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On the rational recursive sequence ${x}_{n+1}=\frac{a{x}_{n-1}}{b+c{x}_{n}{x}_{n-1}}$. (English) Zbl 1212.39008

Consider the difference equation

${x}_{n+1}=\frac{a{x}_{n-1}}{b+c{x}_{n}{x}_{n-1}},\phantom{\rule{1.em}{0ex}}n=0,1,\cdots ,$

with $a,b,c$ positive real numbers, and nonnegative initial conditions $\left({x}_{-1},{x}_{0}\right)$.

By a change of variables, it is reduced to

${y}_{n+1}=\frac{{y}_{n-1}}{p+{y}_{n}{y}_{n-1}},\phantom{\rule{1.em}{0ex}}n=0,1,\cdots ,\phantom{\rule{2.em}{0ex}}\left(1\right)$

where $p=b/a$, ${x}_{n}={\left(a/c\right)}^{1/2}{y}_{n}$.

Since this equation is semiconjugate to a Möbius transformation [cf. A. Andruch-Sobiło, 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 $p\ge 1$, and converges to a periodic solution of period two if $0. [For recent results concerning the same equation, see H. Sedaghat, J. Difference Equ. Appl. 15, No. 3, 215–224 (2009; Zbl 1169.39006)].

##### MSC:
 39A20 Generalized difference equations 39A22 Growth, boundedness, comparison of solutions (difference equations)