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Handbook of multivalued analysis. Volume II: Applications. (English) Zbl 0943.47037

Mathematics and its Applications (Dordrecht). 500. Dordrecht: Kluwer Academic Publishers. xi, 926 p. (2000).
This is the second volume of the advanced treatise and a reference book on multivalued analysis (for volume 1 see Zbl 0887.47001). This volume is devoted to numerous applications of multivalued analysis, after all, to control theory and mathematical economics. The structure of the volume is the same. Volume 2 consists of 8 chapters, an appendix with basic definitions and results on the theory of Sobolev spaces and vector-valued functions, references, symbols, and an index. Each chapter also 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 literature.
Chapter 1 “Evolution inclusions involving monotone coercive operators” deals with inclusions of type \[ -\dot x(t)\in A(t,x(t))+ F(t,x(t))\quad\text{a.e. on }T\tag{1} \] with convex- and nonconvex valued mappings \(A(t,x)\) acting between \(T\times X\) and \(X^*\) and \(F(t,x)\) between \(T\times H\) and \(2^H\), where \(X\), \(H\), \(X^*\) is an evolution triple, i.e. a Hilbert space \(H\), a Banach space \(X\), and its dual \(X^*\), such that \(X\subseteq H\subseteq X^*\) (such triples are called framed Hilbert spaces). Using the Galerkin method and theory of nonlinear operators of monotone type, the authors present basic existence and relaxation results for the Cauchy problem with such inclusions, describe topological properties of the set of the Cauchy problem, give basic results for the periodic problem for (1), and, finally, they discuss problems on continuous dependence of the solution set with respect to different parameters involving in the right-hand side of the inclusion under consideration. In the end of the chapter the authors consider some concrete parabolic differential inclusions with different multivalued terms.
Chapter 2 “Evolution inclusions of the subdifferential type” repeats Chapter 1, however, for differential inclusions of type \[ -\dot x(t)\in\partial\varphi(t, x(t))+F(t,x(t))\quad\text{a.e. on }T \] and more general inclusions whose right-hand sides consist of two subgradients; differential inclusions of type \[ \partial\varphi(\dot x(t))+ \partial\psi(x(t))\ni f(t) \] are also considered.
Chapter 3 “Special topics in differential and evolution inclusions” is the last one devoted to differential inclusions; it deals with some topics which were not covered in Chapters 1-2 and could not be found in existing books. Here one can find some results about solvability of initial and boundary value problems for differential inclusions with convex and nonconvex multivalued nonlinearities in \(\mathbb{R}^N\), the theory of evolution inclusions involving \(m\)-accretive operators and Volterra integral inclusions, theorems about reachable sets and variational inclusions, and elements of the theory of differential inclusions with discontinuities.
Chapter 4 “Optimal control” is devoted to the classical problem \[ J(x,u)= \int^b_0 L(t,x(t), u(t)) dt\to\inf= m, \]
\[ \dot x(t)+ A(t, x(t))= f(t,x(t), u(t))\text{ a.e. on }T,\;x(0)= x_0\in H,\;u(t)\in U(t,u(t))\text{ a.e. on }T \] in abstract form and as well in important concrete cases, for the most part, in finite-dimensional cases. The authors present basic existence results, discuss in detail problems concerning the relaxation and variational stability property of control systems, and some closed problems. This part of the book seems to be the most original, first of all, in the part related to the relaxation and sensibility analysis.
Chapter 5 “Calculus of variations” is a nice and curious account of this classical field with systematical using of delicate results in functional analysis and theory of multivalued mappings. Here is the contents of this chapter: lower-semicontinuity results, existence theorems, relaxation results, well-posedness result, duality theory, homogenization. One can remark that this chapter effectively demonstrates substantial changes in this classical branch of mathematics associated with multivalued analysis; in turn, one must understand that the explosive development of multivalued analysis owes to problems in classical calculus of variations.
Chapter 6 “Mathematical economics” can be considered as a small book in this branch written on the base of multivalued analysis. And although, the contents of this chapter is classical and routine (existence theory for growth models, price characterization of optimal programs, models with recursive utility, discrete-time stochastic growth models, continuous-time stochastic dynamics, portfolio theory and financial economics, equilibria in exchange economics), the account is original and interesting. There is an essential difference between this chapter and the previous one; if one can find a “smooth” variant of calculus of variations, the most part of mathematical economics cannot be presented in the framework of usual, single-valued, analysis.
Chapter 7 “Stochastic games” gives a useful account of this theory. The central themes of this chapter are the “dynamic programming methodology” going back to R. Bellman and the Markovian nature of most systems. Formally, the chapter is devoted to sequential decision processes under uncertainty in the framework of games of two persons. The last part of the chapter deals with adaptive control problems and theory of Cournot-Nash equilibria.
Chapter 8 “Special topics in mathematical economics and optimization” is devoted to resource allocation problems, convergence of information, cooperative behaviour of firms, revealed preference and expected utility, nonconvex duality, and Pareto optimization.
The list of References in Volume II also contains more than a thousand of items. And it must be repeated all concluding remarks to Volume I: this book 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 the field. The acquaintance with the book by Shouchuan Hu and Nikoas S. Papageorgiu will be useful for everyone who deals with multifunctions, researchers and lecturers; the book can also be used as a reference book in the field; any its chapter can be used as a textbook for the corresponding one or two semester course.
Reviewer: P.Zabreiko (Minsk)

MSC:

47H04 Set-valued operators
47-02 Research exposition (monographs, survey articles) pertaining to operator theory
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
47J30 Variational methods involving nonlinear operators
47J20 Variational and other types of inequalities involving nonlinear operators (general)
91A15 Stochastic games, stochastic differential games
91B62 Economic growth models

Citations:

Zbl 0887.47001
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