# Article

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Keywords:
difference equations; nonlinear; asymptotic behavior; nonoscillatory solutions
Summary:
The authors consider the difference equation $\Delta ^{m} [y_{n} - p_{n} y_{n - k}] + \delta q_{n} y_{\sigma (n + m - 1)} = 0 \qquad \mathrm {(\ast )}$ where $m \ge 2$, $\delta = \pm 1$, $k \in N_0 = \lbrace 0,1, 2, \dots \rbrace$, $\Delta y_{n} = y_{n + 1} - y_{n}$, $q_{n} > 0$, and $\lbrace \sigma (n)\rbrace$ is a sequence of integers with $\sigma (n) \le n$ and $\lim _{n \rightarrow \infty } \sigma (n) = \infty$. They obtain results on the classification of the set of nonoscillatory solutions of ($\ast$) and use a fixed point method to show the existence of solutions having certain types of asymptotic behavior. Examples illustrating the results are included.
References:
Agarwal, R. P.: Difference Equations and Inequalities. Marcel Dekker, New York, 1992. MR 1155840 | Zbl 0925.39001
Erbe, L. H. and Zhang, B. G.: Oscillation of discrete analogues of delay equation. Diff. Integral Equations 2 (1989), 300-309. MR 0983682
Georgiou, D. A., Grove, E. A. and Ladas, G.: Oscillation of neutral difference equations. Appl. Anal. 33 (1989), 243–253. MR 1030111
Georgiou, D.  A., Grove, E.  A. and Ladas, G.: Oscillation of neutral difference equations with variable coefficients. in: “Differential Equations, Stability and Control", S. Elaydi (ed.), Lecture Notes Pure Appl. Math. Vol. 127, Dekker, New York, 1991, pp. 165–173. MR 1096752
Lalli, B. S., Grace, S. R.: Oscillation theorems for second order neutral difference equations. J. Math. Anal. Appl. (to appear).
Lalli, B. S., Zhang, B. G.: On existence of positive solutions and bounded oscillations for netural difference equations. J. Math. Anal. Appl. 166 (1992), 272–278. MR 1159653
Lalli, B. S., Zhang, B. G.: Oscillation and comparison theorems for certain difference equations. J. Aust. Math. Soc. Ser B. 34 (1992), 245–256. MR 1181576
Lalli, B. S., Zhang, B. G.: Oscillation and comparison theorems for certain neutral difference equations. J. Aust. Math. Soc. Ser B. (to appear). MR 1181576
Lalli, B. S., Zhang, B. G. and Li, J. Z.: On the oscillation of solutions and existence of positive solutions of neutral difference equations. J. Math. Anal. Appl. 158 (1991), 213–233. MR 1113411
Lakshmikantham, V., Trigiante, D.: Theory of Difference Equations: Numerical Methods and Applications. Math. in Science and Engineering Vol. 181, Academic Press, New York, 1988. MR 0939611
Moore, R. E.: Computational Functional Analysis. Ellis Harwood Series, Halsted Press, New York, 1985. MR 0783431 | Zbl 0574.46001
Thandapani, E.: Asymptotic and oscillatory behavior of solutions of a second order nonlinear neutral delay difference equation. Riv. Math. Univ. Parma (5) 1 (1992), 105–113. MR 1230602
Thandapani, E., Sundaram, P., Graef, J. R. and Spikes, P. W.: Asymptotic properties of solutions of nonlinear second order neutral delay difference equations. Dynamic Syst. Appl. 4 (1995), 125–136. MR 1312484
Thandapani, E., Sundaram, P., Graef, J. R. and Spikes, P. W.: Asymptotic behavior and oscillation of solutions of neutral delay difference equations of arbitrary order. (to appear). MR 1635228
Thandapani, E., Sundaram, P. and Györi, I.: On the behavior of solutions of first order nonlinear neutral difference equations. (to appear).
Thandapani, E., Sundaram, P. and Györi, I.: Oscillations of second order nonlinear neutral delay difference equations. (to appear). MR 1867518
Zafer, A., Dahiya, R. S.: Oscillation of a neutral difference equation. Appl. Math. Lett. 6 (1993), 71–74. MR 1347777

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