# zbMATH — the first resource for mathematics

Global behavior of solutions to two classes of second-order rational difference equations. (English) Zbl 1177.39016
Summary: For nonnegative real numbers $$\alpha,\beta,\gamma, A, B$$, and $$C$$ such that $$B+C>0$$ and $$\alpha+\beta+\gamma>0$$, the difference equation $x_{n+1}=\frac{\alpha+\beta x_n+\gamma x_{n-1}}{A+Bx_n+Cx_{n-1}},\quad n=0,1,2,\dots$
has a unique positive equilibrium. A proof is given here for the following statements: (1) For every choice of positive parameters $$\alpha,\beta,\gamma, A, B$$, and $$C$$, all solutions to the difference equation
$x_{n+1}=\frac{\alpha+\beta x_n+\gamma x_{n-1}}{A+B x_n+Cx_{n-1}},\quad n=0,1,2,\dots,x-1,\;x_0\in [0,\infty)$
converge to the positive equilibrium or to a prime period-two solution. (2) For every choice of positive parameters $$\alpha,\beta,\gamma$$, $$B$$, and $$C$$, all solutions to the difference equation
$x_{n+1}=\frac{\alpha+\beta x_n+\gamma x_{n-1}}{B x_n+C x_{n-1}}, \quad n=0,1,2,\dots,x_{-1},\;x_0\in(0,\infty)$
converge to the positive equilibrium or to a prime period-two solution.

##### MSC:
 39A23 Periodic solutions of difference equations 39A20 Multiplicative and other generalized difference equations
Full Text:
##### References:
 [2] doi:10.1080/10236199508808020 · Zbl 0858.39002 [4] doi:10.1016/S0893-9659(00)00068-9 · Zbl 0958.39021 [5] doi:10.1080/10236190701388492 · Zbl 1131.39005 [6] doi:10.1080/10236190701761482 · Zbl 1138.39002 [7] doi:10.1155/2007/41541 · Zbl 1149.39002 [8] doi:10.1080/10236190701827945 · Zbl 1153.39015
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.