Geodesic
Asymptotic properties of solutions of nonautonomous difference equations
Archivum mathematicum, Tome 46 (2010) no. 1, pp. 1-11.

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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.
Classification : 39A10
Keywords: difference equation; asymptotic behavior; asymptotically polynomial solution 
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     author = {Migda, Janusz},
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Migda, Janusz. Asymptotic properties of solutions of nonautonomous difference equations. Archivum mathematicum, Tome 46 (2010) no. 1, pp. 1-11. http://geodesic.mathdoc.fr/item/ARM_2010__46_1_a0/