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Global behavior for a fourth-order rational difference equation. (English) Zbl 1083.39007

For the difference equation \[ x_{n+1}= \frac{x_{n-1} x_{n-2} x_{n-3}+ x_{n-1}+ x_{n-2}+ x_{n-3}+ a}{x_{n-1} x_{n-2}+ x_{n-1} x_{n-3}+ x_{n-2} x_{n-3}+ 1+ a} \] with \(a\geq 0\) the semicycle structure of positive, strictly oscillating solutions is investigated, and the global asymptotic stability of the equilibrium 1.

MSC:

39A11 Stability of difference equations (MSC2000)
39A20 Multiplicative and other generalized difference equations
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