Filippov's theorem for impulsive differential inclusions with fractional order

• Main Author: Abdelghani Ouahab
• Format: Article
• Language: English
• Published: University of Szeged 2009-10-01
• Series: Electronic Journal of Qualitative Theory of Differential Equations
• Online Access: http://www.math.u-szeged.hu/ejqtde/periodica.html?periodica=1&paramtipus_ertek=publication&param_ertek=425

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 $t_k$ ($k=1,\ldots,m$).