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Local stability of period two cycles of second order rational difference equation. (English) Zbl 1255.39013
Summary: We consider the second order rational difference equation x n+1 =(α+βx n +γx n-1 )/(A+Bx n +Cx n-1 ),n=0,1,2,, where the parameters α,β,γ,A,B,C are positive real numbers, and the initial conditions x -1 ,x 0 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 E. Camouzis and G. Ladas [Dynamics of third-order rational difference equations with open problems and conjectures. Advances in Discrete Mathematics and its Applications 5. Boca Raton, FL: Chapman & Hall/CRC (2008; Zbl 1129.39002)] which appeared previously in Conjecture 11.4.3 in [M. R. S. Kulenović and G. Ladas, Dynamics of second order rational difference equations. With open problems and conjectures. Boca Raton, FL: Chapman & Hall/CRC (2002; Zbl 0981.39011)].
MSC:
39A30Stability theory (difference equations)