On the Period-Two Cycles of x n + 1 = ( α + β x n + γ x n - k ) / ( A + B x n + C x n - k )

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.