## 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)

### Keywords:

asymptotic stability; linear difference equations
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