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A note on the Cauchy problem for differential inclusions. (English) Zbl 0783.34009
Summary: We shall be concerned with the existence of global solutions to the initial value problem for systems of differential inclusions of the type y (k) (t)F(t,y(t),,y (k-1) (t)) a.e. t[0,T], y (i) (t i )=r i , i=0,,k-1, where F:[0,T]× kn n is a multifunction with nonempty compact values satisfying some conditions of measurability, and upper or lower semi-continuity; t i [0,T] and r i n , i=0,,k-1. For k=1, the above Cauchy problem was treated in our notes [C. R. Acad. Sci., Paris, Sér. I 306, No. 18, 747-750 (1988; Zbl 0643.34015); ibid. 310, No. 12, 819-822 (1990; Zbl 0731.47048)]. However, for k>1, the established results are new even in the case where F is a single valued Carathéodory or a continuous function.
MSC:
34A60Differential inclusions
34A12Initial value problems for ODE, existence, uniqueness, etc. of solutions