Bohl-Perron type stability theorems for linear singular difference equations
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.
scientific article; zbMATH DE number 6911881