Connections between the stability of a Poincare map and boundedness of certain associate sequences

Abstract

Let $m\ge 1$ and $N\ge 2$ be two natural numbers and let ${\mathcal{U}}=\{U(p, q)\}_{p\ge q\ge 0}$ be the $N$-periodic discrete evolution family of $m\times m$ matrices, having complex scalars as entries, generated by ${\mathcal{L}}(\mathbb{C}^m)$-valued, $N$-periodic sequence of $m\times m$ matrices $(A_n).$ We prove that the solution of the following discrete problem $$y_{n+1}=A_ny_n+e^{i\mu n}b,\quad n\in\mathbb{Z}_+,\quad y_0=0$$ is bounded for each $\mu\in\mathbb{R}$ and each $m$-vector $b$ if the Poincare map $U(N, 0)$ is stable. The converse statement is also true if we add a new assumption to the boundedness condition. This new assumption refers to the invertibility for each $\mu\in\mathbb{R}$ of the matrix $V_{\mu}:=\sum\nolimits_{\nu=1}^NU(N, \nu)e^{i\mu \nu}.$ By an example it is shown that the assumption on invertibility cannot be removed. Finally, a strong variant of Barbashin's type theorem is proved.