Dichotomies with no invariant unstable manifolds for autonomous equations (Q416327)

From the abstract: This paper analyzes the existence of exponential dichotomies for a well-posed autonomous differential equation that generates a \(C_0\)-semigroup. The novelty of the work consists in the fact that the authors do not assume \(T(t)\)-invariance of the unstable manifolds. Roughly speaking, it is shown that if the solution of the corresponding inhomogeneous difference equation belongs to any sequence space (on which the right shift is an isometry) for every inhomogeneity from the same class of sequence spaces, then the continuous-time solutions of the autonomous homogeneous differential equation will exhibit a exponential dichotomic behaviour.NEWLINENEWLINEThis approach has many advantages among which the authors emphasize that the aforementioned condition is very general and that from discrete-time conditions, information about the continuous-time behaviour of solutions can be obtained.