Oscillation Tests for Fractional Difference Equations

George E. Chatzarakis 1 , Palaniyappan Gokulraj 2  and Thirunavukarasu Kalaimani 2
  • 1 School of Pedagogical and Technological Education (ASPETE),, Athens, Greece
  • 2 Dhirajlal Gandhi College of Technology, Salem, India


In this paper, we study the oscillatory behavior of solutions of the fractional difference equation of the form


where Δα denotes the Riemann-Liouville fractional difference operator of order α, 0 < α ≤ 1, ℕt0+1−α={t0+1−αt0+2−α…}, t0 > 0 and γ > 0 is a quotient of odd positive integers. We establish some oscillatory criteria for the above equation, using the Riccati transformation and Hardy type inequalities. Examples are provided to illustrate the theoretical results.

If the inline PDF is not rendering correctly, you can download the PDF file here.

  • [1] ABDALLA, B.—ABUDAYA, K.—ALZABUT, J.—ABDELJAWAD, T.: New oscillation criteria for forced nonlinear fractional difference equations, Vietnam J. Math. 45 (2017), 609–618.

  • [2] ABDELJAWAD, T.: On Riemann and Caputo fractional differences, Comput. Math. Appl. 62 (2011), 1602–1611.

  • [3] ALZABUT, J.—ABDELJAWAD, T.: Sufficient conditions for the oscillation of nonlinear fractional difference equations, J. Fract. Calc. Appl. 5 (2014), 177–187.

  • [4] CHATZARAKIS, G. E.—GOKULRAJ, P.—KALAIMANI, T.—SADHASIVAM, V.: Oscillatory solutions of nonlinear fractional difference equations, Int. J. Differ. Equ. 2018 (2018) (to appear).

  • [5] CHEN, F.: Fixed points and asymptotic stability of nonlinear fractional difference equations, Electron. J. Qual. Theory Differ. Equ. 39 (2001), 1–18.

  • [6] CHEN, F.—LIU, Z.: Asymptotic stability results for nonlinear fractional difference equations, J. Appl. Math. 2012 (2012), 1–14.

  • [7] CHEN, F.—LUO, X.—ZHOU, Y.: Existence results for nonlinear fractional difference equation, Adv. Differ. Equ. 2011 (2011), 1–12.

  • [8] DIAZ, J. B.—OLSER, T. J.: Differences of fractional order, Math. Comput. 28 (1974), 185–202.

  • [9] SELVAM, A. G. M.—SAGAYARAJ, M. R.—LOGANATHAN, M. P.: Oscillatory behavior of a class of fractional difference equations with damping, Int. J. Appl. Math. Res. 3 (2014), 220–224.

  • [10] HARDY, G. H.—LITTLEWOOD, J. E.—POLYA, G.: Inequalities. Cambridge University Press, Cambridge, 1959.

  • [11] LI, W. N.: Oscillation results for certain forced fractional difference equations with damping term, Adv. Differ. Equ. 2016 (2016), 1–9.

  • [12] LOGANATHAN, M. P.—SAGAYARAJ, M. R.—SELVAM, A. G. M.: On the oscillation of non-linear fractional difference equations, Math. Aeterna 4 (2014), 91–99.

  • [13] SAGAYARAJ, M. R.—SELVAM, A. G. M.—LOGANATHAN, M. P.: Oscillation criteria for a class of discrete nonlinear fractional equations, Bull. Soc. Math. Serv. Stand. 3 (2014), 27–35.

  • [14] SECER, A.—ADIGUZEL, H.: Oscillation of solutions for a class of nonlinear fractional difference equations, J. Nonlinear Sci. Appl. 9 (2016), 5862–5869.


Journal + Issues