Comparison theorems for the asymptotic behavior of solutions of nonlinear difference equations.

*(English)*Zbl 0929.39002We 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.

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.

Reviewer: E. Thandapani (Salem)

##### MSC:

39A11 | Stability of difference equations (MSC2000) |

##### Keywords:

asymptotic behavior; bounded solutions; nonlinear difference equations; asymptotically equivalent solutions
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\textit{A. Gleska} and \textit{J. Werbowski}, J. Math. Anal. Appl. 226, No. 2, 456--465 (1998; Zbl 0929.39002)

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##### References:

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