On the period-two cycles of \(x_{n + 1} = (\alpha + \beta x_n + \gamma x_{n - k})/(A + Bx_n + Cx_{n - k})\) (Q369752)

Summary: We consider the higher-order nonlinear rational difference equation \(x_{n + 1} = (\alpha + \beta x_n + \gamma x_{n - k})/(A + Bx_n + Cx_{n - k})\), \(n = 0, 1, 2, \dots\), where the parameters \(\alpha, \beta, \gamma, A, B, C\) are positive real numbers and the initial conditions \(x_{-k}, \dots, x_{-1}, x_0\) are nonnegative real numbers, \(k \in \{1, 2, \dots\}\). 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.