Nonoscillation and oscillation criteria for class of higher - order difference equations involving generalized difference operator

Year 2022, Volume: 71 Issue: 1, 188 - 203, 30.03.2022

Aysun Nar Yaşar Bolat Serbun Ufuk Değer Murat Gevgeşoğlu

https://doi.org/10.31801/cfsuasmas.904804

Abstract

In this paper, sufficient conditions are obtained for nonoscillation/oscillation of all solutions of a class of higher-order difference equations involving the generalized difference operator of the form

$\Delta _{a}^{k}(p_{n}\Delta _{a}^{2}y_{n})=f(n,y_{n},\Delta_{a}y_{n},\Delta _{a}^{2}y_{n},...,\Delta _{a}^{k+1}y_{n}),$

where $\Delta _{a}$ is generalized difference operator which is defined as $\Delta _{a}y_{n}=y_{n+1}-ay_{n}, a\neq{0}.$

Keywords

Difference equations, generalized difference operator, oscillation, nonoscillation

References

• Agarwal, R. P., Wong, P. J. Y., Advanced Topics in Difference Equations, Kluwer, Dordrecht, 1997.
• Agarwal, R. P., Grace, S. R., O’Regan, D., Oscillation Theory for Difference and Functional Diffrential Equations, Kluwer, Dordrecht, 2000.
• Agarwal, R. P., Difference Equations and Inequalities: Second Edition, Revised and Expanded, Marcel Dekker, New York, 2000.
• Agarwal, R. P., Grace, S. R., Oscillation of higher order difference equations, Applied Mathematics Letters, 13 (2000), 81-88.
• Agarwal, R. P., Grace, S. R., O’Regan, D., On the oscillation of higher order difference equations, Soochow Journal of Mathematics, 31(2) (2005), 245-259.
• Alzabut, J., Bolat, Y., Oscillation criteria for nonlinear higher-order forced functional difference equations, Vietnam Journal of Mathematics 43(3) (2014), 1-12. https://doi.org/10.1007/s10013-014-0106-y
• Bolat, Y., Alzabut, J., On the oscillation of higher-order half-linear delay difference equations, Applied Mathematics & Information Sciences, 6(3) (2012), 423-427.
• Bolat, Y., Alzabut, J., On the oscillation of even-order half-linear functional difference equations with damping term, International Journal of Differential Equations, 2014, Article ID 791631 (2014), 6 pages. https://doi.org/10.1155/2014/791631
• Köprübaşı T., Ünal, Z., Bolat, Y., Oscillation criteria for higher-order neutral type difference equations, Turkish Journal of Mathematics, 44 (2020), 729-738. https://doi.org/10.3906/mat- 1703-6
• Parhi, N., Panda, A., Nonoscillation and oscillation of solutions of a class of third order difference equations, J. Math. Anal. Appl. 336 (2007), 213-223.
• Patula, W. T, Growth, oscillation and comparison theorems for second-order linear difference equations, SIAM J. Math. Anal., 10(6) (1979), 1272-1279.
• Popenda, J., Oscillation and nonoscillation theorems for second-order difference equations, J. Math. Anal. Appl., 123 (1987), 34-38.
• Saker, S. H., Alzabut, J., Mukheimer, A., On the oscillatory behavior for a certain class of third order nonlinear delay difference equations, Electron. J. Qual. Theory Differ. Equ., 67 (2010), 1-16.
• Szafrranski, Z., On some oscillation criteria for difference equations of second order, Fasc. Math., 11 (1979), 135-142.
• Szmanda, B., Oscillation theorems for nonlinear second-order difference equations, J. Math. Anal. Appl., 79 (1981), 90-95.
• Tan, M., Yang, E., Oscillation and nonoscillation theorems for second order nonlinear difference equations, J. Math. Anal. Appl. , 276 (2002), 239-247.