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Zbl 1187.34108
Mophou, GisÃ¨le M.
Existence and uniqueness of mild solutions to impulsive fractional differential equations.
(English)
[J] Nonlinear Anal., Theory Methods Appl., Ser. A, Theory Methods 72, No. 3-4, A, 1604-1615 (2010). ISSN 0362-546X

Summary: Our aim in this paper is to study the existence and the uniqueness of the solution for the fractional semilinear differential equation: $$\cases D^\alpha_t x(t)=Ax(t)+f(t,x(t)),\quad t\in I=[0,T],\ t\ne t_k,\\ x(0)=x_0\in X,\\ \Delta x|_{t=t_k}=l_k(x(t^-_k)),\quad k=1,\dots,m,\endcases\tag1$$ where $0<\alpha<1$, the operator $A:D(A)\subset X\to X$ is a generator of ${\cal C}_0$-semigroup $(T(t))_{t\ge 0}$ on a Banach space $\Bbb X$, $D^\alpha_t$ is the Caputo fractional derivative, $f:I\times \Bbb X\to \Bbb X$ is a given continuous function $I_k:\Bbb X\to \Bbb X$, $0=t_0<t_1<\cdots<t_m<t_{m+1}=T$. $\Delta x|_{t=t_k}=x(t^+_k)-x(t^-_k)$, $x(t^+_k)=\lim_{h\to0^+}x(t_k+h)$ 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 2000:
*34K30 Functional-differential equations in abstract spaces
34K05 General theory of functional-differential equations
34K37
34K45 Equations with impulses

Keywords: Cauchy problem; fractional abstract differential equation; impulsive equations; mild solutions; nonlocal conditions

Cited in: Zbl 1242.34011

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