Some Results on the Oscillatory Behavior of Integro-differential Equations

This article was retracted on June 30, 2022. https://dergipark.org.tr/en/pub/tjmcs/article/1138321

Abstract

In this paper, we investigate the oscillation of a class of generalized proportional fractional integro-differential equations with forcing term. We present sufficient conditions to prove some oscillation criteria in both of the Riemann-Liouville and Caputo cases. Besides, we present some numerical examples for applicability of our results.

Keywords

• Oscillation
• Fractional integro-differential equations
• Proportional Derivative

References

1. [1] Abdalla, B., On the oscillation of q-fractional difference equations, Adv. Difference Equ., 2017:254(2017), 11 pp.
2. [2] Abdalla, B., Oscillation of differential equations in the frame of nonlocal fractional derivatives generated by conformable derivatives, Adv. Difference Equ., 2018:107(2018), 15 pp.
3. [3] Abdalla, B., Abdeljawad, T., On the oscillation of Hadamard fractional differential equations, Adv. Difference Equ., 2018:409(2018), 13 pp.
4. [4] Abdalla, B., Abdeljawad, T., On the oscillation of Caputo fractional differential equations with Mittag-Leffler nonsingular kernel, Chaos, Solitons and Fractals, 127(2019), 173–177.
5. [5] Abdalla, B., Abdeljawad, T., Oscillation criteria for kernel function dependent fractional dynamic equations, Discrete Contin. Dyn. Syst. Ser. S, 14(2021), 3337–3349.
6. [6] Abdeljawad, T., On conformable fractional calculus, J. Comput. Appl. Math., 279(2015), 57–66.
7. [7] Alzabut, J., Abdeljawad, T., Sufficient conditions for the oscillation of nonlinear fractional difference equations, J. Fract. Calc. Appl., 5(2014), 177–187.
8. [8] Anderson, D.R., Ulness, D.J., Newly defined conformable derivatives, Adv. Dyn. Syst. Appl., 10(2015), 109–137.

9. [9] Aslıyüce, S., Güvenilir, A.F., Zafer, A., Oscillation criteria for a certain class of fractional order integro-differential equations, Hacet. J. Math. Stat., 46(2017), 199-207.
10. [10] Aphithana, A., Ntouyas, S.K., Tariboon, J., Forced oscillation of fractional differential equations via conformable derivatives with damping term, Bound. Value Probl., 2019:47(2019), 16 pp.
11. [11] Atangana, A., Fractal-fractional differentiation and integration: Connecting fractal calculus and fractional calculus to predict complex system, Chaos, Solitons and Fractals, 102(2017), 396–406.
12. [12] Atangana, A., Baleanu, D., New fractional derivatives with nonlocal and non-singular kernel:Theory and application to heat transfer model, Therm. Sci., 20(2016), 763–769.
13. [13] Bolat, Y., On the oscillation of fractional order delay differential equations with constant coefficients, Commun. Nonlinear Sci. Numer. Simul., 19(2014), 3988–3993.
14. [14] Chen, D.X., Oscillation criteria of fractional differential equations, Adv. Difference Equ., 2012:33(2012), 10 pp.
15. [15] Chen, D.X., Qu, P.X., Lan, Y.H., Forced oscillation of certain fractional differential equations, Adv. Difference Equ., 2013:125(2013), 10 pp.
16. [16] Grace, S.R., Agarwal, R.P., Wong, P.J.Y., Zafer, A., On the oscillation of fractional differential equations, Fract. Calc. Appl. Anal., 15(2012), 222–231.
17. [17] Hardy, G.H., Littlewood, J.E., Polya, G., Inequalities, Cambridge University Press, 1988.
18. [18] Jarad, F., Abdeljawad, T., Alzabut, J., Generalized fractional derivatives generated by a class of local proportional derivatives, Eur. Phys. J. Special Topics, 226(2017), 3457–3471.
19. [19] Khalil, R., Al Horani, M., Yousef, A., Sababheh, M., A new definition of fractional derivative, J. Comput. Appl. Math., 264(2014), 65–70.
20. [20] Kilbas, A.A., Srivastava, M.H., Trujillo, J.J., Theory and Application of Fractional Differential Equations, North Holland Mathematics Studies, 204, 2006.
21. [21] Samko, S.G., Kilbas, A.A., Marichev, O.I., Fractional Integrals and Derivatives: Theory and Applications, Gordon and Breach, Yverdon, 1993.
22. [22] Singh, J., Kumar, D., Baleanu, D., New aspects of fractional Biswas–Milovic model with Mittag–Leffler law, Math. Model. Nat. Phenom., 14(2019), 23 pp.
23. [23] Sudsutad, W., Alzabut, J., Tearnbucha, C., Thaiprayoon, C., On the oscillation of differential equations in frame of generalized proportional fractional derivatives, AIMS Math., 5(2020), 856–871.
24. [24] Zhou, Y., Ahmad, B., Chen, F., Alsaedi, A., Oscillation for fractional partial differential equations, Bull. Malays. Math. Sci. Soc., 42(2019), 449–465.
25. [25] Zhu, P., Xiang, Q., Oscillation criteria for a class of fractional delay differential equations, Adv. Difference Equ., 2018:403(2018), 11 pp.