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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:
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


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