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Periodicity and stability of linear Volterra difference systems. (English) Zbl 0796.39004
A sufficient condition for the existence of a unique $$N$$-periodic solution of the difference equation $$x_{n+1} = A_ nx_ n + \sum^ n_{i=- \infty} B_{ni} x_ i + c_ n$$, $$n \geq 0$$, where $$N$$ is a positive integer, $$(A_ n)$$, $$(B_{ni})$$ are $$N$$-periodic sequences of $$k$$-dimensional matrices with $$\sum^ n_{i=- \infty} | B_{ni} |<\infty$$ for all $$n \geq 0$$ and where $$(c_ n)$$ is an $$N$$-periodic sequence of $$k$$-dimensional column vectors, is given. Moreover, a sufficient condition for the uniform asymptotic stability of the zero solution of the difference equation $$x_{n+1} = A_ n x_ n + \sum^ n_{i=0} B_{ni} x_ i$$, $$n \geq 0$$, where $$A_ n$$, $$B_{ni}$$ are $$k$$-dimensional matrices, is given.
Reviewer: H.Länger (Wien)

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