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Existence and uniqueness of solutions to impulsive fractional differential equations. (English) Zbl 1178.34004

From the introduction: This article studies the existence and uniqueness of solutions for the initial value problems for fractional order differential equations
\[ ^cD^\alpha y(t)=f(t,y),\quad t\in J=[0,T],\;t\neq t_k, \]
\[ \Delta y|_{t=t_k}=I_k(y(t^-_k)), \]
\[ y(0)=y_0, \]
where \(k=1,\dots,m\), \(0<\alpha\leq 1\), \(^cD^\alpha\) is the Caputo fractional derivative, \(f:J\times \mathbb R\to \mathbb R\) is a given function \(I_k:\mathbb R\to\mathbb R\), and \(y_0\in\mathbb R\), \(0=t_0<t_1<\cdots<t_m<t_{m+1}=T\), \(\Delta y|_{t=t_k}=y(t^+_k)-y(t^-_k)\), \(y(t^+_k)=\lim_{h\to 0^+}y(t_k+h)\) and \(y(t^-_k)\lim_{h\to 0^-}y(t_k+h)\) represent the right and left limits of \(y(t)\) at \(t=t_k\).

MSC:

34A08 Fractional ordinary differential equations
34A37 Ordinary differential equations with impulses
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