# Asymptotic properties of solutions of nonautonomous difference equations

## Janusz Migda

Address: Faculty of Mathematics and Computer Science A. Mickiewicz University ul. Umultowska 87, 61-614 Poznań, Poland

E-mail: migda@amu.edu.pl

Abstract: Asymptotic properties of solutions of difference equation of the form
\[ \Delta ^m x_n=a_n\varphi _n(x_{\sigma (n)})+b_n \]
are studied. Conditions under which every (every bounded) solution of the equation $\Delta ^m y_n=b_n$ is asymptotically equivalent to some solution of the above equation are obtained. Moreover, the conditions under which every polynomial sequence of degree less than $m$ is asymptotically equivalent to some solution of the equation and every solution is asymptotically polynomial are obtained. The consequences of the existence of asymptotically polynomial solution are also studied.

AMSclassification: primary 39A10.

Keywords: difference equation, asymptotic behavior, asymptotically polynomial solution.