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

MSC:

34A60 Ordinary differential inclusions
34A12 Initial value problems, existence, uniqueness, continuous dependence and continuation of solutions to ordinary differential equations
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