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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 $x_{n+1} = (\alpha + \beta x_n + \gamma x_{n-1})/(A + Bx_n + Cx_{n-1}), n = 0, 1 ,2 ,\dots$, where the parameters $\alpha, \beta, \gamma, A, B, C$ 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 {\it E. Camouzis} and {\it 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 [{\it M. R. S. Kulenović} and {\it 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)
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Full Text: DOI
References:
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