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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.

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