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On the Cushing-Henson conjecture, delay difference equations and attenuant cycles. (English) Zbl 1158.39004
Consider a difference equation of the form
\[ x_{n+1}=f(K_{n},x_{n-h_{1}(n)},\dots,x_{n-h_{r}(n)}), \quad n\in \{0,1,2,\dots\}, \tag{1} \] where \(r\) is a fixed positive integer, \(\{ K_{n}\} _{n=0}^\infty\) is a positive \(k\)-periodic sequence, each \(h_{q}\) is a nonnegative sequence with period \(l\) such that the function \(H_{q}\) defined by \( H_{q}(n)= (n-h_{q}(n))\bmod l\) is a one to one map of the set \(\{ 0,1,\dots,l-1\}\) onto itself, and \(f:[0,\infty)^{r+1}\rightarrow [0,\infty)\) is a continuous function satisfying
\[ f(K_n,K_n,\dots,K_n)= K_n, \] and
\[ f(K_n,x,\dots,x)\begin{cases} >x &\text{if }0<x<K_n,\\ <K_n &\text{if }x>K_n.\end{cases} \] A sufficient condition (based on a concavity assumption on \(f\)) is obtained for a positive \(m\)-cycle of (1) to be attenuant, where an \(m\)-cycle of a periodic solution \(\left\{ x_{n}\right\} \) of (1) is said to be attenuant if
\[ \frac{1}{m}\sum_{j=1}^{m}x_{j}<\frac{1}{k}\sum_{j=1}^{k}K_{j}. \] A sufficient condition for the global attractivity of the positive \(p\)-periodic solution of the so called Beverton-Holt equation
\[ x_{n+1}= \frac{\lambda x_{n}}{1+(\lambda -1)(x_{n-p}/K_n)}, \quad n\in \{0,1,2,\dots\}, \] where \(\lambda >1\) and \(\{K_n\}\) is a positive \(p\)-periodic sequence, is also obtained.

MSC:
39A11 Stability of difference equations (MSC2000)
92D25 Population dynamics (general)
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