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

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. (*).

### MSC:

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

### Keywords:

linear difference equations; asymptotic stability
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