Filippov's theorem for impulsive differential inclusions with fractional order
In this paper, we present an impulsive version of Filippov's Theorem for fractional differential inclusions of the form: $$ \begin{array}{rlll} D^{\alpha}_*y(t) &\in& F(t,y(t)), &\hbox{ a.e. } \, t\in J\backslash \{t_{1},\ldots,t_{m}\},\ \alpha\in(1,2],\\ y(t^+_{k})-y(t^-_k)&...
| Main Author: | |
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| Format: | Article |
| Language: | English |
| Published: |
University of Szeged
2009-10-01
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| Series: | Electronic Journal of Qualitative Theory of Differential Equations |
| Online Access: | http://www.math.u-szeged.hu/ejqtde/periodica.html?periodica=1¶mtipus_ertek=publication¶m_ertek=425 |
| Summary: | In this paper, we present an impulsive version of Filippov's Theorem for fractional differential inclusions of the form:
$$ \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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| ISSN: | 1417-3875 1417-3875 |