A sharp version of Henry's theorem on small solutions

AbstractA small solution of a linear autonomous retarded functional differential equation (rfde) is a solution that goes to zero faster than any exponential. Henry's theorem on small solutions states that there exists a time T—depending on the dimension and the delay of the equation—such that all small solutions vanish a.e. for t ⩾ T. In this paper we shall give an explicit characterisation for the smallest possible time T, in terms of properties of the specific kernel. This characterisation helps to establish new results concerning completeness and F-completeness of the generalized eigenfunctions of the infinitesimal generator of the C0-semigroup associated with the linear autonomous rfde