Oscillatory Criteria of Nonlinear Higher Order $\Psi$-Hilfer Fractional Differential Equations

Year 2021, Volume: 4 Issue: 2, 134 - 142, 01.06.2021

Tuğba Yalçın Uzun

https://doi.org/10.33401/fujma.888390

Cited By: 3

Abstract

In this paper, we study the forced oscillatory theory for higher order fractional differential equations with damping term via $\Psi$-Hilfer fractional derivative. We get sufficient conditions which ensure the oscillation of all solutions and give an illustrative example for our results. The $\Psi$-Hilfer fractional derivative according to the choice of the $\Psi$ function is a generalization of the different fractional derivatives defined earlier. The results obtained in this paper are a generalization of the known results in the literature, and present new results for some fractional derivatives.

Keywords

$\Psi$-Hilfer fractional derivative, Damping term, Forced oscillation, Fractional differential equations

References

• [1] T. Li, N. Pintus, G. Viglialoro, Properties of solutions to porous medium problems with different sources and boundary conditions, Z. Angew. Math. Phys., 70(3) (2019), Art. 86, pp. 1-18.
• [2] T. Li, G. Viglialoro, Boundedness for a nonlocal reaction chemotaxis model even in the attraction-dominated regime, Differ. Integral Equ., 34(5-6) (2021), 315-336.
• [3] M. Javadi, M. A. Noorian, S. Irani, Stability analysis of pipes conveying fluid with fractional viscoelastic model, Meccanica 54 (2019), 399–410. https://doi.org/10.1007/s11012-019-00950-3
• [4] I. S. Jesus, J. A. Tenreiro Machado, Application of Integer and Fractional Models in Electrochemical Systems, Math. Prob. Eng., 2012 (2012), Article ID 248175.
• [5] F. Ali, N. A. Sheikh, I. Khan, M. Saqib, Magnetic field effect on blood flow of Casson fluid in axisymmetric cylindrical tube: A fractional model, J. Magn. Magn. Mater., 423 (2017), 327-336.
• [6] Y. Tang, Y. Zhen, B. Fang, Nonlinear vibration analysis of a fractional dynamic model for the viscoelastic pipe conveying fluid, Appl. Math. Modell., 56 (2018), 123-136.
• [7] J. Hadamard, Essai sur letude des fonctions donn´ees par leur d´eveloppement de taylor, Jour. Pure and Appl. Math., 4(8) (1892), 101–186.
• [8] A. A. Kilbas, H. M. Srivastava, J. J. Trujillo, Theory and Applications of Fractional Differential Equations, Volume 204 (North-Holland Mathematics Studies). Elsevier Science Inc., USA, 2006.
• [9] I. Podlubny, Fractional Differential Equations: An Introduction to Fractional Derivatives, Fractional Differential Equations, to Methods of Their Solution and Some of Their Applications, Academic Press, San Diego, CA, 1998.
• [10] S. Samko, A. A. Kilbas, O. I. Marichev, Fractional Integrals and Derivatives: Theory and Applications, Gordon and Breach Science Publishers, Switzerland, 1993.
• [11] D.-X. Chen, Oscillatory behavior of a class of fractional differential equations with damping, U.P.B. Sci. Bull. Ser. A, 75(1) (2013), 107–118.
• [12] D.-X. Chen, P.-X. Qu, Y.-H. Lan, Forced oscillation of certain fractional differential equations, Adv. Difference Equ., 2013(1) (2013), 125.
• [13] Q. Feng, A. Liu, Oscillation for a class of fractional differential equation, J. Appl. Math. Phys., 7(07) (2019), 1429.
• [14] S. Grace, R. Agarwal, P. Wong, A. Zafer, On the oscillation of fractional differential equations, Fract. Calc. Appl. Anal., 15(06) (2012), 222–231.
• [15] Z. Han, Y. Zhao, Y. Sun, C. Zhang, Oscillation for a class of fractional differential equation Discrete Dyn. Nat. Soc., 2013 (2013).
• [16] H. Qin, B. Zheng. Oscillation of a class of fractional differential equations with damping term, Sci. World J., 2013 (2013).
• [17] T. Yalçın Uzun, H. Büyükçavuşoğlu Erçolak, M. K. Yıldız, Oscillation criteria for higher order fractional differential equations with mixed nonlinearities, Konuralp J. Math., 7 (2019), 203–207.
• [18] J. Yang, A. Liu, T. Liu, Forced oscillation of nonlinear fractional differential equations with damping term, Adv. Difference Equ., 2015(1) (2015), 1.
• [19] B. Zheng, Oscillation for a class of nonlinear fractional differential equations with damping term, J. Adv. Math. Stud., 6(1) (2013), 107–109.
• [20] R. P. Agarwal, M. Bohner, T. Li, Oscillatory behavior of second-order half-linear damped dynamic equations, Appl. Math. Comput., 254 (2015), 408-418.
• [21] M. Bohner, T. Li, Kamenev-type criteria for nonlinear damped dynamic equations, Sci. China Math., 58(7) (2015), 1445-1452.
• [22] J. Dzurina, S. R. Grace, I. Jadlovsk´a, T. Li, Oscillation criteria for second-order Emden-Fowler delay differential equations with a sublinear neutral term, Math. Nachr., 293(5) (2020), 910-922.
• [23] T. Li, Yu. V. Rogovchenko, On the asymptotic behavior of solutions to a class of third-order nonlinear neutral differential equations, Appl. Math. Lett., 105 (2020), Art. 106293, pp. 1-7.
• [24] D. Vivek, E. Elsayed, K. Kanagarajan, On the oscillation of fractional differential equations via y-hilfer fractional derivative, Eng. Appl. Sci. Lett., 2(3) (2019), 1–6.
• [25] R. P. Agarwal, M. Bohner, W.-T. Li, Nonoscillation and oscillation theory for functional differential equations, volume 267. CRC Press, 2004.
• [26] R. Hilfer, P. Butzer, U. Westphal, An introduction to fractional calculus, Appl. Fract. Calc. Phys., World Scientific, (2010), 1–85.
• [27] J. V. d. C. Sousa, E. C. de Oliveira, On the y-hilfer fractional derivative, Commun. Nonl. Sci. Numer. Simul., 60 (2018), 72–91.
• [28] U. Katugampola, A new approach to generalized fractional derivatives, B. Math. Anal. App., 6(4) (2014), 1–15.
• [29] U. Katugampola, Existence and uniqueness results for a class of generalized fractional differential equations, (2014), arXiv:1411.5229 [math.CA].