| Title:
|
Global attractivity of the equilibrium of a nonlinear difference equation (English) |
| Author:
|
Graef, J. R. |
| Author:
|
Qian, C. |
| Language:
|
English |
| Journal:
|
Czechoslovak Mathematical Journal |
| ISSN:
|
0011-4642 (print) |
| ISSN:
|
1572-9141 (online) |
| Volume:
|
52 |
| Issue:
|
4 |
| Year:
|
2002 |
| Pages:
|
757-769 |
| Summary lang:
|
English |
| . |
| Category:
|
math |
| . |
| Summary:
|
The authors consider the nonlinear difference equation \[ x_{n+1}=\alpha x_n + x_{n-k}f(x_{n-k}), \quad n=0, 1,\dots .1 \text{where} \alpha \in (0, 1),\hspace{5.0pt}k \in \lbrace 0, 1, \dots \rbrace \hspace{5.0pt}\text{and}\hspace{5.0pt}f\in C^1[[0, \infty ),[0, \infty )] \qquad \mathrm{(0)}\] with $f^{\prime }(x)<0$. They give sufficient conditions for the unique positive equilibrium of (0.1) to be a global attractor of all positive solutions. The results here are somewhat easier to apply than those of other authors. An application to a model of blood cell production is given. (English) |
| Keyword:
|
nonlinear difference equation |
| Keyword:
|
global attractivity |
| Keyword:
|
oscillation |
| MSC:
|
37N25 |
| MSC:
|
39A10 |
| MSC:
|
39A11 |
| MSC:
|
39A12 |
| MSC:
|
92D25 |
| idZBL:
|
Zbl 1014.39003 |
| idMR:
|
MR1940057 |
| . |
| Date available:
|
2009-09-24T10:56:27Z |
| Last updated:
|
2020-07-03 |
| Stable URL:
|
http://hdl.handle.net/10338.dmlcz/127762 |
| . |
| Reference:
|
[1] J. R. Graef and C. Qian: Global stability in a nonlinear difference equation.J. Differ. Equations Appl. 5 (1999), 251–270. MR 1697059, 10.1080/10236199908808186 |
| Reference:
|
[2] A. F. Ivanov: On global stability in a nonlinear discrete model.Nonlinear Anal. 23 (1994), 1383–1389. Zbl 0842.39005, MR 1306677, 10.1016/0362-546X(94)90133-3 |
| Reference:
|
[3] G. Karakostas, Ch. G. Philos and Y. G. Sficas: The dynamics of some discrete population models.Nonlinear Anal. 17 (1991), 1069–1084. MR 1136230, 10.1016/0362-546X(91)90192-4 |
| Reference:
|
[4] V. L. Kocic and G. Ladas: Global Behavior of Nonlinear Difference Equations of Higher Order with Applications.Kluwer Academic Publishers, Dordrecht, 1993. MR 1247956 |
| Reference:
|
[5] M. C. Mackey and L. Glass: Oscillation and chaos in physiological control systems.Science 197 (1977), 287–289. 10.1126/science.267326 |
| . |
