## Differential inclusions. Set-valued maps and viability theory.(English)Zbl 0538.34007

Grundlehren der Mathematischen Wissenschaften, 264. Berlin etc.: Springer-Verlag. XIII, 342 p. DM 118.00; \$ 44.10 (1984).
This is a complete introduction to the subject which has been recently largely developed - the differential inclusions $$\dot x\in F(t,x)$$. After the introductory chapter the first chapter is devoted to the multifunctions. Continuity, semicontinuity and measurability of multifunctions are described and many theorems on single-valued selections are given. The second chapter contains all the main versions of theorems on existence of solutions and different methods of proofs are presented - the solutions are understood in the sense of Carathéodory. The fundamental properties of families of solutions (unicity is here something strange) are also described. All of the third chapter is devoted to a very special but important differential inclusion, where F is a maximal monotone operator; existence and uniqueness of solutions are proved and an important special case of gradient inclusions, being a generalization of gradient equations, is treated. Among the fields where differential inclusions appear and are applied the authors have chosen to present in detail the viability theory and its numerous implications for the economy. Roughly speaking, the viability means the possibility of keeping the solutions in a given set. One chapter is devoted to the nonconvex viability sets and another to the convex ones. Applications are given, among them a dynamical analogue of the Walras equilibrium theory. The last chapter contains the applications of Lyapunov functions to differential inclusions.
Reviewer: T.Rzezuchowski

### MSC:

 34A60 Ordinary differential inclusions 34-02 Research exposition (monographs, survey articles) pertaining to ordinary differential equations 34A12 Initial value problems, existence, uniqueness, continuous dependence and continuation of solutions to ordinary differential equations 34D20 Stability of solutions to ordinary differential equations