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Global asymptotic stability of a higher order rational difference equation. (English) Zbl 1121.39008

The authors consider the following nonlinear difference equation

${x}_{n+1}=\frac{f\left({x}_{n-{r}_{1}},\cdots ,{x}_{n-{r}_{k}}\right)g\left({x}_{n-{m}_{1}},\cdots ,{x}_{n-{m}_{l}}\right)+1}{f\left({x}_{n-{r}_{1}},\cdots ,{x}_{n-{r}_{k}}\right)+g\left({x}_{n-{m}_{1}},\cdots {x}_{n-{m}_{l}}\right)},\phantom{\rule{1.em}{0ex}}n=0,1,\cdots ,$

where $f\in C\left({ℝ}_{+}^{k},{ℝ}_{+}\right)$, $g\in C\left({ℝ}_{+}^{l},{ℝ}_{+}\right)$ with $k,l\in \left\{1,2,\cdots \right\}$, $0\le {r}_{k}\le \cdots \le {r}_{k}$ and $0\le {m}_{1}\le \cdots \le {m}_{l}$, and the initial values are positive real numbers.

The main result of this note gives sufficient conditions under which the unique equilibrium $x=1$ of the above equation is globally asymptotically stable.

MSC:
 39A11 Stability of difference equations (MSC2000) 39A20 Generalized difference equations
Keywords:
nonlinear difference equations