Oscillatory behavior of higher order nonlinear difference equations
Abstract
The authors present some new oscillation criteria for higher order nonlinear difference equations with nonnegative real coefficients of the form 
Both of the cases n even and n odd are considered. They give examples to illustrate their results.
Keyword : oscillation, higher order, difference equations
How to Cite
Grace, S. R., & Graef, J. R. (2020). Oscillatory behavior of higher order nonlinear difference equations. Mathematical Modelling and Analysis, 25(4), 522-530. https://doi.org/10.3846/mma.2020.11447
This work is licensed under a Creative Commons Attribution 4.0 International License.
References
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R.P. Agarwal, M. Bohner, S.R. Grace and D. O’Regan. Discrete Oscillation Theory. Hindawi, New York, 2005. https://doi.org/10.1155/9789775945198
R.P. Agarwal and S.R. Grace. Oscillation of higher order difference equations. Appl. Math. Letters, 13:81–88, 2000. https://doi.org/10.1016/S0893-9659(99)00149-4
R.P. Agarwal, S.R. Grace and D. O’Regan. Oscillation Theory for Difference and Functional Differential Equations. Kluwer, Dordrecht, 2000. https://doi.org/10.1007/978-94-015-9401-1
C. Dharuman, J.R. Graef, E. Thandapani and K.S. Vidhyaa. Oscillation of second order difference equations with a sub-linear neutral term. J. Math. Appl., 40:59–67, 2017. https://doi.org/10.7862/rf.2017.4
H.A. El-Morshedy. Oscillation and nonoscillation criteria for half-linear second order difference equations. Dynam. Systems Appl., 15:429–450, 2006.
H.A. El-Morshedy. New oscillation criteria for second order linear difference equations with positive and negative coefficients. Comput. Math. Appl., 58:1988– 1997, 2009. https://doi.org/10.1016/j.camwa.2009.07.078
H.A. El-Morshedy and S.R. Grace. Comparison theorems for second order nonlinear difference equations. J. Math. Anal. Appl., 306:106–121, 2005. https://doi.org/10.1016/j.jmaa.2004.12.024
S.R. Grace, R.P. Agarwal, M. Bohner and D. O’Regan. Oscillation of second-order strongly superlinear and strongly sublinear dynamic equations. Commun. Nonlinear Sci. Numer. Simul., 14:3463–3471, 2009. https://doi.org/10.1016/j.cnsns.2009.01.003
S.R. Grace, M. Bohner and R.P. Agarwal. On the oscillation of second-order half-linear dynamic equations. J. Difference Equ. Appl., 15:451–460, 2009. https://doi.org/10.1080/10236190802125371
S.R. Grace and H.A. El-Morshedy. Oscillation criteria of comparison type for second order difference equations. J. Appl. Anal., 6:87–103, 2000. https://doi.org/10.1515/JAA.2000.87
S.R. Grace and J.R. Graef. Oscillatory behavior of second order nonlinear differential equations with a sublinear neutral term. Math. Model. Anal., 23:217–226, 2018. https://doi.org/10.3846/mma.2018.014
J.R. Graef, S.R. Grace and E. Tunç¸. Oscillatory behavior of even-order nonlinear differential equations with a sublinear neutral term. Opuscula Math., 39:39–47, 2019. https://doi.org/10.7494/OpMath.2019.39.1.39
I. Györi and G. Ladas. Oscillation Theory of Delay Differential Equations and Applications. Oxford University Press, Oxford, 1991.
G. Ladas and I.P. Stavroulakis. Oscillation caused by several retarded and advanced arguments. J. Differ. Equations, 44:134–152, 1982. https://doi.org/10.1016/0022-0396(82)90029-8
Ch.G. Philos. On the existence of nonoscillatory solutions tending to zero at ∞ for differential equations with positive delays. Arch. Math. (Basel), 36:168–178, 1981. https://doi.org/10.1007/BF01223686
S. Selvarangam, M. Madhan and E. Thandapani. Oscillation theorems for second order nonlinear neutral type difference equations with positive and negative coefficients. Rom. J. Math. Comput. Sci., 7:1–10, 2017.
