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

Kuruklis, Spiridon A.

J. Math. Anal. Appl. 188, No. 3, 719-731 (1994).

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 \(k\).

Reviewer: Bernd Aulbach (Augsburg)

MSC:

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

Keywords:

linear difference equations; asymptotic stability; delay