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A stability result for differential inclusions in Banach spaces. (English) Zbl 0594.34016
In this paper one existence theorem for the differential inclusion (1) x ˙F(t,x), x(t 0 )=x 0 in a separable Banach space is proved. The multifunction F has nonempty, compact, convex values and satisfies the Caratheodory-type conditions. The proof uses the Ky Fan fixed point theorem and some properties of integral of multifunctions. Next the continuous dependence of solutions to (1) on the right-hand side is studied where the convergence of F n to F is unerstand in Kuratowski-Mosco sense. At the end one theorem on convergence of the sets of fixed points of some sequence of multifunctions, say {F n }, (with Lipschitz constants smaller than 1) to the set of fixed points of the limit of F n in a Banach space with Frechet-differentiable norm is proved. A theorem of the same type is proved for the set of integrable selectors of a sequence of multifunctions.
Reviewer: Z.Wyderka
MSC:
34A60Differential inclusions
28B20Set-valued set functions and measures; integration of set-valued functions; measurable selections
54C65Continuous selections
49R50Variational methods for eigenvalues of operators (MSC2000)
93B05Controllability