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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 ${y}^{\left(k\right)}\left(t\right)\in F\left(t,y\left(t\right),\cdots ,{y}^{\left(k-1\right)}\left(t\right)\right)$ a.e. $t\in \left[0,T\right]$, ${y}^{\left(i\right)}\left({t}_{i}\right)={r}_{i}$, $i=0,\cdots ,k-1$, where $F:\left[0,T\right]×{ℝ}^{kn}\to {ℝ}^{n}$ is a multifunction with nonempty compact values satisfying some conditions of measurability, and upper or lower semi-continuity; ${t}_{i}\in \left[0,T\right]$ and ${r}_{i}\in {ℝ}^{n}$, $i=0,\cdots ,k-1$. 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:
 34A60 Differential inclusions 34A12 Initial value problems for ODE, existence, uniqueness, etc. of solutions