Oscillation of two-dimensional difference systems. (English) Zbl 0964.39012

The paper presents some oscillation results for the two-dimensional difference system \[ \begin{aligned} \triangle x_n&=b_ng(y_n),\\ \triangle y_{n-1}&=-a_nf(x_n), \end{aligned} \qquad n\in\mathbb N (n_0)=\{n_0,n_0+1,\dots \}, \tag{1} \] where \(n_0\in\mathbb N\), \(\{a_n\}, \{b_n\}\), \(n\in\mathbb N (n_0)\) are real sequences and \(f, g\) are real continuous functions with the sign property \(uf(u)>0\) and \(ug(u)>0\) for all \(u\neq 0\). Assuming both sequences \(\{a_n\}, \{b_n\}\) are nonnegative and not identically zero for infinitely many values of \(n\) the authors establish the sufficient conditions for all solutions to the system (1) to be oscillatory. Moreover, some of their results do not require the sign condition on \(\{a_n\}\), i.e., \(\{a_n\}\) can assume both positive and negative values. The difference system (1) can be reduced via a special choice of \(f, g\) and \(b_n\) to the second order superlinear or sublinear equation. It is shown that the derived oscillation results generalize the known oscillation criteria for this type of difference equations.


39A11 Stability of difference equations (MSC2000)
Full Text: DOI


[1] Kocic, V.; Ladas, G., Global behavior of nonlinear difference equations of higher order with applications, (1993), Kluwer Publisher Dordrecht · Zbl 0787.39001
[2] Potts, R., Exact solution of a difference approximation to Duffing’s equation, J. austral. math. soc. (series B), 23, 64-77, (1981) · Zbl 0475.34008
[3] Hooker, J.W.; Patula, W.T., A second-order nonlinear difference equation: oscillation and asymptotic behavior, J. math. anal. appl., 91, 9-29, (1983) · Zbl 0508.39005
[4] Zhang, B.G.; Chen, G.D., Oscillation of second-order nonlinear difference equations, J. math. anal. appl., 199, 827-841, (1996) · Zbl 0855.39011
[5] Thandapani, E.; Arul, R., Oscillation and nonoscillation theorems for a class of second-order quasilinear difference equations, Z. anal. anwendungen, 16, 1-11, (1997) · Zbl 0883.39007
[6] Thandapani, E.; Graef, J.R.; Spikes, P.W., On the oscillation of solutions of second-order quasilinear difference equations, Nonlinear world, 3, 545-565, (1996) · Zbl 0897.39002
[7] Thandapani, E.; Manuel, M.M.S.; Agarwal, R.P., Oscillation and nonoscillation theorems for second-order quasilinear difference equations, Facta univ. ser. math. inform., 11, 49-65, (1996) · Zbl 1014.39004
[8] Thandapani, E.; Györi, I.; Lalli, B.S., An application of discrete inequality to second-order nonlinear oscillation, J. math. anal. appl., 186, 200-208, (1994) · Zbl 0823.39004
[9] Szafranski, Z.; Szmanda, B., Oscillatory properties of solutions of some difference systems, Rad. mat., 6, 205-214, (1990) · Zbl 0762.39007
[10] Agarwal, R.P., Difference equations and inequalities, (1992), Marcel Dekker New York · Zbl 0784.33008
[11] Agarwal, R.P.; Wong, P.J.Y., Advanced topics in difference equations, (1997), Kluwer Publisher Dordrecht · Zbl 0914.39005
[12] Waltman, P., An oscillation criterion for a nonlinear second-order equation, J. math. anal. appl., 10, 439-441, (1965) · Zbl 0131.08902
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. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.