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Dynamics of the nonlinear rational difference equation \({x_{n + 1}} = \frac{Ax_{n - \alpha}x_{n - \beta} + Bx_{n - \gamma}} {Cx_{n - \alpha}x_{n - \beta} + Dx_{n - \gamma}} \). (English) Zbl 1429.39007

Summary: In this article, we study the global stability and the asymptotic properties of the non-negative solutions of the non-linear difference equation: \[x_{n + 1} = \frac{Ax_{n - \alpha}x_{n - \beta} + Bx_{n - \gamma}} {Cx_{n - \alpha}x_{n - \beta} + Dx_{n - \gamma}}\quad n = 0,1, \ldots,\] where \(\alpha, \beta, \gamma\) are positive integers, \(A, B, C, D\) are positive real numbers and the initial conditions \(x_{- p}, x_{- p+1}, \dots, x_{-1}, x_0\) for \(p = \max\{\alpha, \beta, \gamma \}\) are arbitrary positive real numbers.

MSC:

39A20 Multiplicative and other generalized difference equations
39A30 Stability theory for difference equations
39A22 Growth, boundedness, comparison of solutions to difference equations
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