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Stable periodic solutions in a discrete periodic logistic equation. (English) Zbl 1049.39017
Consider the discrete logistic equation \[ x(n+1)=x(n) \exp \left[r(n)(1-\frac{x(n)}{K(n)})\right],\quad n\in N,\tag{\(*\)} \] where \(x(0)>0,\) \(\{r(n)\}\) and \(\{K(n)\}\) are strictly positive sequences of real numbers defined for \(n\in N=\{0,1,2,\dots\}.\) In addition there exist positive constants \(r_{*},r^{*},K_{*},\) and \(K^{*}\) such that \(0<r_{*}\leq r(n)\leq r^{*},\,0\leq K_{*}\leq K(n)\leq K^{*}\), \(n\in N.\) The equation (\(*\)) has been considered by S. Mohamad and K. Gopalsamy [Tohoku Math. J., II. Ser. 52, No. 1, 107–125 (2000; Zbl 0954.39005), correction ibid. 53, No. 4, 629–631 (2001)].
The authors give counterexamples to show that the results obtained by Mohamad and Gopalsamy are false. In addition, they correct some of those results. The authors prove the following: Assume that \(\{r(n)\}\) and \(\{K(n)\}\) are positive periodic sequences with a common positive period \(\omega\). Then
(i) there exists an \(\omega\)-periodic solution for equation (\(*\))
(ii) for every positive solution \(\{x(n)\}\,\) of (\(*\)), it is true that \(\lim_{n\rightarrow \infty }(x(n)-\widetilde{x}(n))=0,\) where \(\{\widetilde{x}(n)\}\) is a periodic solution of equation (\(*\)) provided that \(\frac{K^{*}}{K_{*}}\exp (r^{*}-1)\leq 2,\) where \(r^{*}=\max_{n\in N}\{r(n)\}\,,K_{*}=\min_{n\in N}\{K(n)\},\) and \(K^{*}=\max_{n\in N}\{K(n)\}.\)
Reviewer: Fozi Dannan (Doha)

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