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Differential inclusions in Banach spaces. (English) Zbl 0609.34013
Let F be a multifunction from an open set $$\Omega\subset {\mathbb{R}}\times E$$ (E a Banach space) to the space $${\mathcal K}_ c(E)$$ of all nonempty bounded closed subsets of E, endowed with the Pompeiu-Hausdorff metric. In this paper, the authors establish the existence of solutions for the Cauchy problem $$\dot x\in F(t,x)$$, $$x(t_ 0)=x_ 0$$ under hypotheses on F that exclude compactness entirely. In fact, they suppose that for each (t,x)$$\in \Omega$$, the closed convex hull of F(t,x) has a nonempty interior. If, in addition, the multifunction $$F: \Omega \to {\mathcal K}_ c(E)$$ is continuous and the Banach space E is reflexive and separable, then they prove that the above Cauchy problem has solutions.
Reviewer: N.L.Maria

##### MSC:
 34A60 Ordinary differential inclusions 34G10 Linear differential equations in abstract spaces
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##### References:
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