Geodesic
Global attractivity of the equilibrium of a nonlinear difference equation
Czechoslovak Mathematical Journal, Tome 52 (2002) no. 4, pp. 757-769

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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.
Classification : 37N25, 39A10, 39A11, 39A12, 92D25
Keywords: nonlinear difference equation; global attractivity; oscillation 
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     author = {Graef, J. R. and Qian, C.},
     title = {Global attractivity of the equilibrium of a nonlinear difference equation},
     journal = {Czechoslovak Mathematical Journal},
     pages = {757--769},
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Graef, J. R.; Qian, C. Global attractivity of the equilibrium of a nonlinear difference equation. Czechoslovak Mathematical Journal, Tome 52 (2002) no. 4, pp. 757-769. http://geodesic.mathdoc.fr/item/CMJ_2002__52_4_a8/