Abstract
In this paper, we study the oscillation of the impulsive Riemann–Liouville fractional differential equation
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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Armando, H.A., Romo, M.P., Roberto, M.T.: Response spectra generation using a fractional differential model. Soil Dyn. Earthq. Eng. 115, 719–729 (2018)
Jiang, Y., Xia, B., Zhao, X., Nguyen, T., Mi, C., Callafon, R.A.: Data-based fractional differential models for non-linear dynamic modeling of a lithiumion battery. Energy 135, 171–181 (2017)
Ortega, A., Rosales, J.J., Cruz-Duarte, J.M., Guia, M.: Fractional model of the dielectric dispersion. Opt. Int. J. Light Electron Opt. 180, 754–759 (2019)
Jiang, C., Zhang, F., Li, T.: Synchronization and antisynchronization of N-coupled fractional—order complex chaotic systems with ring connection. Math. Methods Appl. Sci. 41, 2625–2638 (2018)
Guo, T.: Controllability and observability of impulsive fractional linear time-invariant system. Comput. Math. Appl. 64(10), 3171–3182 (2012)
Wang, J., Li, X., Wei, W.: On the natural solution of an impulsive fractional differential equation of order \(q\in (1,2)\). Commun. Nonlinear Sci. Numer. Simul. 17, 4384–4394 (2012)
Stamova, I.: Global stability of impulsive fractional differential equations. Appl. Math. Comput. 237, 605–612 (2014)
Xu, L., Li, J., Ge, S.: Impulsive stabilization of fractional differential systems. ISA Trans. 70, 125–131 (2017)
Agarwal, R.P., Bohner, M., Li, T., Zhang, C.: A Philos-type theorem for third-order nonlinear retarded dynamic equations. Appl. Math. Comput. 249, 527–531 (2014)
Agarwal, R.P., Zhang, C., Li, T.: New Kamenev-type oscillation criteria for second-order nonlinear advanced dynamic equations. Appl. Math. Comput. 225, 822–828 (2013)
Bohner, M., Hassan, T.S., Li, T.: Fite-Hille-Wintner-type oscillation criteria for second-order half-linear dynamic equations with deviating arguments. Indag. Math. 29, 548–560 (2018)
Bohner, M., Li, T.: Kamenev-type criteria for nonlinear damped dynamic equations. Sci. China Math. 58, 1445–1452 (2015)
Wang, P., Li, C., Zhang, J., Li, T.: Quasilinearization method for first-order impulsive integro-differential equations. Electron. J. Differ. Equ. 2019, 1–14 (2019)
Graef, J.R., Shen, J.H., Stavroulakis, I.P.: Oscillation of impulsive neutral delay differential equations. J. Math. Anal. Appl. 268(1), 310–333 (2002)
Tariboon, J., Ntouyas, S.K.: Oscillation of impulsive conformable fractional differential equations. Open Math. 14, 497–508 (2016)
Raheem, A., Maqbul, M.: Oscillation criteria for impulsive partial fractional differential equations. Comput. Math. Appl. 73, 1781–1788 (2017)
Benchohra, M., Hamani, S., Zhou, Y.: Oscillation and nonoscillation for Caputo–Hadamard impulsive fractional differential inclusions. Adv. Differ. Equ. 74, 1–15 (2019)
Ma, Q., Liu, A.: Oscillation criteria of nonlinear fractional differential equations with damping term. Math. Appl. 29(2), 291–297 (2016)
Sugie, J., Ishihara, K.: Philos-type oscillation criteria for linear differential equations with impulsive effects. J. Math. Anal. Appl. 470, 911–930 (2019)
Kilbas, A.A., Srivastava, H.M., Trujillo, J.J.: Theory and Applications of Fractional Differential Equations. Elsevier, Amsterdam (2006)
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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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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