Filippov's theorem for impulsive differential inclusions with fractional order

A. Ouahab, Université de Sidi Bel Abbés, Sidi Bel Abbés, Algérie

E. J. Qualitative Theory of Diff. Equ., Spec. Ed. I, 2009 No. 23., pp. 1-23.

Abstract: In this paper, we present an impulsive version of Filippov's Theorem for fractional differential inclusions of the form:
$$ \begin{array}{rlll}
D^{\alpha}_*y(t) &\in& F(t,y(t)), &\hbox{ a.e. } \, t\in
J\backslash
\{t_{1},\ldots,t_{m}\},\ \alpha\in(1,2],\\
y(t^+_{k})-y(t^-_k)&=&I_{k}(y(t_{k}^{-})), &k=1,\ldots,m,\\
y'(t^+_{k})-y'(t^-_k)&=&\overline{I}_{k}(y'(t_{k}^{-})), &k=1,\ldots,m,\\
y(0)&=&a,\ y'(0)=c,\
\end{array}$$
where $J=[0,b],$ $D^{\alpha}_*$ denotes the Caputo fractional derivative and $F$ is a set-valued map. The functions $I_k,\overline{I}_k$ characterize the jump of the solutions at impulse points $t_k$ ($k=1,\ldots,m$).

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