# Article

 Title: Oscillation theorems for third order nonlinear delay difference equations (English) Author: Vidhyaa, Kumar S. Author: Dharuman, Chinnappa Author: Thandapani, Ethiraju Author: Pinelas, Sandra Language: English Journal: Mathematica Bohemica ISSN: 0862-7959 (print) ISSN: 2464-7136 (online) Volume: 144 Issue: 1 Year: 2019 Pages: 25-37 Summary lang: English . Category: math . 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. (English) Keyword: third order delay difference equation Keyword: property ${(\rm A)}$ Keyword: comparison theorem MSC: 39A10 idZBL: Zbl 07088834 idMR: MR3934196 DOI: 10.21136/MB.2018.0019-17 . Date available: 2019-03-21T12:30:28Z Last updated: 2020-07-01 Stable URL: http://hdl.handle.net/10338.dmlcz/147637 . Reference: [1] Agarwal, R. P.: Difference Equations and Inequalities. Theory, Methods and Applications.Pure and Applied Mathematics 228. Marcel Dekker, NewYork (2000). Zbl 0952.39001, MR 1740241 Reference: [2] Agarwal, R. P., Bohner, M., Grace, S. R., O'Regan, D.: Discrete Oscillation Theory.Hindawi Publishing, New York (2005). Zbl 1084.39001, MR 2179948 Reference: [3] Agarwal, R. P., Grace, S. R.: Oscillation of certain third-order difference equations.Comput. Math. Appl. 42 (2001), 379-384. Zbl 1003.39006, MR 1837999, 10.1016/S0898-1221(01)00162-6 Reference: [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. Zbl 1107.39004, MR 2201689, 10.1155/ADE.2005.345 Reference: [5] Alzabut, J., Bolat, Y.: Oscillation criteria for nonlinear higher-order forced functional difference equations.Vietnam J. Math. 43 (2015), 583-594. Zbl 1326.39008, MR 3386063, 10.1007/s10013-014-0106-y Reference: [6] Artzrouni, M.: Generalized stable population theory.J. Math. Biol. 21 (1985), 363-381. Zbl 0567.92013, MR 0804157, 10.1007/BF00276233 Reference: [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 Reference: [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. Zbl 1291.39032, MR 3214492, 10.1155/2014/791631 Reference: [9] Došlá, Z., Kobza, A.: Global asymptotic properties of third-order difference equations.Comput. Math. Appl. 48 (2004), 191-200. Zbl 1068.39006, MR 2086796, 10.1016/j.camwa.2003.05.008 Reference: [10] Došlá, Z., Kobza, A.: On third-order linear difference equations involving quasi-differences.Adv. Difference Equ. (2006), Article ID 65652, 13 pages. Zbl 1133.39007, MR 2209669, 10.1155/ADE/2006/65652 Reference: [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. Zbl 1224.39016, MR 2499304, 10.2298/AADM0901027G Reference: [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. Zbl 1141.39301, MR 1745309 Reference: [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. Zbl 1215.39015, MR 2767893 Reference: [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. Zbl 1208.39019, MR 2735028, 10.14232/ejqtde.2010.1.67 Reference: [15] Smith, B.: Oscillatory and asymptotic behavior in certain third-order difference equations.Rocky Mt. J. Math. 17 (1987), 597-606. Zbl 0632.39002, MR 0908266, 10.1216/RMJ-1987-17-3-597 Reference: [16] B. Smith, W. E. Taylor, Jr.: Nonlinear third-order difference equation: Oscillatory and asymptotic behavior.Tamkang J. Math. 19 (1988), 91-95. Zbl 0688.39001, MR 1010642 Reference: [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. Zbl 1154.39302, MR 2329832 Reference: [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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