×

Global attractivity of the equilibrium of a nonlinear difference equation. (English) Zbl 1014.39003

Summary: The authors consider the nonlinear difference equation \[ x_{n+1}=\alpha x_n + x_{n-k}f(x_{n-k}), \quad n=0, 1,\dots \] with \(f'(x)<0\). They give sufficient conditions for the unique positive equilibrium 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.

MSC:

39A11 Stability of difference equations (MSC2000)
39A12 Discrete version of topics in analysis
92D25 Population dynamics (general)
PDF BibTeX XML Cite
Full Text: DOI EuDML

References:

[1] J. R. Graef and C. Qian: Global stability in a nonlinear difference equation. J. Differ. Equations Appl. 5 (1999), 251-270. · Zbl 0945.39002
[2] A. F. Ivanov: On global stability in a nonlinear discrete model. Nonlinear Anal. 23 (1994), 1383-1389. · Zbl 0842.39005
[3] G. Karakostas, Ch. G. Philos and Y. G. Sficas: The dynamics of some discrete population models. Nonlinear Anal. 17 (1991), 1069-1084. · Zbl 0760.92019
[4] V. L. Kocic and G. Ladas: Global Behavior of Nonlinear Difference Equations of Higher Order with Applications. Kluwer Academic Publishers, Dordrecht, 1993. · Zbl 0787.39001
[5] M. C. Mackey and L. Glass: Oscillation and chaos in physiological control systems. Science 197 (1977), 287-289. · Zbl 1383.92036
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.