S. Selvarangam, E. Thandapani and S. Pinelas. Oscillation theorems for second order nonlinear neutral difference equations. J. Inequal. Appl., 2014(417):1–15, 2014. https://doi.org/10.1186/1029-242X-2014-417
M.K. Yildiz and H. Ögünmez. Oscillation results of higher order nonlinear neutral delay difference equations with a nonlinear neutral term. Hacet. J. Math. Stat., 43:809–814, 2014.
R.P. Agarwal, M. Bohner, S.R. Grace and D. O’Regan. Discrete Oscillation Theory. Hindawi, New York, 2005. https://doi.org/10.1155/9789775945198
R.P. Agarwal and S.R. Grace. Oscillation of higher order difference equations. Appl. Math. Letters, 13:81–88, 2000. https://doi.org/10.1016/S0893-9659(99)00149-4
R.P. Agarwal, S.R. Grace and D. O’Regan. Oscillation Theory for Difference and Functional Differential Equations. Kluwer, Dordrecht, 2000. https://doi.org/10.1007/978-94-015-9401-1
C. Dharuman, J.R. Graef, E. Thandapani and K.S. Vidhyaa. Oscillation of second order difference equations with a sub-linear neutral term. J. Math. Appl., 40:59–67, 2017. https://doi.org/10.7862/rf.2017.4
H.A. El-Morshedy. Oscillation and nonoscillation criteria for half-linear second order difference equations. Dynam. Systems Appl., 15:429–450, 2006.
H.A. El-Morshedy. New oscillation criteria for second order linear difference equations with positive and negative coefficients. Comput. Math. Appl., 58:1988– 1997, 2009. https://doi.org/10.1016/j.camwa.2009.07.078
H.A. El-Morshedy and S.R. Grace. Comparison theorems for second order nonlinear difference equations. J. Math. Anal. Appl., 306:106–121, 2005. https://doi.org/10.1016/j.jmaa.2004.12.024
S.R. Grace, R.P. Agarwal, M. Bohner and D. O’Regan. Oscillation of second-order strongly superlinear and strongly sublinear dynamic equations. Commun. Nonlinear Sci. Numer. Simul., 14:3463–3471, 2009. https://doi.org/10.1016/j.cnsns.2009.01.003
S.R. Grace, M. Bohner and R.P. Agarwal. On the oscillation of second-order half-linear dynamic equations. J. Difference Equ. Appl., 15:451–460, 2009. https://doi.org/10.1080/10236190802125371
S.R. Grace and H.A. El-Morshedy. Oscillation criteria of comparison type for second order difference equations. J. Appl. Anal., 6:87–103, 2000. https://doi.org/10.1515/JAA.2000.87
S.R. Grace and J.R. Graef. Oscillatory behavior of second order nonlinear differential equations with a sublinear neutral term. Math. Model. Anal., 23:217–226, 2018. https://doi.org/10.3846/mma.2018.014
J.R. Graef, S.R. Grace and E. Tunç¸. Oscillatory behavior of even-order nonlinear differential equations with a sublinear neutral term. Opuscula Math., 39:39–47, 2019. https://doi.org/10.7494/OpMath.2019.39.1.39
I. Györi and G. Ladas. Oscillation Theory of Delay Differential Equations and Applications. Oxford University Press, Oxford, 1991.
G. Ladas and I.P. Stavroulakis. Oscillation caused by several retarded and advanced arguments. J. Differ. Equations, 44:134–152, 1982. https://doi.org/10.1016/0022-0396(82)90029-8
Ch.G. Philos. On the existence of nonoscillatory solutions tending to zero at ∞ for differential equations with positive delays. Arch. Math. (Basel), 36:168–178, 1981. https://doi.org/10.1007/BF01223686
S. Selvarangam, M. Madhan and E. Thandapani. Oscillation theorems for second order nonlinear neutral type difference equations with positive and negative coefficients. Rom. J. Math. Comput. Sci., 7:1–10, 2017.
S. Selvarangam, E. Thandapani and S. Pinelas. Oscillation theorems for second order nonlinear neutral difference equations. J. Inequal. Appl., 2014(417):1–15, 2014. https://doi.org/10.1186/1029-242X-2014-417
M.K. Yildiz and H. Ögünmez. Oscillation results of higher order nonlinear neutral delay difference equations with a nonlinear neutral term. Hacet. J. Math. Stat., 43:809–814, 2014.