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Regularization of closed-valued multifunctions in a non-metric setting. (English) Zbl 0816.28009
Let $$F: T\times X\to Y$$ be a Carathéodory type closed-valued multifunction. Scorza-Dragoni type theorems for $$F$$ are useful in the study of solutions to the differential inclusion $$\dot x\in F(t, x)$$. Suppose that $$F$$ is only upper semicontinuous in $$x$$ for almost every fixed $$t$$. In this case J. Jarník and J. Kurzweil [Časopis Pěst. Mat. 102, 334-349 (1977; Zbl 0369.34002)] used another tool, namely the regularization of $$F$$. The multifunction $$F$$ is replaced by a more regular map so that the set of solutions of the differential inclusion remains unchanged. T. Rzeżuchowski [Bull. Acad. Pol. Sci., Ser. Sci. Math. 28, 61-66 (1980; Zbl 0459.28007)] worked along this line of ideas assuming $$T$$, $$X$$, and $$Y$$ to be metric spaces. In the present paper the Rzeżuchowski existence and uniqueness theorems for regularizations are extended and proved in a more general framework.
Reviewer: H.-A.Klei (Paris)

##### MSC:
 28B20 Set-valued set functions and measures; integration of set-valued functions; measurable selections 54C60 Set-valued maps in general topology 26E25 Set-valued functions 34A60 Ordinary differential inclusions
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