Oscillation of delay difference equation. (English) Zbl 0937.39012
Consider the following delay difference equation
\[
x_{n+1}-x_n+\sum^m_{i=1}p_i(n)X_{n-k_i}=0,\;n=0,1,\dots,
\]
where \(p_i(n)\geq 0\) for \(n\geq 0\), \(i=1,2, \dots,m\) and \(k_i\) are positive integers. Every solution of this equation is oscillatory if there exists an integer \(l\geq 1\) such that
\[
\sum_{n=0 }^\infty \biggl\{p(n) \bigl[(k+1)/k\bigr]^\ell \bigl(p^l(n) \bigr)^{ 1/(k +1)}-1 \biggr\}= \infty.
\]
Some illustrative examples are included. Some mistakes in the papers by other authors are pointed out.
Reviewer: D.Bobrowski (Poznań)
MSC:
| 39A11 | Stability of difference equations (MSC2000) |
References:
| [1] | Agarwal, R. P., Difference Equations and Inequalities (1992), Marcel Dekker: Marcel Dekker New York · Zbl 0784.33008 |
| [2] | Chen, M. P.; Yu, J. S., Oscillations for delay difference equations with variable coefficients, (Elaydi, S. N.; etal., Proceedings of the First International Conference on Difference Equations (1995), Gordon and Breach), 105-114 · Zbl 0860.39022 |
| [3] | Erbe, L. H.; Zhang, B. G., Oscillation of discrete analogue of delay equations, Differential Integral Equations, 2, 300-309 (1989) · Zbl 0723.39004 |
| [4] | Kelley, W. G.; Peterson, A. C., Difference Equations with Applications (1991), Acad. Press: Acad. Press New York · Zbl 0733.39001 |
| [5] | Ladas, G.; Philos, Ch. G.; Sficas, Y. G., Sharp condition for the oscillation of delay difference equations, J. Appl. Math., 2, 101-112 (1989) · Zbl 0685.39004 |
| [6] | Ladas, G., Explicit condition for the oscillation of difference equations, J. Math. Anal. Appl., 153, 276-286 (1990) · Zbl 0718.39002 |
| [7] | Ladas, G., Recent developments in the oscillation of delay difference equations, (International Conference on Differential Equations, Stability and Control (1990), Dekker: Dekker New York) · Zbl 0731.39002 |
| [8] | Tang, X. H., Oscillations of delay differential equations with variable coefficients, J. Central South University of Technology, 29, 3, 287-288 (1998), (Chinese) |
| [9] | Tang, Q. G.; Deng, Y. B., Oscillation for difference equations with several delays, J. Hunan University, 25, 2, 1-3 (1998), (Chinese) · Zbl 0901.39003 |
| [10] | Yan, J.; Qian, C., Oscillation and comparison result for delay difference equations, J. Math. Anal. Appl., 165, 346-360 (1992) · Zbl 0818.39002 |
| [11] | Yu, J. S.; Zhang, B. G.; Wang, Z. C., Oscillation of delay difference equation, Appl. Anal., 133, 117-124 (1994) · Zbl 0729.39003 |
| [12] | Yu, J. S.; Wang, Z. C., Asymptopic behavior and oscillation in delay difference equations, Funkcial. Ekvac., 37, 241-248 (1994) · Zbl 0808.34083 |
| [13] | Yu, J. S.; Zhang, B. G.; Qian, X. Z., Oscillations of delay difference equations with deviating coefficients, J. Math. Anal. Appl., 177, 432-444 (1993) · Zbl 0787.39004 |
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.
