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Comparison theorems for the asymptotic behavior of solutions of nonlinear difference equations. (English) Zbl 0929.39002
We consider the asymptotic behavior of solutions to the nonlinear difference equation $\Delta^my_n+f(n,y_n,y_{n+1},\dots,y_{n+m-1})=b_n,\tag{E}$ where $$m\geq 2$$, $$n\in N_{n_0}$$, $$b:N_{n_0}\to\mathbb{R}$$, and $$f:N_{n_0}\times\mathbb{R}^m\to\mathbb{R}$$. Here $$y_n=y(n)$$, $$b_n=b(n)$$.
It is known that in the case when the function $$f$$ is “small” in some sense, then the solutions of Eq. (E) are asymptotically equivalent with the solutions of the following difference equation: $$\Delta^mz_n=b_n$$ as $$n\to\infty$$.
The purpose of this paper is to consider the above problem using a comparison method differing in a number of ways from those previously applied. Namely, we reduce the problem of asymptotically equivalent solutions to a difference equation of higher order to the boundedness of solutions to some difference equation of first order. In consequence, our conditions imposed on the function $$f$$ are less restrictive than those previously required by other authors.

##### MSC:
 39A11 Stability of difference equations (MSC2000)
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##### References:
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