On bounded solutions of a difference equation with jumps of the operator coefficient (Q1707069)

The authors study the difference equations \[ x_{n+1}=Ax_n+y_n,\;\;n\geq 1, \] \[ x_{n+1}=Bx_n+y_n,\;\;n\leq 0, \] on a finite-dimensional complex Banach space \(X\). They obtain necessary and sufficient conditions for the operators \(A\) and \(B\) such that for any bounded sequence \(\{y_n, n\in \mathbb{Z}\}\) in \(X\), the above difference equations have a unique bounded solution \(\{x_n, n\in \mathbb{Z}\}\) in the space \(X\).