## 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

Zbl 0954.39005
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### References:

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