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A periodically forced Beverton-Holt equation. (English) Zbl 1023.39013
The authors present an open problem: to prove or disprove the following assertions for \(p\geq 3\).
(a) The periodically forced Beverton-Holt equation \[ x_{n+1}=\frac{rK_n}{K_n+(r-1)x_n}x_n, \quad n=0, 1, 2, \ldots, \] has a positive \(p\)-periodic solution \(\{y_n\}\), and it is globally attracting for \(x_0>0\), where \(r>1\) and \(\{K_n\}\) is a sequence of positive numbers with a base period \(p\geq 1\).
(b) If \(p>2\), the strict inequality \(av(y_n)<av(K_n)\) holds. Here \(av(y_n)=\frac{1}{p}\sum_{i=0}^{p-1}y_i\).

MSC:
39B05 General theory of functional equations and inequalities
39A11 Stability of difference equations (MSC2000)
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[1] DOI: 10.1080/10236190108808308 · Zbl 1002.39003 · doi:10.1080/10236190108808308
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