On the difference equation of higher order (Q2870953)
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scientific article; zbMATH DE number 6248706
| Language | Label | Description | Also known as |
|---|---|---|---|
| English | On the difference equation of higher order |
scientific article; zbMATH DE number 6248706 |
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21 January 2014
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rational difference equation
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global asymptotic stability
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equilibrium point
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On the difference equation of higher order (English)
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The authors study the asymptotic behavior of the solutions of the difference equation NEWLINE\[NEWLINE x_{n+1}=\frac{a+a^{(1-\alpha)/2}\,x_nx_{n-k}^\alpha}{x_n+a^{(1-\alpha)/2}\,x_{n-k}^\alpha}, \quad n\geq0, NEWLINE\]NEWLINE where \(k\in{\mathbb N}\), \(a\) and \(\alpha\) are positive, and the initial conditions \(x_0, x_{-1}, \dots, x_{-k}\) are positive. The main result states that the positive equilibrium \(\bar x=\sqrt{a}\) is globally asymptotically stable.
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