zbMATH — the first resource for mathematics

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.

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
Full Text: EMIS