A sharp version of Henry's theorem on small solutions

Abstract A 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 C 0 -semigroup associated with the linear autonomous rfde.