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Global behavior of the difference equation \(x_{n+1}=(p+x_{n-1})/(qx_n+x_{n-1})\). (English) Zbl 1203.39007

Summary: We study the following difference equation \(x_{n+1}=(p+x_{n-1})/(qx_n+x_{n-1})\), \(n=0,1,\dots\), where \(p,q\in (0,+\infty)\) and the initial conditions \(x_{-1},x_0\in (0,+\infty)\). We show that every positive solution of the above equation either converges to a finite limit or to a two cycle, which confirms that the Conjecture 6.10.4 proposed by 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)] is true.

MSC:

39A20 Multiplicative and other generalized difference equations
39A22 Growth, boundedness, comparison of solutions to difference equations

Citations:

Zbl 0981.39011
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References:

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