Abstract
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, ... , where the parameters alpha, beta, gamma, A, B, C are positive real numbers and the initial conditions x(-k), ... , x(-1), x(0) are nonnegative real numbers, k is an element of {1, 2, ...}. 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.