## 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.

### MSC:

 39A11 Stability of difference equations (MSC2000)

### Keywords:

nonlinear difference system; oscillation
Full Text:

### References:

 [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
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