## Abstract

We consider the higher order nonlinear rational difference equation ${x}_{n+1}=(\alpha +\beta {x}_{n}+\gamma {x}_{n-k})/(A+B{x}_{n}+C{x}_{n-k}),\mathrm{}\mathrm{}\mathrm{}\mathrm{}n=\mathrm{0,1},\mathrm{2},\dots \mathrm{}$, 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, $k\in \{\mathrm{1,2},\dots \}$. 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.

## Citation

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

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