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Dynamics of a \(k\)th order rational difference equation. (English) Zbl 1108.39004

This paper is concerned with the solutions, stability character and semicycle behavior of the rational difference equation \[ x_{n+1}=\frac{x_{n-k}}{A+x_{n-k}x_{n}}, \] where \(x_{-k}, x_{-k+1}, \dots, x_{0}>0\), and \(A>0\). The authors generalize the existing results in the reference. For the special case \(k=1\), they give an explicit solution of the difference equation \[ x_{n+1}=\frac{x_{n-1}}{A+x_{n-1}x_{n}}. \]

MSC:

39A11 Stability of difference equations (MSC2000)
39A20 Multiplicative and other generalized difference equations
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References:

[1] Aloqeili, M., Dynamics of a rational difference equation, Applied Mathematics and Computation, 176, 2, 773-779 (2006)
[2] Çinar, C., On the positive solutions of the difference equation \(x_{n + 1} = \frac{x_{n - 1}}{1 + x_{n - 1} x_n} \), Journal of Applied Mathematics and Computation, 150, 21-24 (2004) · Zbl 1050.39005
[3] Stević, S., More on a rational recurrence relation, Applied Mathematics E-Notes, 4, 80-84 (2004) · Zbl 1069.39024
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