Asymptotic properties of solutions of nonautonomous difference equations

Janusz Migda

Archivum Mathematicum (2010)

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

Abstract

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Asymptotic properties of solutions of difference equation of the form Δ m x n = a n ϕ n ( x σ ( n ) ) + b n are studied. Conditions under which every (every bounded) solution of the equation Δ 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
  2. Gleska, A., Werbowski, J., 10.1006/jmaa.1998.6094, J. Math. Anal. Appl. 226 (2) (1998), 456–465. (1998) Zbl0929.39002MR1650201DOI10.1006/jmaa.1998.6094
  3. Li, Wan-Tong, Agarwal, R. P., Positive solutions of higher-order nonlinear delay difference equations, Comput. Math. Appl. 45 (6-7) (2003), 1203–1211. (2003) Zbl1054.39006MR2000590
  4. Migda, J., Asymptotic properties of solutions of higher order difference equations, submitted. Zbl0702.39002
  5. Migda, J., Asymptotically linear solutions of second order difference equations, submitted. 
  6. Migda, J., Asymptotic behavior of solutions of nonlinear difference equations, Math. Bohem. 129 (4) (2004), 349–359. (2004) Zbl1080.39501MR2102609
  7. Migda, M., Migda, J., 10.1016/S0362-546X(01)00581-8, Nonlinear Anal. 47 (7) (2001), 4687–4695. (2001) Zbl1042.39509MR1975862DOI10.1016/S0362-546X(01)00581-8
  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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