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Oscillations and global attractivity in a discrete delay logistic model. (English) Zbl 0799.39004
The paper is devoted to the study of the discrete analogue of the logistic equation (1) $$x_{n + 1} = \alpha x_ n/(1 + \beta x_{n - k})$$, $$n = 0, 1, 2, \dots$$, where (2) $$\alpha \in (1,\infty)$$, $$\beta \in (0,\infty)$$, and $$k \in N = \{0, 1, 2, \dots\}$$. Conditions for the oscillation and asymptotic stability of all positive solutions of equation (1) about its positive equilibrium $$(\alpha - 1)/ \beta$$ are obtained. It is shown that all positive solutions of equation (1) are bounded and that for $$k=0$$ and $$k=1$$ the positive equilibrium $$(\alpha- 1)/ \beta$$ is a global attractor.

MSC:
 39A10 Additive difference equations 39A11 Stability of difference equations (MSC2000)
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