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Mophou, Gisèle M.

Existence and uniqueness of mild solutions to impulsive fractional differential equations. (English) Zbl 1187.34108

Nonlinear Anal., Theory Methods Appl., Ser. A, Theory Methods 72, No. 3-4, 1604-1615 (2010).
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 \(t=t_k\).

Cited in 1 Review
Cited in 107 Documents

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

Keywords:

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

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