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
\[
x_{n+1}=\frac{x_{n-1}}{A+x_{n-1}x_{n}}.
\]
Reviewer: Zhiming Guo (Guangzhou)
MSC:
39A11 | Stability of difference equations (MSC2000) |
39A20 | Multiplicative and other generalized difference equations |
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 |
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. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.