A note on asymptotic stability conditions for delay difference equations. (English) Zbl 1079.39006

Kaewong, T.; Lenbury, Y.; Niamsup, P.

Int. J. Math. Math. Sci. 2005, No. 7, 1007-1013 (2005).

The authors establish the following necessary and sufficient condition for the asymptotic stability of the zero solution of the linear difference equation: \[ x_{n+1}+p\sum_{j=1}^N x_{n-k+(j-1)l}=0,\tag{*} \] where \(n=0,1,2,\dots\), \(p\) is a real number, and \(k,~l\), and \(N\) are positive integers such that \(k\geq (N-1)l\).
Theorem. Let \(k,~l\), and \(N\) be positive integers with \(k\geq (N-1)l\). Then the zero solution of Eq. (*) is asymptotically stable if and only if \[ -\frac{1}{N} < p < p_{\min}, \] where \(p_{\min}\) is the smallest positive real value of \(p\) for which the characteristic equation, \(F(z)=z^{k+1} +p \left( z^{(N-1)l}+ \cdots +z^l+1 \right)=0\), associated with Eq. (*) has a root on the unit circle.
Also, they prove the following result.
Lemma: If \(z\) is a root of the characteristic equation associated with Eq. (*) which lies on the unit circle, then \[ z=e^{w_n i}\quad \text{and} \quad p=(-1)^{m+1} \frac{\sin(lw_m/2)}{\sin (Nlw_m/2)} \] for some \(m=0,1,\dots,M-1\), where \(w_m=(2m/M)\pi\) and \(M=2k+2-(N-1)l\). Conversely, if \(p\) is as given above, then \(z=e^{w_n i}\) is a root of the characteristic equation associated with Eq. (*).

Reviewer: Raghib Abu-Saris (Sharjah)

MSC:

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

Keywords:

linear difference equations; asymptotic stability