On the oscillation of higher order fractional difference equations with mixed nonlinearities

Abstract

Based on certain mathematical inequalities and Volterra sum equations, we establish oscillation criteria for higher order fractional difference equations with mixed nonlinearities. The problem is addressed for equations involving Riemann-Liouville and Caputo operators. Two examples are constructed to demonstrate the validity of the proposed assumptions. Our results improve those obtained in the previous works.

Keywords

• Oscillation of solutions,Fractional difference equations,Mixed nonlinearities

References

1. R. P. Agarwal, S. R. Grace, D. O’Regan: Oscillation Theory for Second Order Linear, Half–Linear, Superlinear and Sublinear Dynamic Equations, Kluwer Academic Publishers, 2002.
2. R. P. Agarwal, M. Bohner, W.-T. Li: Nonoscillation and Oscillation Theory for Functional Differential Equations, Marcel Dekker, New York, 2004.
3. R. P. Agarwal, M. Bohner, S. R. Grace, D. O’Regan: Discrete Oscillation Theory, Hindawi Publishing Corporation, 2005.
4. S. H. Saker: Oscillation Theory of Delay Differential and Difference Equations: Second and Third Orders, Verlag Dr. Müller, 2010.
5. Y. Sun, J. S. W. Wong: Oscillation criteria for second order forced ordinary differential equations with mixed nonlinearities, J. Math. Anal. Appl. 334 (2007), 549–560.
6. R. P. Agarwal, M. Bohner, W.-S. Cheung, S. R. Grace: Oscillation criteria for forced second order differential equations with mixed nonlinearities, Math. Comput. Modelling 45 (2007), 965–973.
7. Z. Zhang, X.Wang, H. Han: Oscillation criteria for forced second order differential equations with mixed nonlinearities, Appl. Math. Lett. 22 (7) (2009), 1096–1101.
8. L. Erbe, T. S. Hassan, A. Peterson: Oscillation of third–order functional dynamic equations with mixed nonlinearities on time scales, J. Appl. math. Comput. 34 (2010), 353–371.

9. T. S. Hassan, L. Erbe, A. Peterson: Forced oscillation of second order differential equations with mixed nonlinearities, Acta Math. Sci. Ser. B Engl. Ed. 31 (2) (2011). 613–626.
10. A. Özbekler, A. Zafer: Second order oscillation of mixed nonlinear dynamic equations with several positive and negative coefficients, Duscrete Contin. Dyn. Syst. Suppl. 2011, 1167– 1175.
11. Z. Xu, A. Cheng: Oscillation of second order differential equations with mixed nonlinearities, Turk J. Math. 38 (2014), 688–705.
12. S. G. Samko, A. A. Kilbas, O. I. Marichev: Fractional Intergrals and Derivatives: Theory and Applications, Gordon and Beach Science, Yverdon, Switzerland, 1993.
13. I. Podlubny: Fractional Diffeential Equations, vol. 198 of Mathematics in Science and Engineering, Academiv Press, San Diego, Calif., USA, 1999.
14. R. Hilfer: Applications of Fractional Calculus in Physics, World Scientific Publishing Company, Singapore, 2000.
15. S. R. Grace, R. P. Agarwal, P. J. Y. Wong, A. Zafer: On the oscillation of fractional differential equations, Fract. Calc. Appl. Anal. 15 (2) (2012), 222–231.
16. D.-X. Chen: Oscillation criteria of fractional differential equations, Adv. Difference Equ. 2012, 2012:33.
17. Q. Feng, F. Meng: Oscillation of solutions to nonlinear forced fractional differential equations, Electon. J. Differential Equations, Vol. 2013 (2013), No. 169, pp. 110.
18. H. Qin, B. Zheng: Oscillation of a class of fractional differential equations with damping term, The Sci. World Journal Volume 2013, Article ID 685621, 9 pages.
19. R. Xu: Oscillation criteria for nonlinear fractional differential equations, J. Appl. Math. Volume 2013, Article ID 971357, 7 pages.
20. J. Yang, A. Liu, T. Liu: Forced oscillation of nonlinear fractional differential equations with damping term, Adv. Difference Equ. (2015) 2015:1.
21. Y.-Z. Wang, Z.-L. Han, P. Zhao, S.-R. Sun: Oscillation theorems for fractional neutral differential equations, Hacet. J. Math. Stat. 44 (6) (2015), 1477-1488.
22. Y. Pan, R. Xu: Some new oscillation criteria for a class of nonlinear fractional differential equations, Fract. Diff. Calc. 6 (1) (2016), 17–33.
23. J. O. Alzabut, T. Abdeljawad: Sufficient conditons for oscillation of nonlinear fractional difference equations, J. Fract. Calc. Appl. 5 (1) (2014), 177–187.
24. S. Kisalar, M. K. Yıldız, E. Aktopark: Oscillation of higher order fractional nonlinear difference equations, Internat. J. Differ. Equations 10 (2) (2015), 201–212.
25. B. Abdalla, K. Abudaya, J. Alzabut, T. Abdeljawad: New oscillation criteria for forced nonlinear fractional difference equations, Vietnam J. Math. 45 (2017), 609-–618.
26. J. Alzabut, T. Abdeljawad, H. Alrabaiah: Oscillation criteria for forced and damped nabla fractional difference equations, J. Comput. Anal. Appl. 24 (8) (2018), 1387–1394.
27. B. Abdalla: On the oscillation of q–fractional difference equations, Adv. Difference Equ. 2017(1). Article Number: 254.
28. G. H. Hardy, J. E. Littlewood, G. Polya: Inequalities, Cambridge University Press, Cambridge 1988.
29. G. E. Andrews, R. Roy: Special Functions, Cambridge University Press, Cambridge 1999.