# Article

 Title: Oscillatory and asymptotic behaviour of perturbed quasilinear second order difference equations (English) Author: Thandapani, E. Author: Ramuppillai, L. Language: English Journal: Archivum Mathematicum ISSN: 0044-8753 (print) ISSN: 1212-5059 (online) Volume: 34 Issue: 4 Year: 1998 Pages: 455-466 Summary lang: English . Category: math . Summary: This paper deals with oscillatory and asymptotic behaviour of solutions of second order quasilinear difference equation of the form $\Delta (a_{n-1}| \Delta y_{n-1}|^{\alpha -1} \Delta y_{n-1})+ F(n, y_n)= G(n, y_n, \Delta y_n), \quad n\in N(n_0) \qquad \mathrm {(E)}$ where $\alpha >0$. Some sufficient conditions for all solutions of (E) to be oscillatory are obtained. Asymptotic behaviour of nonoscillatory solutions of (E) are also considered. (English) Keyword: perturbed quasilinear difference equation Keyword: oscillatory solution MSC: 39A11 idZBL: Zbl 0969.39004 idMR: MR1679640 . Date available: 2009-02-17T10:15:58Z Last updated: 2012-05-10 Stable URL: http://hdl.handle.net/10338.dmlcz/107673 . Reference: [1] Agarwal, R. P.: Difference Equations and Inequalities.Marcel Dekker, New York, 1992. Zbl 0952.39001, MR 1155840 Reference: [2] Bing Liu, Yan, J.: Oscillatory and asymptotic behaviour of second order nonlinear difference equations.Proc. Edin. Math. Soc. 39(1996), 525-533. MR 1417694 Reference: [3] Bing Liu, Cheng, S. S.: Positive solutions of second order nonlinear difference equation.J. Math. Anal. Appl. 204(1996), 482-493. MR 1421461 Reference: [4] Cheng, S..,S., Li, H. J.: Bounded and zero convergent solutions of second order difference equations.J. Math. Anal. Appl. 14(1989), 141-149. MR 1009057 Reference: [5] Hardy, G. H., Littlewood, J. E., Polya, G.: Inequalities.2nd Edition, Cambridge University Press, 1988. MR 0944909 Reference: [6] He, H. Z.: Oscillatory and asymptotic behaviour of second order nonlinear difference equations.J. Math. Anal. Appl. 175(1993), 482-498. Zbl 0780.39001, MR 1219191 Reference: [7] Szmanda, B.: Nonoscillation, oscillation and growth of solutions of nonlinear difference equations of second order.J. Math. Anal. Appl. 109(1985), 22-30. Zbl 0589.39003, MR 0796040 Reference: [8] Thandapani, E.: Oscillation theorems for perturbed nonlinear second order difference equations.Computers Math. Appl. 28(1994), 309-316. Zbl 0807.39002, MR 1284245 Reference: [9] Thandapani, E., Arul, R.: Oscillation and nonoscillation theorems for a class of second order quasilinear difference equations.ZAA, 16 (1997), 749-759. MR 1472729 Reference: [10] Thandapani, E., Arul, R.: Oscillation theory for a class of second order quasilinear difference equations.Tamkang J. Math.Tamkang J. Math. 28 (1997), 229-238. MR 1486791 Reference: [11] Trench, W. F.: Asymptotic behaviour of solutions of Emden-Fowler difference equations with oscillating coefficients.J. Math. Anal. Appl. 179(1993), 135-153. MR 1244954 Reference: [12] Wong, P. J. Y., Agarwal, R. P.: Oscillation theorems and existence of positive monotone solutions for second order nonlinear difference equations.Math. Comput. Modelling 21(1995), 63-84. MR 1316120 Reference: [13] Wong, P. J. Y., Agarwal, R. P.: Oscillation and monotone solutions of second order quasilinear difference equations.Funk. Ekva. 39(1996), 491-517. MR 1433914 Reference: [14] Yu, Y. H.: Higher type oscillation criterion and Sturm type comparison theorem.Math. Nachr. 153(1991), 485-496. MR 1131938 .

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