Asymptotic properties of solutions of nonautonomous difference equations

summary: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