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Zbl 1201.34008
Deng, Jiqin; Ma, Lifeng
Existence and uniqueness of solutions of initial value problems for nonlinear fractional differential equations.
(English)
[J] Appl. Math. Lett. 23, No. 6, 676-680 (2010). ISSN 0893-9659

Considered are existence and uniqueness of solutions of the following initial value problems $$D^\alpha x(t)=f(t,D^\beta x(t)),\, 0<t\leq 1;\quad x^{(k)}=n_k,\,k=0,1,\dots,m-1,$$ where $m-1<\alpha<m,$ $n-1<\beta<n$ $(m,n\in \mathbb{N},\,m-1>n)$, $D^\alpha$ stands for the Caputo derivative of order $\alpha$ and $f$ is a continuous function defined on $[0,1]\times \mathbb{R}$. The proofs are achieved by means of the contraction mapping principle.
[Gisèle M. Mophou (Pointe-à-Pitre)]
MSC 2000:
*34A08
34A12 Initial value problems for ODE

Keywords: Caputo derivative; fractional integral; Banach's contraction principle

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