## 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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