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Existence results for nonlinear fractional differential equations in \(C[0,T)\). (English) Zbl 1395.34013

Summary: In this paper, we investigate the existence results for fractional differential equations of the form \[ \begin{aligned} D_{c}^{q}x(t)&=f(t,x(t))\quad t\in [0, T)\left( 0<T\leq \infty \right), \quad q \in (1,2),\\ x(0)&=a_{0}, \quad x^{'}(0)=a_{1}, \\ \text{}\\ D_{c}^{q}x(t)&=f(t,x(t)) \quad t\in [0, T), \quad q \in (0,1),\\ x(0)&=a_{0}, \end{aligned}{\text{and}} \] where \(D_{c}^{q}\) is the Caputo fractional derivative. We prove the above equations have solutions in \(C[0,T)\). Particularly, we present the existence and uniqueness results for the above equations on \([0,+\infty)\).

MSC:

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