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Philos-type oscillation criteria for impulsive fractional differential equations

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Abstract

In this paper, we study the oscillation of the impulsive Riemann–Liouville fractional differential equation

$$\begin{aligned} {\left\{ \begin{array}{ll} [r(t)D_{ t_k^+}^\alpha x(t)]'+q(t)f\left( d+\int _{t_k^+}^t(t-s)^{-\alpha }x(s)ds\right) =0,\quad t\in (t_k,t_{k+1}],\ k=0,1,2\ldots ,\\ \frac{1}{d}{D_{t_{k}^+}^\alpha x(t_k^+)}-\frac{D_{t_{k-1}^+}^\alpha x(t_k^-)}{d+\int _{t_{k-1}^+}^{t_{k}^-}(t_{k}^--s)^{-\alpha }x(s)ds}=-b_k,\ \ k=1,2,\ldots \ \end{array}\right. } \end{aligned}$$

Philos-type oscillation criteria of the equation are obtained. We are interested in finding adequate impulsive controls to make the fractional system with Riemann–Liouville derivatives oscillate. An example of the change from non-oscillation to oscillation under the impulsive conditions is found.

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Acknowledgements

The authors sincerely thank the reviewers for their valuable suggestions and useful comments that have led to the present improved version of the original manuscript.

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Correspondence to Zhenlai Han.

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This research is supported by the Natural Science Foundation of China (61703180, 61803176), and supported by Shandong Provincial Natural Science Foundation (ZR2017MA043).

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Feng, L., Sun, Y. & Han, Z. Philos-type oscillation criteria for impulsive fractional differential equations. J. Appl. Math. Comput. 62, 361–376 (2020). https://doi.org/10.1007/s12190-019-01288-5

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  • DOI: https://doi.org/10.1007/s12190-019-01288-5

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