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On the system of high order rational difference equations \(x_{n}=\frac {a}{y_{n-p}}, y_{n}=\frac {by_{n-p}}{x_{n-q}{y_{n-q}}}\). (English) Zbl 1093.39013
The paper deals with the system of difference equations \[ x_n = \frac{a}{y_{n-p}}, \;y_n = \frac{by_{n-p}}{x_{n-q}y_{n-q}}, \;n=1,2,..., \eqno(1) \] where \(a\) and \(b\) are positive constants, \(p\) and \(q\) are positive integers, \(q\) is divisible by \(p\). The authors study monotonicity properties of positive solutions of (1). If \(a=b\) then every positive solution of (1) is periodic with period \(2q\).

MSC:
39A11 Stability of difference equations (MSC2000)
39A20 Multiplicative and other generalized difference equations, e.g., of Lyness type
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[1] Cinar, C., On the positive solutions of the difference equation system \(x_{n + 1} = \frac{1}{y_n}\), \(y_{n + 1} = \frac{y_n}{x_{n - 1} y_{n - 1}}\), Applied mathematics and computation, 158, 303-305, (2004) · Zbl 1066.39006
[2] Clark, D.; Kulenovic, M.R.S., A coupled system of rational difference equations, Computers and mathematics with applications, 43, 849-867, (2002) · Zbl 1001.39017
[3] Clark, D.; Kulenovic, M.R.S.; Selgrade, J.F., Global asymptotic behavior of a two-dimensional difference equation modeling competition, Nonlinear analysis, 52, 1765-1776, (2003) · Zbl 1019.39006
[4] Grove, E.A.; Ladas, G.; McGrath, L.C.; Teixeira, C.T., Existence and behavior of solutions of a rational system, Communications on applied nonlinear analysis, 8, 1-25, (2001) · Zbl 1035.39013
[5] Papaschinopoulos, G.; Schinas, C.J., On a system of two nonlinear difference equations, Journal of mathematical analysis and applications, 219, 415-426, (1998) · Zbl 0908.39003
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