On the Period-Two Cycles of xn+1=(α+βxn+γxn-k)/(A+Bxn+Cxn-k)

Abstract

We consider the higher order nonlinear rational difference equation xn+1=(α+βxn+γxn-k)/(A+Bxn+Cxn-k),n=0,1,2,…, where the parameters α,β,γ,A,B,C are positive real numbers and the initial conditions x-k,…,x-1,x0 are nonnegative real numbers, k∈{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.