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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 $y\sp{(k)}(t)\in F(t,y(t),\dots, y\sp{(k-1)}(t))$ a.e. $t\in [0,T]$, $y\sp{(i)}(t\sb i)=r\sb i$, $i=0,\dots,k-1$, where $F:[0,T]\times\bbfR\sp{kn} \to \bbfR\sp n$ is a multifunction with nonempty compact values satisfying some conditions of measurability, and upper or lower semi-continuity; $t\sb i\in[0,T]$ and $r\sb i\in\bbfR\sp n$, $i=0,\dots,k-1$. 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.

34A60Differential inclusions
34A12Initial value problems for ODE, existence, uniqueness, etc. of solutions