| Popis: |
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. |
