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On the Cauchy problem for a class of differential inclusions with applications. (English) Zbl 1453.34026

The paper studies differential inclusions of the form \[ x'\in F(t,x), \] where \(F:\mathbb{R}\times \mathbb{R}^n\to \mathcal{P}(\mathbb{R}^n)\) is a set-valued map with non-empty compact values.
Under the hypothesis that there exists a set \(U\subset \mathbb{R}\times \mathbb{R}^n\) with certain geometric properties such that the restriction of \(F\) to \(\mathbb{R}\times \mathbb{R}^n\backslash U\) is bounded lower semicontinuous with non-empty closed values it is proved the existence of an upper semicontinuous set-valued map \(G:\mathbb{R}\times \mathbb{R}^n\to \mathcal{P}(\mathbb{R}^n)\) with non-empty compact values such that every solution of the differential inclusion \(x'\in G(t,x)\) is also solution of the differential inclusion \(x'\in F(t,x)\).
Several applications of this result are also provided.

MSC:

34A60 Ordinary differential inclusions
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