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Zbl 1198.39025
Al-Shabi, M.A.; Abo-Zeid, R.
Global asymptotic stability of a higher order difference equation.
(English)
[J] Appl. Math. Sci., Ruse 4, No. 17-20, 839-847 (2010). ISSN 1312-885X; ISSN 1314-7552/e

The article deals with the equation $$x_{n+1} = \frac{Ax_{n-2r-1}}{B + Cx_{n-2l}x_{n-2k}}, \quad n = 0,1,2,\ldots,\tag1$$ where $A, B, C$ are nonnegative reals, $l, r, k$ nonnegative integers, $l, r \le k$. Equation (1) has a zero equilibrium point and, if $\gamma = BA^{-1} < 1$, a nonzero equilibrium point $\bar{y} = \sqrt{1 - \gamma}$. The main results are the following: If $\gamma > 1$ then the zero is a locally asymptotic stable equilibrium point; if $\gamma < 1$ then both equilibrium points are unstable. The case $r = k$ and $\gamma = 1$ is also considered; in this case there exist periodic solutions with the prime period $2(k + 1)$ and every solution of (1) converges to a periodic solution of (1) with the period $2(k + 1)$.
[Peter Zabreiko (Minsk)]
MSC 2000:
*39A30
39A23
39A20 Generalized difference equations

Keywords: rational difference equations; periodic solutions; asymptotical stability; equilibrium points

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