On a neutral functional differential equation in a fading memory space

AbstractThe linear autonomous, neutral system of functional differential equations ddt (μ ∗ x(t) + ƒ(t)) = v ∗ s(t) + g(t) (t ⩾ o), (∗) x(t) = ϑ(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 ƒ, g, and ϑ are Cn-valued, continuous or locaily integrable functions, bounded with respect to a fading memory norm. Conditions which imply that solutions of (∗) 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 (∗) into a stable part and an unstable part