Bohl-Perron type stability theorems for linear singular difference equations. (English) Zbl 1406.39022
The paper deals with the linear singular difference equation
\[
E_n y(n+1) = A_n y(n) + q_n, \qquad n = n_0,n_0 + 1,\dots,
\]
where \(q_n \in \mathbb R^d,\) \(E_n\) and \(A_n\) are \(d \times d\)-matrices, \({\mathrm{rank}} (E_n) = r = {\mathrm{const}} < d\) for all \(n \geq n_0.\)
Using a projector-based approach the authors define the Cauchy operator associated with the corresponding homogeneous system and give definitions of stability and exponential stability.
The authors prove three Bohi-Perron type theorems about the relation between the exponential stability of homogeneous system and the boundedness of solutions of the considered system.
Using a projector-based approach the authors define the Cauchy operator associated with the corresponding homogeneous system and give definitions of stability and exponential stability.
The authors prove three Bohi-Perron type theorems about the relation between the exponential stability of homogeneous system and the boundedness of solutions of the considered system.
Reviewer: Victor I. Tkachenko (Kyïv)
MSC:
| 39A30 | Stability theory for difference equations |
| 39A06 | Linear difference equations |
| 39A22 | Growth, boundedness, comparison of solutions to difference equations |
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