## Oscillation theorems for third order nonlinear delay difference equations.(English)Zbl 1474.39024

Sufficient conditions for the third-order nonlinear delay difference equation of the form $\Delta\left( a_{n}(\Delta(b_{n}(\Delta y_{n})^{\alpha}))\right)+q_{n}f(y_{\sigma(n)})=0,$ to have property (A) or to be oscillatory are established. Two examples illustrating the results are given.

### MSC:

 39A21 Oscillation theory for difference equations 39A10 Additive difference equations
Full Text:

### References:

 [1] Agarwal, R. P., Difference Equations and Inequalities. Theory, Methods and Applications, Pure and Applied Mathematics 228. Marcel Dekker, NewYork (2000) [2] Agarwal, R. P.; Bohner, M.; Grace, S. R.; O’Regan, D., Discrete Oscillation Theory, Hindawi Publishing, New York (2005) [3] Agarwal, R. P.; Grace, S. R., Oscillation of certain third-order difference equations, Comput. Math. Appl. 42 (2001), 379-384 [4] Agarwal, R. P.; Grace, S. R.; O’Regan, D., On the oscillation of certain third-order difference equations, Adv. Difference Equ. 2005 (2005), 345-367 [5] Alzabut, J.; Bolat, Y., Oscillation criteria for nonlinear higher-order forced functional difference equations, Vietnam J. Math. 43 (2015), 583-594 [6] Artzrouni, M., Generalized stable population theory, J. Math. Biol. 21 (1985), 363-381 [7] Bolat, Y.; Alzabut, J., On the oscillation of higher-order half-linear delay difference equations, Appl. Math. Inf. Sci. 6 (2012), 423-427 [8] Bolat, Y.; Alzabut, J., On the oscillation of even-order half-linear functional difference equations with damping term, Int. J. Differ. Equ. 2014 (2014), Article ID 791631, 6 pages [9] Došlá, Z.; Kobza, A., Global asymptotic properties of third-order difference equations, Comput. Math. Appl. 48 (2004), 191-200 [10] Došlá, Z.; Kobza, A., On third-order linear difference equations involving quasi-differences, Adv. Difference Equ. (2006), Article ID 65652, 13 pages [11] Grace, S. R.; Agarwal, R. P.; Graef, J. R., Oscillation criteria for certain third order nonlinear difference equation, Appl. Anal. Discrete Math. 3 (2009), 27-38 [12] Graef, J. R.; Thandapani, E., Oscillatory and asymptotic behavior of solutions of third order delay difference equations, Funkc. Ekvacioj, Ser. Int. 42 (1999), 355-369 [13] Saker, S. H.; Alzabut, J. O., Oscillatory behavior of third order nonlinear difference equations with delayed argument, Dyn. Contin. Discrete Impuls. Syst. Ser. A Math. Anal. 17 (2010), 707-723 [14] Saker, S. H.; Alzabut, J. O.; Mukheimer, A., On the oscillatory behavior for a certain class of third order nonlinear delay difference equations, Electron. J. Qual. Theory Differ. Equ. 2010 (2010), Paper No. 67, 16 pages [15] Smith, B., Oscillatory and asymptotic behavior in certain third-order difference equations, Rocky Mt. J. Math. 17 (1987), 597-606 [16] B. Smith; W. E. Taylor, Jr., Nonlinear third-order difference equation: Oscillatory and asymptotic behavior, Tamkang J. Math. 19 (1988), 91-95 [17] Thandapani, E.; Pandian, S.; Balasubramanian, R. K., Oscillatory behavior of solutions of third order quasilinear delay difference equations, Stud. Univ. Žilina, Math. Ser. 19 (2005), 65-78 [18] Wang, X.; Huang, L., Oscillation for an odd-order delay difference equations with several delays, Int. J. Qual. Theory Differ. Equ. Appl. 2 (2008), 15-23
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.