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The asymptotic stability of \(x_{n+1}-ax_ n+bx_{n-k}=0\). (English) Zbl 0842.39004

For linear difference equations of the form \[ x_{n+1}= ax_n+ bx_{n-k}, \qquad n\in \mathbb{N}\cup \{0\} \] necessary and sufficient conditions for the asymptotic stability are presented in terms of the real coefficients \(a\), \(b\) and the delay \(k\in \mathbb{N}\cup \{0\}\). The proof of the main result is based on a careful analysis of the dependence of the roots of the characteristic polynomial on the parameters \(a\), \(b\) and \(k\).

MSC:

39A11 Stability of difference equations (MSC2000)
39A10 Additive difference equations
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