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.
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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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/

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