# zbMATH — the first resource for mathematics

Stability and periodicity of difference equations with finite delay. (English) Zbl 0819.39006
Using a Lyapunov type discrete functional or function the authors give sufficient conditions for the zero solution of the difference system $x(n + 1) = F \biggl( n, x(n), x \bigl( n - \tau_ 1 (n) \bigr), \dots, x \bigl( n - \tau_ m (n) \bigr) \biggr) \tag{+}$ with finite delays $$\tau_ i$$, to be uniformly asymptotically stable or a global attractor for (+).
In the second section some kind of Alekseev formula is presented for the solution $$y(n, n_ 0, \varphi)$$ of the system $\Delta y(n) = \sum^ n_{j = n - r} B(n,j)y(j) + h(n). \tag{++}$ Moreover, the existence and uniqueness of a periodic solution of (++) are established provided that the homogeneous system is uniformly asymptotically stable.

##### MSC:
 39A10 Additive difference equations 39A11 Stability of difference equations (MSC2000)