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Necessary conditions for the solutions of second order nonlinear neutral delay difference equations to be oscillatory or tend to zero. (English) Zbl 1148.39010

Necessary conditions for every solution of the neutral delay difference equation \[ \Delta(r_n\Delta(y_n- p_n y_{n-m}))+ q_n G(y_{n-k})= f_n \] to oscillate or to tend to zero as \(n\to\infty\) are found. Here \(\Delta\) is the forward difference operator \(\Delta x_n= x_{n+1}- x_n\), and \(p_n\), \(q_n\), \(r_n\) are sequences of real numbers with \(q_n\geq 0\), \(r_n> 0\).

MSC:

39A11 Stability of difference equations (MSC2000)
39A10 Additive difference equations
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References:

[1] S. N. Elaydi, An Introduction to Difference Equations, Undergraduate Texts in Mathematics, Springer, New York, NY, USA, 1996. · Zbl 0840.39002
[2] W. G. Kelley and A. C. Peterson, Difference Equations: An Introduction with Applications, Academic Press, Boston, Mass, USA, 1991. · Zbl 0733.39001
[3] R. E. Mickens, Difference Equations, Van Nostrand Reinhold, New York, NY, USA, 1987. · Zbl 1235.70006
[4] R. P. Agarwal, Difference Equations and Inequalities, vol. 228 of Monographs and Textbooks in Pure and Applied Mathematics, Marcel Dekker, New York, NY, USA, 2nd edition, 2000. · Zbl 0952.39001
[5] R. P. Agarwal, M. M. S. Manuel, and E. Thandapani, “Oscillatory and nonoscillatory behavior of second order neutral delay difference equations,” Mathematical and Computer Modelling, vol. 24, no. 1, pp. 5-11, 1996. · Zbl 0856.34077 · doi:10.1016/0895-7177(96)00076-3
[6] R. P. Agarwal, M. M. S. Manuel, and E. Thandapani, “Oscillatory and nonoscillatory behavior of second-order neutral delay difference equations II,” Applied Mathematics Letters, vol. 10, no. 2, pp. 103-109, 1997. · Zbl 0905.34065 · doi:10.1016/S0893-9659(97)00019-0
[7] S. S. Cheng and W. T. Patula, “An existence theorem for a nonlinear difference equation,” Nonlinear Analysis: Theory, Methods & Applications, vol. 20, no. 3, pp. 193-203, 1993. · Zbl 0774.39001 · doi:10.1016/0362-546X(93)90157-N
[8] P. Das and N. Misra, “A necessary and sufficient condition for the solutions of a functional-differential equation to be oscillatory or tend to zero,” Journal of Mathematical Analysis and Applications, vol. 205, no. 1, pp. 78-87, 1997. · Zbl 0874.34058 · doi:10.1006/jmaa.1996.5143
[9] L. H. Erbe, Q. Kong, and B. G. Zhang, Oscillation Theory for Functional-Differential Equations, vol. 190 of Monographs and Textbooks in Pure and Applied Mathematics, Marcel Dekker, New York, NY, USA, 1995. · Zbl 0821.34067
[10] M. R. S. Kulenović and S. Had, “Existence of nonoscillatory solution of second order linear neutral delay equation,” Journal of Mathematical Analysis and Applications, vol. 228, no. 2, pp. 436-448, 1998. · Zbl 0919.34067 · doi:10.1006/jmaa.1997.6156
[11] B. S. Lalli, B. G. Zhang, and J. Z. Li, “On the oscillation of solutions and existence of positive solutions of neutral difference equations,” Journal of Mathematical Analysis and Applications, vol. 158, no. 1, pp. 213-233, 1991. · Zbl 0732.39002 · doi:10.1016/0022-247X(91)90278-8
[12] N. Parhi and A. K. Tripathy, “Oscillation criteria for forced nonlinear neutral delay difference equations of first order,” Differential Equations and Dynamical Systems, vol. 8, no. 1, pp. 81-97, 2000. · Zbl 0978.39006
[13] N. Parhi and A. K. Tripathy, “On asymptotic behaviour and oscillation of forced first order nonlinear neutral difference equations,” Fasciculi Mathematici, no. 32, pp. 83-95, 2001. · Zbl 0994.39011
[14] N. Parhi and R. N. Rath, “On oscillation and asymptotic behaviour of solutions of forced first order neutral differential equations,” Proceedings of Indian Academy of Sciences, vol. 111, no. 3, pp. 337-350, 2001. · Zbl 0995.34058 · doi:10.1007/BF02829600
[15] N. Parhi and A. K. Tripathy, “Oscillation of forced nonlinear neutral delay difference equations of first order,” Czechoslovak Mathematical Journal, vol. 53(128), no. 1, pp. 83-101, 2003. · Zbl 1016.39011 · doi:10.1023/A:1022975525370
[16] N. Parhi and A. K. Tripathy, “Oscillation of a class of nonlinear neutral difference equations of higher order,” Journal of Mathematical Analysis and Applications, vol. 284, no. 2, pp. 756-774, 2003. · Zbl 1037.39002 · doi:10.1016/S0022-247X(03)00298-1
[17] N. Parhi and R. N. Rath, “On oscillation of solutions of forced nonlinear neutral differential equations of higher order,” Czechoslovak Mathematical Journal, vol. 53(128), no. 4, pp. 805-825, 2003. · Zbl 1080.34522 · doi:10.1007/s10587-004-0805-8
[18] N. Parhi and R. N. Rath, “Oscillatory behaviour of solutions of nonlinear higher order neutral differential equations,” Mathematica Bohemica, vol. 129, no. 1, pp. 11-27, 2004. · Zbl 1069.34093
[19] R. N. Rath and N. Misra, “Necessary and sufficient conditions for oscillatory behaviour of solutions of a forced nonlinear neutral equation of first order with positive and negative coefficients,” Mathematica Slovaca, vol. 54, no. 3, pp. 255-266, 2004. · Zbl 1078.34528
[20] R. N. Rath, L. N. Padhy, and N. Misra, “Oscillation of solutions of non-linear neutral delay differential equations of higher order for p(t)=\pm 1,” Archivum Mathematicum, vol. 40, no. 4, pp. 359-366, 2004. · Zbl 1116.34332
[21] R. N. Rath and L. N. Padhy, “Necessary and sufficient conditions for oscillation of solutions of a first order forced nonlinear difference equation with several delays,” Fasciculi Mathematici, no. 35, pp. 99-113, 2005. · Zbl 1093.39011
[22] R. N. Rath, N. Misra, and L. N. Padhy, “Oscillatory and asymptotic behaviour of a nonlinear second order neutral differential equation,” Mathematica Slovaca, vol. 57, no. 2, pp. 157-170, 2007. · Zbl 1150.34026 · doi:10.2478/s12175-007-0006-7
[23] E. Thandapani, P. Sundaram, J. R. Graef, and P. W. Spikes, “Asymptotic behaviour and oscillation of solutions of neutral delay difference equations of arbitrary order,” Mathematica Slovaca, vol. 47, no. 5, pp. 539-551, 1997. · Zbl 0941.39006
[24] I. Györi and G. Ladas, Oscillation Theory of Delay Differential Equations, Oxford Mathematical Monographs, The Clarendon Press, Oxford University Press, New York, NY, USA, 1991. · Zbl 0780.34048
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