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On the asymptotics of some difference equations. (English) Zbl 1252.39007
In the first part of the paper, the authors prove two theorems regarding a comparison of certain solutions to the equations $$ \align y_n &=\frac{f(y_{n-1},\dots,y_{n-k})}{g(y_{n-1},\dots,y_{n-k})}, \\ x_n &=(1-\varepsilon)f(x_{n-1},\dots,x_{n-k}),\\ z_n &=(1+\varepsilon)f(z_{n-1},\dots,z_{n-k}),\endalign$$ where $f,g:\mathbb{R}_+^k\to\mathbb{R}_+$ with $f$ being nondecreasing in all arguments, and $\varepsilon\in(0,1).$ Then, in order to apply these theorems, some stability results are recalled and proved. The final part is devoted to applications of these results in examination of some concrete difference equations, for example, $$ y_n=\frac{y_{n-1}y_{n-2}\cdots y_{n-2m}}{g(y_{n-1}+1,y_{n-2}+1,\dots,y_{n-2m}+1)}. $$

39A20Generalized difference equations
39A22Growth, boundedness, comparison of solutions (difference equations)
39A30Stability theory (difference equations)
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