On the Period-Two Cycles ofxn+1=(α+βxn+γx

Abstract

We consider the higher order nonlinear rational difference equationxn+1=(α+βxn+γxn-k)/(A+Bxn+Cxn-k),n=0,1,2,…, where the parametersα,β,γ,A,B,Care positive real numbers and the initial conditionsx-k,…,x-1,x0are 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.