### Abstract

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).

Original language | English |
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Article number | 969813 |

Journal | Discrete Dynamics in Nature and Society |

Volume | 2012 |

DOIs | |

Publication status | Published - 2012 |

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### ASJC Scopus subject areas

- Modelling and Simulation

### Cite this

*Discrete Dynamics in Nature and Society*,

*2012*, [969813]. https://doi.org/10.1155/2012/969813

**Local stability of period two cycles of second order rational difference equation.** / Atawna, S.; Abu-Saris, R.; Hashim, Ishak.

Research output: Contribution to journal › Article

*Discrete Dynamics in Nature and Society*, vol. 2012, 969813. https://doi.org/10.1155/2012/969813

}

TY - JOUR

T1 - Local stability of period two cycles of second order rational difference equation

AU - Atawna, S.

AU - Abu-Saris, R.

AU - Hashim, Ishak

PY - 2012

Y1 - 2012

N2 - 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).

AB - 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).

UR - http://www.scopus.com/inward/record.url?scp=84871373675&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=84871373675&partnerID=8YFLogxK

U2 - 10.1155/2012/969813

DO - 10.1155/2012/969813

M3 - Article

VL - 2012

JO - Discrete Dynamics in Nature and Society

JF - Discrete Dynamics in Nature and Society

SN - 1026-0226

M1 - 969813

ER -