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Nonautonomous Beverton-Holt equations and the Cushing-Henson conjectures. (English) Zbl 1084.39005
The authors consider the nonautonomous Beverton–Holt equation \[ x_{n+1}=\frac{\mu_n K_nx_n}{K_n+(\mu_n-1)x_n}, \quad n=0,1,\ldots \eqno(1) \] where \(\{\mu_n\}\) and \(\{K_n\}\) are sequences with minimal common period \(p \geq 2\). For a positive \(p\)-periodic solution \(\{x_n\}\) the authors establish the inequality \[ \frac{1}{p}\sum^{p-1}_{k=0}x_k < \frac{\mu^*(\mu^*-1)}{\mu_*(\mu_*-1)} \frac{1}{p}\sum^{p-1}_{k=0}K_k, \eqno(2) \] where \(\mu^* = \max(\mu_n)\), \(\mu_* = \min(\mu_n)\). The obtained result is an extension of the conjecture formulated by J. M. Cushing and S. M. Henson [J. Difference Equ. Appl. 8, No. 12, 1119-1120 (2002; Zbl 1023.39013)] in the case when \(\mu_n\) is constant. In the present paper a refinement of (2) is given for \(p=2\). The authors construct an example having a geometric cycle with minimal period \(r < p\). In the case of (1) with constant \(\mu_n\) the authors prove that any geometric cycle must have the same minimal period \(p\) as (1).

MSC:
39A11 Stability of difference equations (MSC2000)
39A12 Discrete version of topics in analysis
39A20 Multiplicative and other generalized difference equations, e.g., of Lyness type
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