for everyone who deals with multifunctions, as researcher so also as a lecturer; it is clear that this book can be used as a reference book in the field; however, any its chapter can be used as a textbook for the corresponding one or two semester course.

Reviewer: P.Zabreiko (Minsk)

### MSC:

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

47H04 | Set-valued operators |

00A20 | Dictionaries and other general reference works |

47-00 | General reference works (handbooks, dictionaries, bibliographies, etc.) pertaining to operator theory |

60G46 | Martingales and classical analysis |

47H06 | Nonlinear accretive operators, dissipative operators, etc. |

47H11 | Degree theory for nonlinear operators |

47H40 | Random nonlinear operators |

47H05 | Monotone operators and generalizations |

47H10 | Fixed-point theorems |