| Title:
|
On the difference equation $x_{n+1}=\dfrac{a_{0}x_{n}+a_{1}x_{n-1}+\dots +a_{k}x_{n-k}}{b_{0}x_{n}+b_{1}x_{n-1}+\dots +b_{k}x_{n-k}} $ (English) |
| Author:
|
Elabbasy, E. M. |
| Author:
|
El-Metwally, H. |
| Author:
|
Elsayed, E. M. |
| Language:
|
English |
| Journal:
|
Mathematica Bohemica |
| ISSN:
|
0862-7959 (print) |
| ISSN:
|
2464-7136 (online) |
| Volume:
|
133 |
| Issue:
|
2 |
| Year:
|
2008 |
| Pages:
|
133-147 |
| Summary lang:
|
English |
| . |
| Category:
|
math |
| . |
| Summary:
|
In this paper we investigate the global convergence result, boundedness and periodicity of solutions of the recursive sequence \[ x_{n+1}=\frac{a_{0}x_{n}+a_{1}x_{n-1}+\dots +a_{k}x_{n-k}}{b_{0}x_{n}+b_{1}x_{n-1}+\dots +b_{k}x_{n-k}},\,\,\,n=0,1,\dots \,\ \] where the parameters $ a_{i}$ and $b_{i}$ for $i=0,1,\dots ,k$ are positive real numbers and the initial conditions $x_{-k},x_{-k+1},\dots ,x_{0}$ are arbitrary positive numbers. (English) |
| Keyword:
|
stability |
| Keyword:
|
periodic solution |
| Keyword:
|
difference equation |
| MSC:
|
39A10 |
| MSC:
|
39A11 |
| MSC:
|
39A20 |
| MSC:
|
39A22 |
| MSC:
|
39A23 |
| MSC:
|
39A30 |
| idZBL:
|
Zbl 1199.39028 |
| idMR:
|
MR2428309 |
| DOI:
|
10.21136/MB.2008.134057 |
| . |
| Date available:
|
2009-09-24T22:35:25Z |
| Last updated:
|
2020-07-29 |
| Stable URL:
|
http://hdl.handle.net/10338.dmlcz/134057 |
| . |
| Reference:
|
[1] E. M. Elabbasy, H. El-Metwally, E. M. Elsayed: On the periodic nature of some max-type difference equation.Int. J. Math. Math. Sci. 14 (2005), 2227–2239. MR 2177819 |
| Reference:
|
[2] E. M. Elabbasy, H. El-Metwally, E. M. Elsayed: On the difference equation $x_{n+1}=\frac{\alpha x_{n-k}}{\Bigl (\beta +\gamma \prod _{i=0}^{k}x_{n-i}\Bigr )}$.J. Conc. Appl. Math. 5 (2007), 101–113. MR 2292704 |
| Reference:
|
[3] H. El-Metwally, E. A. Grove, G. Ladas, H. D. Voulov: On the global attractivity and the periodic character of some difference equations.J. Difference Equ. Appl. 7 (2001), 837–850. MR 1870725, 10.1080/10236190108808306 |
| Reference:
|
[4] H. El-Metwally, E. A. Grove, G. Ladas, L. C. McGrath: On the difference equation $y_{n+1}= \frac{(y_{n-(2k+1)}+p)}{(y_{n-(2k+1)}+qy_{n-2l})}$.Proceedings of the 6th ICDE, Taylor and Francis, London, 2004. MR 2092580 |
| Reference:
|
[5] G. Karakostas: Asymptotic 2-periodic difference equations with diagonally self-invertible responses.J. Difference Equ. Appl. 6 (2000), 329–335. Zbl 0963.39020, MR 1785059, 10.1080/10236190008808232 |
| Reference:
|
[6] G. Karakostas: Convergence of a difference equation via the full limiting sequences method.Diff. Equ. Dyn. Sys. 1 (1993), 289–294. Zbl 0868.39002, MR 1259169 |
| Reference:
|
[7] V. L. Kocic, G. Ladas: Global Behavior of Nonlinear Difference Equations of Higher Order with Applications.Kluwer Academic Publishers, Dordrecht, 1993. MR 1247956 |
| Reference:
|
[8] W. A. Kosmala, M. R. S. Kulenovic, G. Ladas, C. T. Teixeira: On the recursive sequence $y_{n+1}= \frac{p+y_{n-1}}{qy_{n}+y_{n-1}}$.J. Math. Anal. Appl. 251 (2001), 571–586. MR 1794759 |
| Reference:
|
[9] M. R. S. Kulenovic, G. Ladas, N. R. Prokup: A rational difference equation.Comput. Math. Appl. 41 (2001), 671–678. MR 1822594, 10.1016/S0898-1221(00)00311-4 |
| Reference:
|
[10] M. R. S. Kulenovic, G. Ladas, N. R. Prokup: On the recursive sequence $x_{n+1}=(\alpha x_{n}+\beta x_{n-1})/{(A+x_{n})}$.J. Difference Equ. Appl. 6 (2000), 563–576. MR 1802447 |
| Reference:
|
[11] M. R. S. Kulenovic, G. Ladas, W. S. Sizer: On the recursive sequence $x_{n+1}=(\alpha x_{n}+\beta x_{n-1})/{(\gamma x_{n}+\delta x_{n-1})}$.Math. Sci. Res. Hot-Line 2 (1998), 1–16. MR 1623643 |
| Reference:
|
[12] Wan-Tong Li, Hong-Rui Sun: Dynamics of a rational difference equation.Appl. Math. Comp. 163 (2005), 577–591. MR 2121812, 10.1016/j.amc.2004.04.002 |
| . |
