##
**Extreme stability and almost periodicity in a discrete logistic equation.**
*(English)*
Zbl 0954.39005

Tohoku Math. J., II. Ser. 52, No. 1, 107-125 (2000); correction ibid. 53, No. 4, 629-631 (2001).

In this interesting paper, the authors obtain a sufficient condition for the existence of a globally asymptotically stable positive almost periodic solution to the discrete logistic equation
\[
x(n+1)= x(n) \exp \left[r(n) \left(1-{x(n) \over K(n)}\right) \right],\quad n\in Z= \{0,1,2, \dots\};\;x(0)>0,\tag{1}
\]
where \(\{r(n)\}\), \(\{K(n)\}(n\in Z)\) are strictly positive sequences of real numbers. The following main results are established in the paper:

Theorem 3.2. Suppose that \[ 0<\inf_{n\in Z}r(n)\leq r(n)\leq \sup_{n\in Z}r(n)< \infty, \quad 0<\inf_{n\in Z}K(n)\leq K(n)\leq\sup_{n\in Z}K(n) <\infty \] hold. Then equation (1) is extremely stable in the sense of T. Yoshizawa, i.e., for any two positive solutions \(\{x(n)\}\), \(\{y(n)\}\) to (1) we have \(\lim_{n\to\infty}|x(n)-y(n)|=0\).

Theorem 4.1. Suppose that the strictly positive sequences \(\{r(n)\}\), \(\{K(n)\}(n\in Z)\) in Theorem 3.2 are almost periodic and the additional condition \(\sup_{n\in Z}r(n)<2\) holds. Then equation (1) has a unique globally asymptotically stable almost periodic solution.

The last existence condition is also necessary for the corresponding autonomous equation of equation (1). The authors conjectured in the paper that the condition on \(\{r(n)\}\) can be replaced by \[ 0<\lim_{n \to\infty} \left[{1\over n}\sum^{n-1}_{j=0} r(j)\right]<2. \]

Theorem 3.2. Suppose that \[ 0<\inf_{n\in Z}r(n)\leq r(n)\leq \sup_{n\in Z}r(n)< \infty, \quad 0<\inf_{n\in Z}K(n)\leq K(n)\leq\sup_{n\in Z}K(n) <\infty \] hold. Then equation (1) is extremely stable in the sense of T. Yoshizawa, i.e., for any two positive solutions \(\{x(n)\}\), \(\{y(n)\}\) to (1) we have \(\lim_{n\to\infty}|x(n)-y(n)|=0\).

Theorem 4.1. Suppose that the strictly positive sequences \(\{r(n)\}\), \(\{K(n)\}(n\in Z)\) in Theorem 3.2 are almost periodic and the additional condition \(\sup_{n\in Z}r(n)<2\) holds. Then equation (1) has a unique globally asymptotically stable almost periodic solution.

The last existence condition is also necessary for the corresponding autonomous equation of equation (1). The authors conjectured in the paper that the condition on \(\{r(n)\}\) can be replaced by \[ 0<\lim_{n \to\infty} \left[{1\over n}\sum^{n-1}_{j=0} r(j)\right]<2. \]

Reviewer: Yang En-Hao (Guangzhou)

### MSC:

39A11 | Stability of difference equations (MSC2000) |

92D25 | Population dynamics (general) |

92D40 | Ecology |

### Keywords:

global asymptotic stability; almost periodic solution; discrete logistic equation; positive solutions
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\textit{S. Mohamad} and \textit{K. Gopalsamy}, Tôhoku Math. J. (2) 52, No. 1, 107--125 (2000; Zbl 0954.39005)

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