Existence and uniqueness of solutions of initial value problems for nonlinear fractional differential equations. (English) Zbl 1201.34008

Deng, Jiqin; Ma, Lifeng

Appl. Math. Lett. 23, No. 6, 676-680 (2010).

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.

Reviewer: Gisèle M. Mophou (Pointe-à-Pitre)

MSC:

34A08	Fractional ordinary differential equations
34A12	Initial value problems, existence, uniqueness, continuous dependence and continuation of solutions to ordinary differential equations

Keywords:

Caputo derivative; fractional integral; Banach’s contraction principle

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