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Existence and uniqueness of mild solutions to impulsive fractional differential equations. (English) Zbl 1187.34108
Summary: Our aim in this paper is to study the existence and the uniqueness of the solution for the fractional semilinear differential equation:
$\begin{cases} D^\alpha_t x(t)=Ax(t)+f(t,x(t)),\quad t\in I=[0,T],\;t\neq t_k,\\ x(0)=x_0\in X,\\ \Delta x|_{t=t_k}=l_k(x(t^-_k)),\quad k=1,\dots,m,\end{cases}\tag{1}$
where $$0<\alpha<1$$, the operator $$A:D(A)\subset X\to X$$ is a generator of $${\mathcal C}_0$$-semigroup $$(T(t))_{t\geq 0}$$ on a Banach space $$\mathbb X$$, $$D^\alpha_t$$ is the Caputo fractional derivative, $$f:I\times \mathbb X\to \mathbb X$$ is a given continuous function $$I_k:\mathbb X\to \mathbb X$$, $$0=t_0<t_1<\cdots<t_m<t_{m+1}=T$$. $$\Delta x|_{t=t_k}=x(t^+_k)-x(t^-_k)$$, $$x(t^+_k)=\lim_{h\to0^+}x(t_k+h)$$ and $$x(t^-_k)=\lim_{h\to 0}-x(t_k+h)$$ represent respectively the right and left limits of $$x(t)$$ at $$t=t_k$$.

##### MSC:
 34K30 Functional-differential equations in abstract spaces 34K05 General theory of functional-differential equations 34K37 Functional-differential equations with fractional derivatives 34K45 Functional-differential equations with impulses
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