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Asymptotic properties of the solutions to the second-order difference equation. (English) Zbl 0969.39002

The authors investigate the asymptotic properties of solutions to the second-order nonlinear difference equation (with a perturbed argument) \[ \Delta ^2 x_n=a_n\varphi (x_{n+k}). \tag{\(*\)} \] A typical result is the following statement.
{Theorem.} Suppose that \(\sum ^{\infty }n |a_n |<\infty \) and \(\varphi :\mathbb{R}\to \mathbb{R}\) is a continuous function. Then for every \(c\in \mathbb{R}\) and every \(k\in \mathbf N\cup \{0\}\) there exists a solution \(\{x_n\}\) of \((*)\) such that \(\lim x_n=c\).
The proofs of the presented results are based on an application of the Schauder fixed point theorem to certain operator associated with (\(*\)) acting in the Banach space \(l_{\infty }\). Essentially, equation (\(*\)) is viewed as a perturbation of the linear equation \(\Delta ^2 x_n=0\) and it is shown that if the “perturbation” \(a_n\varphi (\cdot)\) is small, then the solutions of this equation behave asymptotically as those of the unperturbed equation \(\Delta ^2 x_n=0\).

MSC:

39A11 Stability of difference equations (MSC2000)
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