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Elements of probabilistic analysis with applications. Transl. from the Romanian by Victor Giurgiuţiu. (English) Zbl 0694.60002
Mathematics and Its Applications. East European Series, 36. Dordrecht etc.: Kluwer Academic Publishers; Bucureşti: Editura Academiei. xi, 476 p. Dfl. 320.00; $ 149.00; £99.00 (1989).
This book is mainly concerned with fixed point theorems in probabilistic metric spaces (PM-spaces), random fixed point theorems, and their applications. PM-spaces were introduced by K. Menger [Proc. Nat. Acad. Sci. USA 28, 535-537 (1942)]. The first forty years in the development of this theory were summarized by the monograph of B. Schweizer and A. Sklar [Probabilistic metric spaces. (1983; Zbl 0546.60010)]. In the fifties the Prague school of probabilists around A. Špaček and O. Hanš initiated the study of random operators and their applications. The book of A. T. Bharucha-Reid, Random integral equations. (1972; Zbl 0327.60040), and his survey of random fixed point theorems [Bull. Amer. Math. Soc. 82, 641-657 (1976; Zbl 0339.60061)] revived the interest in this theory and stimulated its further development.
The book under review is devoted to these two parts of stochastic analysis. It consists of seven chapters: 1. Probabilistic structures and related topics. 2. Measurable multivalued mappings and randomness. 3. Fixed point theory in probabilistic structures. 4. Random fixed point theorems. 5. Applications of the fixed point theory to random operator equations. 6. Some analysis problems and methods for random equations. 7. Some applications of random operators. The book is endowed with 437 references.
Chapter 1 provides basic definitions and results on PM-spaces and related structures. Special attention is paid to probabilistic measures of noncompactness. In the second chapter the authors survey some results on random operators and measurable multifunctions. Chapter 3 is concerned with fixed point theorems in PM-spaces, random normed spaces, and probabilistic locally convex spaces. The authors give a comprehensive survey of results obtained up to the middle eighties.
Chapter 4 is devoted to random fixed point theorems for single-valued and multi-valued random operators. In particular, it discusses the application of probabilistic structures in proving of such theorems, and connections between these two types of fixed point theorems. Special attention is paid to random operators with stochastic domains. In chapter 5, some applications of random fixed point theorems to different types of random differential and integral equations are presented. This chapter also covers the results of the Czech school of stochastic approximation. Chapter 6 focusses attention on approximate methods for solving random equations. It is concluded with the discussion of nonlinear random equations with monotone operators.
Chapter 7 is a survey of different applications of random equations. The authors present random models for wave propagation, turbulent fluid motion, free vibrations of elastic strings and bars, chemotherapy, population growth, and telephone traffic. The final section is devoted to stochastic optimization.
The book is a comprehensive survey on research literature, including the results obtained by the authors and their coworkers from the University of Timişoara. It gives a good view of the “state of the art” in the theory, and a wide sample of applications. The book will be useful for researchers and students interested in stochastic analysis and its applications.
Reviewer’s remarks: The authors included some false “theorems” (Th. 2.2.10, Prop. 2.2.5, Th. 4.1.6, Th. 4.2.12) and uncorrect proofs (Corollary to Th. 4.2.12, Remark on p. 243, Prop. 4.4.2). The text contains numerous misprints.
Reviewer: A.Nowak

60-02 Research exposition (monographs, survey articles) pertaining to probability theory
60H25 Random operators and equations (aspects of stochastic analysis)
47H10 Fixed-point theorems
54H25 Fixed-point and coincidence theorems (topological aspects)