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Connections between the stability of a Poincaré map and boundedness of certain associate sequences. (English) Zbl 1281.39009
Summary: Let $$m\geq 1$$ and $$N\geq 2$$ be two natural numbers and let $${\mathcal{U}}=\{U(p, q)\}_{p\geq q\geq 0}$$ 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 Poincaré 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.

MSC:
 39A22 Growth, boundedness, comparison of solutions to difference equations 34D05 Asymptotic properties of solutions to ordinary differential equations 47A50 Equations and inequalities involving linear operators, with vector unknowns
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