Oscillation criteria for first-order nonlinear delay difference equations. (English) Zbl 1038.39005
Two oscillation criteria are obtained for the delay difference equation
\[
x_{n+1}- x_n+ p_n f(x_{n-l_1},\dots, x_n- l_m)= 0,
\]
where \(p_n\geq 0\) and \(f(y_1,\dots, y_m)/\sum^m_{j=1}| y_j|^{\alpha_j}\text{sgn\,}y_j\to 0\) as \(\max_{1\leq j\leq m}| y_j|\to 0\) for \(\alpha_j> 0\), \(j= 1,\dots, n\), such that \(\sum^m_{j=1} \alpha_j= 1\). For the case where \(f(y_1,\dots, y_m):= f(x_l)\), the results in this paper cover some recent work by X. H. Tang and J. S. Yu [J. Math. Anal. Appl. 249, No. 2, 476–490 (2000; Zbl 0963.39021)].
Reviewer’s remark: The term \(\prod^m_{j=1} u^{\alpha_j}_j\) in \((H_2)\), (iv) and in (9) should be replaced by
\(\prod^m_{j=1}| u_j|^{\alpha_j}\text{sgn\,}u_j\).
Reviewer’s remark: The term \(\prod^m_{j=1} u^{\alpha_j}_j\) in \((H_2)\), (iv) and in (9) should be replaced by
\(\prod^m_{j=1}| u_j|^{\alpha_j}\text{sgn\,}u_j\).
Reviewer: Quingkai Kong (DeKalb)
MSC:
| 39A11 | Stability of difference equations (MSC2000) |
Citations:
Zbl 0963.39021References:
| [1] | Agarwal, R. P., Difference Equations and Inequalities (1992), Marcel Dekker: Marcel Dekker New York · Zbl 0784.33008 |
| [2] | Erbe, L. H.; Zhang, B. G., Oscillation of discrete analogue of delay equations, Differen. Integ. Equat., 2, 300-309 (1989) · Zbl 0723.39004 |
| [3] | Ladas, G.; Philos, Ch. G.; Sficas, Y. G., Sharp condition for the oscillation of delay difference equation, J. Appl. Math. Simulat., 2, 101-112 (1989) · Zbl 0685.39004 |
| [4] | Ladas, G., Explicit condition for the oscillation of difference equations, J. Math. Anal. Appl., 153, 276-286 (1990) · Zbl 0718.39002 |
| [5] | Tang, X. H., Oscillation of delay difference equation with variable coefficients, J. Central South Univ. Tech., 29, 3, 287-288 (1998) |
| [6] | Tang, X. H.; Yu, J. S., Oscillation of delay difference equations, Comput. Math. Appl., 37, 11-20 (1999) · Zbl 0937.39012 |
| [7] | Tang, X. H.; Yu, J. S., A further result on the oscillation of delay difference equations, Appl. Math. Applic., 38, 229-237 (1999) · Zbl 0976.39005 |
| [8] | Tang, X. H.; Yu, J. S., Oscillation of delay difference equations in a critical state, Appl. Math. Lett., 13, 9-15 (2000) · Zbl 0973.39009 |
| [9] | Tang, X. H.; Yu, J. S., Oscillation of nonlinear delay difference equations, J. Math. Anal. Appl., 249, 476-490 (2000) · Zbl 0963.39021 |
| [10] | Yu, J. S.; Zhang, B. G.; Qian, X. Z., Oscillations of delay difference equations with deviating coefficients, J. Math. Anal. Appl., 177, 423-444 (1993) · Zbl 0787.39004 |
| [11] | Jiang, J. C., Oscillation of nonlinear delay difference equations, J. Int. Math. Math. Sci., 28, 301-306 (2001) · Zbl 1004.39004 |
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