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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) $\stackrel{˙}{x}\in F\left(t,x\right)$, $x\left({t}_{0}\right)={x}_{0}$ 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 $\left\{{F}_{n}\right\}$, (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:
 34A60 Differential inclusions 28B20 Set-valued set functions and measures; integration of set-valued functions; measurable selections 54C65 Continuous selections 49R50 Variational methods for eigenvalues of operators (MSC2000) 93B05 Controllability