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Zbl 1212.39008
Andruch-Sobiło, Anna; Małgorzata, Migda
On the rational recursive sequence $x_{n+1}=\frac {ax_{n-1}}{b+cx_nx_{n-1}}$.
(English)
[J] Tatra Mt. Math. Publ. 43, 1-9 (2009). ISSN 1210-3195

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 $(x_{-1},x_0)$. \par 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 $p=b/a$, $x_n=(a/c)^{1/2}y_n$. \par Since this equation is semiconjugate to a Möbius transformation [cf. {\it {A. Andruch-Sobiło}}, {\it {M. Małgorzata}}, 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 {\it {H. Sedaghat}}, J. Difference Equ. Appl. 15, No.~3, 215--224 (2009; Zbl 1169.39006)].
[Eduardo Liz (Vigo)]
MSC 2000:
*39A20 Generalized difference equations
39A22

Keywords: rational difference equation; equilibrium point; boundedness; asymptotic behaviour; positive solution

Citations: Zbl 1131.39003; Zbl 1169.39006

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