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Existence of solutions of four-point boundary value problems for fractional differential equations at resonance

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Abstract

In this paper, we consider the following four-point boundary value problem (BVP) for fractional differential equation

$$\begin{aligned} \left\{ \begin{array}{ll} D^{\alpha }_{0^{+}}u(t)=f(t,u(t),u^{\prime }(t)), &{} 0 \le t \le 1,\\ u^{\prime }(0)-\beta u(\xi )=0, &{} u^{\prime }(1)+\gamma u(\eta ) = 0,\\ \end{array} \right. \end{aligned}$$

where \(D^{\alpha }_{0^{+}}\) denotes the Caputo fractional derivative, \(1<\alpha \le 2\). By using the coincidence degree theory, a new result on the existence of solutions for above fractional BVP is obtained. These results extend the corresponding ones of ordinary differential equations of integer order. Finally, an example is inserted to illustrate the validity and practicability of our main results.

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Acknowledgments

This research was supported by the NSF of China (61402335), the Youth NSF of Jiangxi Province (20114BAB211015), the Youth NSF of the Education Department of Jiangxi Province (GJJ11180). The author would like to thank the referee for his or her careful reading and some comments on improving the presentation of this paper.

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  1. College of Mathematics and Physics, Jinggangshan University, Ji’an, 343009, Jiangxi, People’s Republic of China

    Xiaosong Tang

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  1. Xiaosong Tang
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Tang, X. Existence of solutions of four-point boundary value problems for fractional differential equations at resonance. J. Appl. Math. Comput. 51, 145–160 (2016). https://doi.org/10.1007/s12190-015-0896-4

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  • DOI: https://doi.org/10.1007/s12190-015-0896-4

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