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Filippov’s theorem for impulsive differential inclusions with fractional order. (English) Zbl 1214.34006
This paper deals with an impulsive version of Filippov’s theorem for the following fractional differential inclusions: $$\aligned D_{\ast }^{\alpha }y(t) & \in F(t,y(t)) \text{ a.e. }t\in \lbrack 0,b]\setminus \{t_{1},\dots,t_{m}\},\text{ }1<\alpha \leq 2, \\ \Delta y|_{t=t_{k}} &=I_{k}(y(t_{k})),\quad k=1,\dots,m,\\ \Delta y'|_{t=t_{k}}& =\overline{I} _{k}(y'(t_{k})),\quad k =1,\dots,m,\\ y(0)& =a,\quad y'(0)=c, \endaligned$$ where $D_{\ast }^{\alpha }$ is the Caputo fractional derivative, $ F:[0,b]\times \mathbb{R}\rightarrow \mathcal{P}(\mathbb{R})$ is a Carathéodory-Lipschitz multivalued map with compact values. The author also give an impulsive version of Filippov’s Theorem on the half-line.

34A08Fractional differential equations
34A60Differential inclusions
34A37Differential equations with impulses
Full Text: EMIS EuDML