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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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