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Global stability of a higher order rational recursive sequence. (English) Zbl 1105.39004
The authors investigate stability, periodicity, and oscillatory behavior of solutions of the rational difference equation $$ x_{n+1}=\frac{\beta ~x_{n-k+1}+\gamma~ x_{n-2k+1}}{A+B~x_{n-k+1}},\quad n=0,1,2,.... $$ or equivalently, via the transformation $x_n=Ay_n/B$, $$ y_{n+1}=\frac{p~y_{n-k+1}+q~y_{n-2k+1}}{1+y_{n-k+1}},\quad n=0,1,2,... $$ where the initial conditions $x_{-2k+1},...,x_{-1},x_0$ are positive, $k\in\{1,2,...\}$, and the parameters $\beta,~\gamma,~A,~B$ are positive. It is worth mentioning that if we introduce the change of variables $$ z_m=y_{mk}, $$ then $z_m$ satisfies the difference equation $$ z_{m+1}=y_{(m+1)k}=\frac{p~y_{mk}+q~y_{(m-1)k}}{1+y_{mk}}=\frac{p~z_m+q~z_{m-1}}{1+z_m}, $$ which has been investigated by several authors, as mentioned by the authors of the paper under review.

MSC:
39A11Stability of difference equations (MSC2000)
39A20Generalized difference equations
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Full Text: DOI
References:
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