Opuscula Math. 40, no. 5 (2020), 549-568
On the nonoscillatory behavior of solutions of three classes of fractional difference equations
Abstract. In this paper, we study the nonoscillatory behavior of three classes of fractional difference equations. The investigations are presented in three different folds. Unlike most existing nonoscillation results which have been established by employing Riccati transformation technique, we employ herein an easily verifiable approach based on the fractional Taylor's difference formula, some features of discrete fractional calculus and mathematical inequalities. The theoretical findings are demonstrated by examples. We end the paper by a concluding remark.
Keywords: Caputo difference operator, nonoscillation criteria, fractional difference equation, mathematical inequalities.
Mathematics Subject Classification: 34A08, 39A21.
- B. Abdalla, T. Abdeljawad, On the oscillation of Caputo fractional differential equations with Mittag-Leffler nonsingular kernel, Chaos Solitons Fract. 127 (2019), 173-177.
- B. Abdalla, K. Abodayeh, T. Abdeljawad, J. Alzabut, New oscillation criteria for forced nonlinear fractional difference equations, Vietnam J. Math. 45 (2017), 609-618.
- B. Abdalla, J. Alzabut, T. Abdeljawad, On the oscillation of higher order fractional difference equations with mixed nonlinearities, Hacet. J. Math. Stat. 47 (2018), 207-217.
- T. Abdeljawad, On Riemann and Caputo fractional differences, Comput. Math. Appl. 62 (2011), 1602-1611.
- J. Alzabut, T. Abdeljawad, H. Alrabaiah, Oscillation criteria for forced and damped nabla fractional difference equations, J. Comput. Anal. Appl 24 (2018) 8, 1387-1394.
- G.A. Anastassiou, Discrete fractional calculus and inequalities, arXiv:0911.3370, (2009).
- A. Aphithana, S.K. Ntouyas, J. Tariboon, Forced oscillation of fractional differential equations via conformable derivatives with damping term, Bound. Value Probl. 2019 (2019), 47.
- F.M. Atici, P.W. Eloe, A transform method in discrete fractional calculus, Intern. J. Difference Equ. 2 (2007), 165-176.
- F.M. Atici, P.W. Eloe, Initial value problems in discrete fractional calculus, Proc. Amer. Math. Soc. 137 (2009), 981-989.
- F.M. Atici, S. Sengul, Modeling with fractional difference equations, J. Math. Anal. Appl. 369 (2010) 1, 1-9.
- Z. Bai, R. Xu, The asymptotic behavior of solutions for a class of nonlinear fractional difference equations with damping term, Discrete Dyn. Nat. Soc. 2018 (2018).
- G.E. Chatzarakis, P. Gokulraj, T. Kalaimani, Oscillation tests for fractional difference equations, Tatra Mt. Math. Publ. 71 (2018) 1, 53-64.
- G.E. Chatzarakis, P. Gokulraj, T. Kalaimani, V. Sadhasivam, Oscillatory solutions of nonlinear fractional difference equations, Int. J. Differ. Equ. 13 (2018), 19-31.
- F. Chen, Fixed points and asymptotic stability of nonlinear fractional difference equations, Electron. J. Qual. Theory Differ. Equ. 36 (2011), 1-18.
- D.X. Chen, Oscillation criteria of fractional differential equations, Adv. Differ. Equ. 2012 (2012), 33.
- F. Chen, Z. Liu, Asymptotic stability results for nonlinear fractional difference equations, J. Appl. Math. 2012 (2012).
- K. Diethelm, The analysis of fractional differential equations, Springer Science & Business Media, 2010.
- C.S. Goodrich, On discrete sequential fractional boundary value problems, J. Math. Anal. Appl. 385 (2012) 1, 111-124.
- C. Goodrich, A.C. Peterson, Discrete Fractional Calculus, Springer, Berlin, 2015.
- S.R. Grace, R.P. Agarwal, P.J.Y. Wong, A. Zafer, On the oscillation of fractional differential equations, Fract. Calc. Appl. Anal. 15 (2012), 222-231.
- S.R. Grace, A. Zafer, On the asymptotic behavior of nonoscillatory solutions of certain fractional differential equations, Eur. Phys. J. Spec. Top. 226 (2017), 3657-3665.
- G.H. Hardy, I.E. Littlewood, G. Polya, Inequalities, University Press, Cambridge, 1959.
- J.M. Holte, Discrete Gronwall lemma and applications, [in:] Proceedings of the MAA-NCS Meeting at the University of North Dakota, (2009).
- A.A. Kilbas, H.M. Srivastava, J.J. Trujillo, Theory and Applications of Fractional Differential Equations, North-Holland Mathematics Studies, vol. 204. Elsevier, Amsterdam, 2006.
- W.N. Li, W. Sheng, Oscillation properties for solutions of a kind of partial fractional differential equations with damping term, J. Nonlinear Sci. Appl 9 (2016), 1600-1608.
- R.E. Mickens, Difference Equations: Theory, Applications and Advanced Topics, Taylor and Francis Group, New York, 2015.
- I. Podlubny, Fractional Differential Equations, vol. 198, Mathematics in Science and Engineering, Academic Press, San Diego, California, 1999.
- V.E. Tarasov, Frcational-order difference equations for physical lattices and some applications, J. Math. Phys. 56 (2015) 10, 1-19.
- E. Tunc, O. Tunc, On the oscillation of a class of damped fractional differential equations, Miskolc Math. Notes 17 (2016), 647-656.
- J. Yang, A. Liu, T. Liu, Forced oscillation of nonlinear fractional differential equations with damping term, Adv. Differ. Equ. 2015 (2015).
- Y. Zhou, A. Alsaedi, B. Ahmad, Oscillation for fractional neutral functional differential systems, J. Comput. Anal. Appl. 25 (2018) 5, 965-974.
- Communicated by Josef Diblík.
- Received: 2020-01-11.
- Revised: 2020-08-20.
- Accepted: 2020-08-28.
- Published online: 2020-10-10.