Set-valued analysis. (English) Zbl 0713.49021

Aubin, Jean-Pierre; Frankowska, Hélène

Systems and Control: Foundations and Applications, 2. Boston etc.: Birkhäuser. xix, 461 p. sFr. 125.00; DM 138.00 (1990).

This book provides a thorough introduction to the multivalued or set- valued analysis, gathering both old and new results in an up-to-date perspective. The book contains ten chapters. Examples in many branches of mathematics, given in the introduction, prevail the reader upon the indispensability to deal with sequences of sets and set-valued maps. The basic tools of classical analysis which proved to be so efficient in the past have now been adapted to the set-valued case and tested in the solution of a wide spectrum of problems. Chapter 1 defines the limits of sets and set-valued continuity while Chapter 2 shows that for the set- valued maps with closed convex cones as their graphs, called closed convex processes, many principles for continuous linear operators, as the open mapping, closed graph and uniform boundedness theorems, remain valid. Going forward to major nonlinear problems, Chapter 3 supplies tools in solving inclusions or studying the stability and approximation of their solutions as extensions of the Banach-Picard successive approximation theorem, to so-called constrained equilibrium theorem and the Brouwer fixed point theorem, Ekeland’s principle, maximal monotone mappings and Yosida approximates. Chapter 4 opens the door of what the authors suggestively call ménagerie of tangents. Refinements of the Clarke tangent cones are obtained. Various definitions of derivative, depending on a choice of a tangent cone, as well as inverse set-valued map theorems are discussed in Chapter 5. The next two chapters deal with epidifferential calculus and graphical convergence, the keystone of the whole book. A unitary view on the concept of subdifferential is realized. Measurability and integrability of set-valued maps are studied in Chapter 8. The last chapters are devoted to selections and parametrization, differential inclusions and viability theory. An extensive bibliography of 468 items concludes the volume. The style is lively and vigorous, the relevant historical comments and suggestive overviews increase the interest for this work. Many results of the book are an outgrowth of the author’s contributions. Graduate students and mathematicians of every persuasion will welcome this unparallel guide to set-valued analysis.

Reviewer: D.Pascali

MSC:

49J52	Nonsmooth analysis
47H04	Set-valued operators
28B20	Set-valued set functions and measures; integration of set-valued functions; measurable selections
47H10	Fixed-point theorems
49-02	Research exposition (monographs, survey articles) pertaining to calculus of variations and optimal control
47Hxx	Nonlinear operators and their properties

Keywords:

set-valued analysis; set-valued continuity; closed convex processes; open mapping; closed graph; uniform boundedness theorems; inclusions; Banach- Picard successive approximation theorem; constrained equilibrium theorem; Brouwer fixed point theorem; Ekeland’s principle; maximal monotone mappings; Yosida approximates; Clarke tangent cones; epidifferential calculus; graphical convergence; subdifferential; selections; parametrization; differential inclusions; viability theory