Smooth robustness of parameterized perturbations of exponential dichotomies (Q710527)

Consider the nonautonomous linear difference equations \[ v_{m+1}=A_{m}v_{m}+B_{m}(\lambda )v_{m}\tag{\(*\)} \] in a Banach space, where \(\lambda \) is a parameter in some open subset of a Banach space, and where \(\lambda \mapsto B_{m}(\lambda )\) is of class \(C^1\) for each \(m\in\mathbb{Z}\). Assuming that the unperturbed dynamics \(v_{m+1}=A_mv_m\) has a nonuniform exponential dichotomy, the authors show that if the maps \(B_{m}\) are of class \(C^1\) in \(\lambda\), then the stable and unstable subspaces of the exponential dichotomies obtained from the perturbation are also of class \(C^1\) in \(\lambda \).