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Keywords:
third order delay difference equation; property ${(\rm A)}$; comparison theorem
Summary:
Sufficient conditions are obtained for the third order nonlinear delay difference equation of the form $$\Delta (a_n(\Delta (b_n(\Delta y_n)^{\alpha })))+q_nf(y_{\sigma (n)})=0$$ to have property ${(\rm A)}$ or to be oscillatory. These conditions improve and complement many known results reported in the literature. Examples are provided to illustrate the importance of the main results.
References:
[1] Agarwal, R. P.: Difference Equations and Inequalities. Theory, Methods and Applications. Pure and Applied Mathematics 228. Marcel Dekker, NewYork (2000). MR 1740241 | Zbl 0952.39001
[2] Agarwal, R. P., Bohner, M., Grace, S. R., O'Regan, D.: Discrete Oscillation Theory. Hindawi Publishing, New York (2005). MR 2179948 | Zbl 1084.39001
[3] Agarwal, R. P., Grace, S. R.: Oscillation of certain third-order difference equations. Comput. Math. Appl. 42 (2001), 379-384. DOI 10.1016/S0898-1221(01)00162-6 | MR 1837999 | Zbl 1003.39006
[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. DOI 10.1155/ADE.2005.345 | MR 2201689 | Zbl 1107.39004
[5] Alzabut, J., Bolat, Y.: Oscillation criteria for nonlinear higher-order forced functional difference equations. Vietnam J. Math. 43 (2015), 583-594. DOI 10.1007/s10013-014-0106-y | MR 3386063 | Zbl 1326.39008
[6] Artzrouni, M.: Generalized stable population theory. J. Math. Biol. 21 (1985), 363-381. DOI 10.1007/BF00276233 | MR 0804157 | Zbl 0567.92013
[7] Bolat, Y., Alzabut, J.: On the oscillation of higher-order half-linear delay difference equations. Appl. Math. Inf. Sci. 6 (2012), 423-427. MR 2970650
[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. DOI 10.1155/2014/791631 | MR 3214492 | Zbl 1291.39032
[9] Došlá, Z., Kobza, A.: Global asymptotic properties of third-order difference equations. Comput. Math. Appl. 48 (2004), 191-200. DOI 10.1016/j.camwa.2003.05.008 | MR 2086796 | Zbl 1068.39006
[10] Došlá, Z., Kobza, A.: On third-order linear difference equations involving quasi-differences. Adv. Difference Equ. (2006), Article ID 65652, 13 pages. DOI 10.1155/ADE/2006/65652 | MR 2209669 | Zbl 1133.39007
[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. DOI 10.2298/AADM0901027G | MR 2499304 | Zbl 1224.39016
[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. MR 1745309 | Zbl 1141.39301
[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. MR 2767893 | Zbl 1215.39015
[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. DOI 10.14232/ejqtde.2010.1.67 | MR 2735028 | Zbl 1208.39019
[15] Smith, B.: Oscillatory and asymptotic behavior in certain third-order difference equations. Rocky Mt. J. Math. 17 (1987), 597-606. DOI 10.1216/RMJ-1987-17-3-597 | MR 0908266 | Zbl 0632.39002
[16] B. Smith, W. E. Taylor, Jr.: Nonlinear third-order difference equation: Oscillatory and asymptotic behavior. Tamkang J. Math. 19 (1988), 91-95. MR 1010642 | Zbl 0688.39001
[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. MR 2329832 | Zbl 1154.39302
[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. Zbl 1263.39009

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