| Contents |
| Asymptotic properties of solutions of difference equations of the form
\[
\Delta^m x_n=a_nf(n,x_{\sigma(n)})+b_n
\]
are studied. Using fixed point theorems we obtain sufficient conditions under which
for any solution $y_n$ of the equation $\Delta^my_n=b_n$ and for any real $s\leq 0$ there exists a solution $x_n$ of the above equation such that $x_n=y_n+\mathrm{o}(n^{s})$. Using a discrete variant of the Bihari lemma and a certain new technique we give also sufficient conditions under which for a given real $s\leq m-1$ all solutions $x_n$ of the equation satisfy the condition $x_n=y_n+\mathrm{o}(n^s)$ where $y_n$ is a solution of the equation $\Delta^my_n=b_n$. In particular, taking $b_n=0$ we obtain asymptotically polynomial
solutions. Similarly, if $b_n$ \ is a periodic sequence, we can obtain asymptotically periodic solutions. |
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