Local Stability of Period Two Cycles of Second Order Rational Difference Equation

Abstract EN

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

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.

In particular, we solve Conjecture 5.201.2 proposed by Camouzis and Ladas in their book (2008) which appeared previously in Conjecture 11.4.3 in Kulenović and Ladas monograph (2002).