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Some theorems of Scorza Dragoni type for multifunctions with application to the problem of existence of solutions for differential multivalued equations. (English) Zbl 0576.49024
Mathematical control theory, Banach Cent. Publ. 14, 625-643 (1985).
[For the entire collection see Zbl 0568.00025.]
This paper presents some measurability and continuity definitions and properties of closed-valued multifunctions defined on separable metrizable or metric spaces and some relationships between them. The main results are of Lusin type and Scorza-Dragoni type theorems under different types of continuity of multifunctions. Next, an existence theorem for differential inclusions $$\dot x\in F(t,x)$$, $$x(t_ o)=x_ o$$ in Euclidean space is proved for F(t,$$\cdot)$$ to be upper continuous, convex-valued or for F(t,$$\cdot)$$ to be lower continuous. Some measurability conditions on F are assumed.
Reviewer: Z.Wyderka

##### MSC:
 93B05 Controllability 28B20 Set-valued set functions and measures; integration of set-valued functions; measurable selections 49J45 Methods involving semicontinuity and convergence; relaxation 34A60 Ordinary differential inclusions 49J15 Existence theories for optimal control problems involving ordinary differential equations 54C60 Set-valued maps in general topology