On the asymptotic behavior of the solutions of autonomous equations without unstable invariant manifolds

Abstract

We investigate strongly continuous semigroups {T(t)}t≥0 on Banach space X by means of discrete time methods applied to a discrete counterpart {T(1)n}n≥0. This kind of approach is not new and can be traced back at least to Henry’s monograph (see Henry (1981) [13]). The semigroup {T(t)}t≥0 is denoted as hyperbolic, if the usual exponential dichotomy conditions are satisfied, i.e. in particular if invertibility is given on the unstable subspace. A weaker version (without assuming the T(t)-invariance of the unstable subspace or even more the invertibility of the operators T(t) on the unstable subspace) is denoted as exponential dichotomy. The latter approach is due to Aulbach and Kalkbrenner dealing with difference equations since in Aulbach and Kalkbrenner (2001) [21] it is clearly indicated that this dichotomy notion lacks L1-robustness. We show that admissibility properties of the sequence spaces (ℓp(N,X),ℓq(N,X)) are sufficient for an exponential dichotomy of {T(t)}t≥0 (see Theorem 3.1). Also if we assume the T(t)-invariance of the unstable subspace we show that the above admissibility condition implies the invertibility of the operators T(t) on the unstable subspace (see Theorem 3.2). Thus, hyperbolicity of {T(t)}t≥0 turns out to be equivalent to admissibility of the pair (ℓp(N,X),ℓq(N,X)) (see Corollary 3.1).