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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.

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