×

On the oscillation of even-order half-linear functional difference equations with damping term. (English) Zbl 1291.39032

Summary: We investigate the oscillatory behavior of solutions of the \(m\)th order half-linear functional difference equations with damping term of the form \(\Delta(p_nQ(\Delta^{m-1}y_n))+q_nQ(\Delta^ {m-1}y_n)+r_nQ(y_{\tau_n})=0\), \(n\geq n_0\) where \(m\) is even and \(Q(s)=|s|^{\alpha|-2}s\), \(\alpha>1\) is a fixed real number. Our main results are obtained via employing the generalized Riccati transformation. We provide two examples to illustrate the effectiveness of the proposed results.

MSC:

39A21 Oscillation theory for difference equations
39A12 Discrete version of topics in analysis
39A10 Additive difference equations
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] P. Rehák, “Oscillatory properties of second order half-linear difference equations,” Czechoslovak Mathematical Journal, vol. 51, no. 126, pp. 303-321, 2001. · Zbl 0982.39004 · doi:10.1023/A:1013790713905
[2] O. Do and P. Rehák, Half-Linear Differential Equations, vol. 202 of Mathematics Studies, North-Holland, Amsterdam, The Netherlands, 2005. · Zbl 1090.34027
[3] R. Agarwal, M. Bohner, S. R. Grace, and D. O’Regan, Discrete Oscillation Theory, Hindawi Publishing Corporation, New York, NY, USA, 2005. · Zbl 1084.39001 · doi:10.1155/9789775945198
[4] M. Cecchi, Z. Do, M. Marini, and I. Vrko\vc, “Asymptotic properties for half-linear difference equations,” Mathematica Bohemica, vol. 131, no. 4, pp. 347-363, 2006. · Zbl 1110.39004
[5] Y. G. Sun and F. W. Meng, “Nonoscillation and oscillation of second order half-linear difference equations,” Applied Mathematics and Computation, vol. 197, no. 1, pp. 121-127, 2008. · Zbl 1141.39010 · doi:10.1016/j.amc.2007.07.037
[6] J. Jiang and X. Tang, “Oscillation of second order half-linear difference equations (II),” Applied Mathematics Letters, vol. 24, no. 9, pp. 1495-1501, 2011. · Zbl 1377.39021 · doi:10.1016/j.aml.2011.03.029
[7] O. Do and S. Fi, “Linearized Riccati technique and (non-)oscillation criteria for half-linear difference equations,” Advances in Difference Equations, vol. 2008, Article ID 438130, 18 pages, 2008. · Zbl 1146.39009 · doi:10.1155/2008/438130
[8] O. Do and S. Fi, “Perturbation principle in discrete half-linear oscillation theory,” Studies of the University of \vZilina. Mathematical Series, vol. 23, no. 1, pp. 19-28, 2009. · Zbl 1221.39001
[9] O. Do and S. Fi, “Variational technique and principal solution in half-linear oscillation criteria,” Applied Mathematics and Computation, vol. 217, no. 12, pp. 5385-5391, 2011. · Zbl 1217.34053 · doi:10.1016/j.amc.2010.12.006
[10] R. P. Agarwal, S. R. Grace, and D. O’Regan, Oscillation Theory for Second Order Linear, Half-Linear, Superlinear and Sublinear Dynamic equations, Kluwer Academic Publishers, Dordrecht, The Netherlands, 2002. · Zbl 1073.34002
[11] O. Do and A. Lomtatidze, “Oscillation and nonoscillation criteria for half-linear second order differential equations,” Hiroshima Mathematical Journal, vol. 36, no. 2, pp. 203-219, 2006. · Zbl 1123.34028
[12] S. Liu, Q. Zhang, and Y. Yu, “Oscillation of even-order half-linear functional differential equations with damping,” Computers & Mathematics with Applications, vol. 61, no. 8, pp. 2191-2196, 2011. · Zbl 1219.34045 · doi:10.1016/j.camwa.2010.09.011
[13] Q. Zhang, S. Liu, and L. Gao, “Oscillation criteria for even-order half-linear functional differential equations with damping,” Applied Mathematics Letters, vol. 24, no. 10, pp. 1709-1715, 2011. · Zbl 1223.34096 · doi:10.1016/j.aml.2011.04.025
[14] C. Zhang, T. Li, B. Sun, and E. Thandapani, “On the oscillation of higher-order half-linear delay differential equations,” Applied Mathematics Letters, vol. 24, no. 9, pp. 1618-1621, 2011. · Zbl 1223.34095 · doi:10.1016/j.aml.2011.04.015
[15] M. Cecchi, Z. Do, and M. Marini, “Positive decreasing solutions of quasi-linear difference equations,” Computers & Mathematics with Applications, vol. 42, no. 10-11, pp. 1401-1410, 2001. · Zbl 1007.39006 · doi:10.1016/S0898-1221(01)00249-8
[16] O. Do and P. Rehák, “Nonoscillation criteria for half-linear second-order difference equations,” Computers & Mathematics with Applications, vol. 42, no. 3-5, pp. 453-464, 2001, Advances in difference equations, III. · Zbl 1006.39012 · doi:10.1016/S0898-1221(01)00169-9
[17] P. Rehák, “Generalized discrete Riccati equation and oscillation of half-linear difference equations,” Mathematical and Computer Modelling, vol. 34, no. 3-4, pp. 257-269, 2001. · Zbl 1038.39002 · doi:10.1016/S0895-7177(01)00059-0
[18] P. Rehák, “Oscillation and nonoscillation criteria for second order linear difference equations,” Fasciculi Mathematici, vol. 31, pp. 71-89, 2001. · Zbl 0999.39006
[19] E. Thandapani, M. M. S. Manuel, J. R. Graef, and P. W. Spikes, “Monotone properties of certain classes of solutions of second-order difference equations,” Computers & Mathematics with Applications, vol. 36, no. 10-12, pp. 291-297, 1998, Advances in difference equations, II. · Zbl 0933.39014 · doi:10.1016/S0898-1221(98)80030-8
[20] E. Thandapani and K. Ravi, “Bounded and monotone properties of solutions of second-order quasilinear forced difference equations,” Computers & Mathematics with Applications, vol. 38, no. 2, pp. 113-121, 1999. · Zbl 0936.39003 · doi:10.1016/S0898-1221(99)00186-8
[21] E. Thandapani and K. Ravi, “Oscillation of second-order half-linear difference equations,” Applied Mathematics Letters, vol. 13, no. 2, pp. 43-49, 2000. · Zbl 0977.39003 · doi:10.1016/S0893-9659(99)00163-9
[22] S. H. Saker, “Oscillation criteria of second-order half-linear delay difference equations,” Kyungpook Mathematical Journal, vol. 45, no. 4, pp. 579-594, 2005. · Zbl 1127.39022
[23] Y. Bolat and J. O. Alzabut, “On the oscillation of higher-order half-linear delay difference equations,” Applied Mathematics & Information Sciences, vol. 6, no. 3, pp. 423-427, 2012.
[24] R. P. Agarwal and S. R. Grace, “Oscillation of certain functional-differential equations,” Computers & Mathematics with Applications, vol. 38, no. 5-6, pp. 143-153, 1999. · Zbl 0935.34059 · doi:10.1016/S0898-1221(99)00221-7
[25] M. Migda, “On the discrete version of generalized Kiguradze’s lemma,” Fasciculi Mathematici, vol. 35, pp. 1-7, 2005. · Zbl 1095.39008
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. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.