## Abstract and Applied Analysis

### On the Period-Two Cycles of ${x}_{n+1}=\left(\alpha +\beta {x}_{n}+\gamma {x}_{n-k}\right)/\left(A+B{x}_{n}+C{x}_{n-k}\right)$

#### Abstract

We consider the higher order nonlinear rational difference equation ${x}_{n+1}=\left(\alpha +\beta {x}_{n}+\gamma {x}_{n-k}\right)/\left(A+B{x}_{n}+C{x}_{n-k}\right),\mathrm{}\mathrm{}\mathrm{}\mathrm{}n=\mathrm{0,1},\mathrm{2},\dots \mathrm{}$, where the parameters $\alpha ,\mathrm{}\beta ,\mathrm{}\gamma ,\mathrm{}A,\mathrm{}B,\mathrm{}C$ are positive real numbers and the initial conditions ${x}_{-k},\dots ,{x}_{-\mathrm{1}},\mathrm{}{x}_{\mathrm{0}}$ are nonnegative real numbers, $k\in \left\{\mathrm{1,2},\dots \right\}$. We give a necessary and sufficient condition for the equation to have a prime period-two solution. We show that the period-two solution of the equation is locally asymptotically stable.

#### Article information

Source
Abstr. Appl. Anal., Volume 2013 (2013), Article ID 179423, 10 pages.

Dates
First available in Project Euclid: 27 February 2014

https://projecteuclid.org/euclid.aaa/1393511884

Digital Object Identifier
doi:10.1155/2013/179423

Mathematical Reviews number (MathSciNet)
MR3055932

Zentralblatt MATH identifier
1275.39004

#### Citation

Atawna, S.; Abu-Saris, R.; Hashim, I.; Ismail, E. S. On the Period-Two Cycles of ${x}_{n+1}=\left(\alpha +\beta {x}_{n}+\gamma {x}_{n-k}\right)/\left(A+B{x}_{n}+C{x}_{n-k}\right)$. Abstr. Appl. Anal. 2013 (2013), Article ID 179423, 10 pages. doi:10.1155/2013/179423. https://projecteuclid.org/euclid.aaa/1393511884