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On the rational recursive sequence \(x_{n+1}=\frac {ax_{n-1}}{b+cx_nx_{n-1}}\). (English) Zbl 1212.39008

Consider the difference equation \[ x_{n+1}=\frac {ax_{n-1}}{b+cx_nx_{n-1}},\quad n=0,1,\dots, \] with \(a,b,c\) positive real numbers, and nonnegative initial conditions \((x_{-1},x_0)\).
By a change of variables, it is reduced to \[ y_{n+1}=\frac {y_{n-1}}{p+y_ny_{n-1}},\quad n=0,1,\dots, \tag{1} \] where \(p=b/a\), \(x_n=(a/c)^{1/2}y_n\).
Since this equation is semiconjugate to a Möbius transformation [cf. {A. Andruch-Sobiło}, {M. Migda}, Opusc. Math. 26, No. 3 387–394 (2006; Zbl 1131.39003)], a formula for the solutions is available in terms of the parameter \(p\) and the initial data. The authors use this formula to prove that every positive solution of (1) converges to zero if \(p\geq 1\), and converges to a periodic solution of period two if \(0<p<1\). [For recent results concerning the same equation, see {H. Sedaghat}, J. Difference Equ. Appl. 15, No. 3, 215–224 (2009; Zbl 1169.39006)].
Reviewer: Eduardo Liz (Vigo)

MSC:

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