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Existence results for nonlinear fractional differential equations in C[0, T)

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Abstract

In this paper, we investigate the existence results for fractional differential equations of the form

$$\begin{aligned} {\left\{ \begin{array}{ll} D_{c}^{q}x(t)=f(t,x(t)) \quad t\in [0, T)\left( 0<T\le \infty \right) , \quad q \in (1,2),\\ x(0)=a_{0},\quad x^{'}(0)=a_{1}, \end{array}\right. } \end{aligned}$$
(0.1)

and

$$\begin{aligned} {\left\{ \begin{array}{ll} D_{c}^{q}x(t)=f(t,x(t)) \quad t\in [0, T), \quad q \in (0,1),\\ x(0)=a_{0}, \end{array}\right. } \end{aligned}$$
(0.2)

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 \([0,+\infty )\).

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Acknowledgements

The research was supported by Scientific Research Foundation of Nanjing Institute of Technology(No: CKJB201508; ZKJ201514).

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Correspondence to Tao Zhu.

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Zhu, T., Zhong, C. & Song, C. Existence results for nonlinear fractional differential equations in C[0, T). J. Appl. Math. Comput. 57, 57–68 (2018). https://doi.org/10.1007/s12190-017-1094-3

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  • DOI: https://doi.org/10.1007/s12190-017-1094-3

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