On the asymptotics of some difference equations. (English) Zbl 1252.39007

Berg, Lothar; Stević, Stevo

J. Difference Equ. Appl. 18, No. 5, 785-797 (2012).

In the first part of the paper, the authors prove two theorems regarding a comparison of certain solutions to the equations \[ \begin{aligned} y_n &=\frac{f(y_{n-1},\dots,y_{n-k})}{g(y_{n-1},\dots,y_{n-k})}, \\ x_n &=(1-\varepsilon)f(x_{n-1},\dots,x_{n-k}),\\ z_n &=(1+\varepsilon)f(z_{n-1},\dots,z_{n-k}),\end{aligned} \] 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, \[ y_n=\frac{y_{n-1}y_{n-2}\cdots y_{n-2m}}{g(y_{n-1}+1,y_{n-2}+1,\dots,y_{n-2m}+1)}. \]

Reviewer: Pavel Rehak (Brno)

MSC:

39A20	Multiplicative and other generalized difference equations
39A22	Growth, boundedness, comparison of solutions to difference equations
39A30	Stability theory for difference equations

Keywords:

positive solutions; asymptotic behaviour; rational difference equations; stability

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