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On the asymptotic stability of a class of linear difference equations. (English) Zbl 0866.39001

The author proves: A sufficient and necessary condition for all roots of the equation \[ z^{n+1}- \alpha z^n+\beta=0, \qquad \alpha,\beta\in\mathbb{R}, \quad n\geq 1 \] are the unit disc \(|z|<1\) is \((\beta,\alpha)\) be in the open region (without borders) bounded by the curves \(\alpha= \beta+1\), \(\alpha= \beta-1\), \((\beta= {{\sin\theta} \over {\sin(n\theta)}}\), \(\alpha= {{\sin[(n+1)\theta]} \over {\sin(n\theta)}}\); \(0<\theta< {\pi\over {n+1}})\), \((\beta=- {{\sin\theta} \over {\sin(n\theta)}}\), \(\alpha=- {{\sin[(n+1)\theta]} \over {\sin(n\theta)}}\); \(0<\theta< {\pi\over {n+1}})\) in the case \(n\) is even; \(\alpha=\beta+1\), \(\alpha=-\beta-1\), \((\beta={{\sin\theta} \over {\sin(n\theta)}}\), \(\alpha= {{\sin[(n+1)\theta]} \over {\sin(n\theta)}}\); \(0<\theta< {\pi\over {n+1}})\), \(\beta= {{\sin\theta} \over {\sin(n\theta)}}\), \(\alpha=- {{\sin[(n+1)\theta]} \over {\sin(n\theta)}}\); \(0<\theta< {\pi\over {n+1}})\) in the case \(n\) is odd.
As a consequence necessary and sufficient conditions are given for all solutions of the difference equation \(x_{k+1}-\alpha x_k+\beta x_{k-n}=0\), \(\alpha,\beta\in \mathbb{R}\), \(n\geq 1\) asymptotically approach zero.

MSC:

39A11 Stability of difference equations (MSC2000)
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