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 $\Delta _{a}y_{n}=y_{n+1}-ay_{n}, a\neq{0}.$
📌 3 Key Takeaways
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Objective & Hypothesis: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 $\Delta _{a}y_{n}=y_{n+1}-ay_{n}, a\neq{0}.$
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Key Findings & Data: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 $\Delta _{a}y_{n}=y_{n+1}-ay_{n}, a\neq{0}.$
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Conclusion & Impact: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 $\Delta _{a}y_{n}=y_{n+1}-ay_{n}, a\neq{0}.$
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 $\Delta _{a}y_{n}=y_{n+1}-ay_{n}, a\neq{0}.$
Nar, A., Bolat, Y., Değer, S. U., Gevgeşoğlu, M. (2022). Nonoscillation and oscillation criteria for class of higher - order difference equations involving generalized difference operator. Communications Faculty of Sciences University of Ankara Series A1 Mathematics and Statistics, Volume: 71 Issue: 1. https://doi.org/10.31801/cfsuasmas.904804
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.
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.
Tan, M., Yang, E., Oscillation and nonoscillation theorems for second order nonlinear difference equations, J. Math. Anal. Appl., 276 (2002), 239-247.