Necessary and Sufcient Conditions for Existence of an Equilibrium and a Periodic of Prime Period 2 Solution of a Certain Rational Difference Equation

This article studies the necessary and sufficient conditions for the existence of positive equilibrium solutions and positive periodic solutions of prime period 2 of the following rational difference equation. $$x_{n+1} = \dfrac{\alpha + \beta x_{n−k}}{A + B_0x_n + B_1x_{n−1} + \cdots + B_kx_{n−k}}, \text{ for } n \in \{0,1,2,\ldots\}$$ where the parameters $\alpha > 0$ and $\beta , A, B_0 , B_1 , \ldots, B_k$ and the initial conditions $x_{−k} , x_{−k+1} , x_{−k+2} ,\ldots, x_{−1} , x_0$ are nonnegative real numbers such that the denominator is always positive.