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Set-valued functions. (Multifuncţii.) (Romanian) Zbl 1112.47041

Oradea: Editura Universităţii din Oradea (ISBN 973-759-122-4/pbk). 139 p. (2006).
The purpose of this book is to present some ideas of set-valued analysis, mainly those ideas necessary to the theory of fixed point for multifunctions. The book contains some classical results, e.g., of C.Berge, [“Espaces topologiques. Fonctions multivoques” (Collection universitaire de mathematiques 3, Dunod, Paris) (1959; Zbl 0088.14703) (\(^2\)1966; Zbl 0164.52902)], as well as recent results from the theory of continuous selections and the theory of fixed points for multifunctions.
In Chapter 1, “Preliminaries”, the author recalls notions and results needed in the following chapters. It contains the following paragraphs: 1.1.Compact and paracompact spaces. 1.2.Regular spaces, normal spaces. 1.3.Quasiconvex (quasiconcave) functions, lower (upper) semicontinuous functions. Chapter 2, “Semicontinuous multifunctions”, consists of the following paragraphs: 2.1.Definitions and some elementary properties. 2.2.Upper semicontinuity. 2.3.Lower semicontinuity. 2.4.Semicontinuous multifunctions. Berge’s theorem. 2.5.Continuous selections. This chapter does not include a study of various forms of semicontinuity for multifunctions, since these are not necessary in the study of fixed points.
The main chapter of this book is Chapter 3, “Fixed point theorems for multifunctions”, that is composed of the following paragraphs: 3.1.KKM principle. KKM multifunctions. 3.2.Kakutani multifunctions. 3.3.Multifunctions with KKM property. 3.4.Fixed point theorem of Fan–Browder. 3.5.Coincidence theorems.
The content of the book is well organized. The terminology is correct and up-to-date. All definitions are sustained by examples. Some important results are proved, other results are only stated, but indicating a reference. Contents and short historical notes on the presented results are also included. The bibliography of the book is well selected and contains classical references, as well as recent papers and books, published over the past 10 years. The book is aimed at graduate students at Masters and Ph.D.level, to researchers in nonlinear analysis, as well as engineers and economists willing to apply set-valued analysis.
In the reviewer’s opinion, a publication of this book in English, French or German would be beneficial.

MSC:

47H04 Set-valued operators
54C60 Set-valued maps in general topology
54H25 Fixed-point and coincidence theorems (topological aspects)
47-01 Introductory exposition (textbooks, tutorial papers, etc.) pertaining to operator theory
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