On a neutral functional differential equation in a fading memory space

The linear autonomous, neutral system of functional differential equations $\text{d}\text{dt}\text{(μ 1 x(t) + ?(t)) = v 1 s(t) + g(t) (t ? o)}$, (1) x(t) = ?(t) (t ? 0), in a fading memory space is studied. Here μ and ν are matrix-valued measures supported on 0, ∞), finite with respect to a weight function, and $\text{?, g,}\text{and}\text{?}$ are Cⁿ-valued, continuous or locaily integrable functions, bounded with respect to a fading memory norm. Conditions which imply that solutions of (1) can be decomposed into a stable part and an unstable part are given. These conditions are of frequency domain type. The usual assumption that the singular part of μ vanishes is not needed. The results can be used to decompose the semigroup generated by (1) into a stable part and an unstable part.