# Asymptotic properties of solutions of nonautonomous difference equations

Archivum Mathematicum (2010)

• Volume: 046, Issue: 1, page 1-11
• ISSN: 0044-8753

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## Abstract

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Asymptotic properties of solutions of difference equation of the form ${\Delta }^{m}{x}_{n}={a}_{n}{\varphi }_{n}\left({x}_{\sigma \left(n\right)}\right)+{b}_{n}$ 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.

## How to cite

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Migda, Janusz. "Asymptotic properties of solutions of nonautonomous difference equations." Archivum Mathematicum 046.1 (2010): 1-11. <http://eudml.org/doc/37649>.

@article{Migda2010,
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.},
author = {Migda, Janusz},
journal = {Archivum Mathematicum},
keywords = {difference equation; asymptotic behavior; asymptotically polynomial solution; difference equation; asymptotic behavior; asymptotically polynomial solution; Schauder fixed point theorem; sequence spaces},
language = {eng},
number = {1},
pages = {1-11},
publisher = {Department of Mathematics, Faculty of Science of Masaryk University, Brno},
title = {Asymptotic properties of solutions of nonautonomous difference equations},
url = {http://eudml.org/doc/37649},
volume = {046},
year = {2010},
}

TY - JOUR
AU - Migda, Janusz
TI - Asymptotic properties of solutions of nonautonomous difference equations
JO - Archivum Mathematicum
PY - 2010
PB - Department of Mathematics, Faculty of Science of Masaryk University, Brno
VL - 046
IS - 1
SP - 1
EP - 11
AB - 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.
LA - eng
KW - difference equation; asymptotic behavior; asymptotically polynomial solution; difference equation; asymptotic behavior; asymptotically polynomial solution; Schauder fixed point theorem; sequence spaces
UR - http://eudml.org/doc/37649
ER -

## References

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1. Drozdowicz, A., Popenda, J., Asymptotic behavior of the solutions of an n-th order difference equations, Comment. Math. Prace Mat. 29 (2) (1990), 161–168. (1990) MR1059121
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8. Migda, M., Migda, J., 10.1016/j.na.2005.02.005, Nonlinear Anal. 63 (2005), 789–799. (2005) Zbl1160.39306DOI10.1016/j.na.2005.02.005
9. Wang, Z., Sun, J., 10.1080/10236190500539352, J. Differ. Equations Appl. 12 (2006), 419–432. (2006) Zbl1098.39006MR2241385DOI10.1080/10236190500539352
10. Zafer, A., 10.1016/0895-7177(95)00005-M, Math. Comput. Modelling 21 (4) (1995), 43–50. (1995) Zbl0820.39001MR1317929DOI10.1016/0895-7177(95)00005-M
11. Zafer, A., 10.1016/S0898-1221(98)00078-9, Comput. Math. Appl. 35 (10) (1998), 125–130. (1998) MR1617906DOI10.1016/S0898-1221(98)00078-9
12. Zhang, B., Sun, Y., 10.1080/1023619021000048841, J. Differ. Equations Appl. 8 (11) (2002), 937–955. (2002) Zbl1014.39009MR1942433DOI10.1080/1023619021000048841

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