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Global behavior of the difference equation \(x_{n + 1} = x_{n - 1} g(x_n)\). (English) Zbl 1474.39026

Summary: We consider the following difference equation \(x_{n + 1} = x_{n - 1} g(x_n)\), \(n = 0,1, \ldots\), where initial values \(x_{- 1}, x_0 \in [0, + \infty)\) and \(g : [0, + \infty) \rightarrow(0,1]\) is a strictly decreasing continuous surjective function. We show the following. (1) Every positive solution of this equation converges to \(a, 0, a, 0, \ldots\), or \(0, a, 0, a, \ldots\) for some \(a \in [0, + \infty)\). (2) Assume \(a \in(0, + \infty)\). Then the set of initial conditions \((x_{- 1}, x_0) \in(0, + \infty) \times(0, + \infty)\) such that the positive solutions of this equation converge to \(a, 0, a, 0, \ldots\), or \(0, a, 0, a, \ldots\) is a unique strictly increasing continuous function or an empty set.

MSC:

39A22 Growth, boundedness, comparison of solutions to difference equations
39A20 Multiplicative and other generalized difference equations
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References:

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