The L^p-version of the generalised Bohl-Perron principle for vector equations with delay

We consider the equation ẏ = Ey, where Ey(t) = ∫ d _τR (t,τ)y(t - τ) (t ≥ 0) with an n x n-matrix-valued function R(t,τ). It is proved that, if for a p ≥ 1, the non-homogeneous equation ẋ = Ex + f with the zero initial condition, for any f ∈ L^p, has a solution x ∈ L^p, then the considered homogeneous equation is exponentially stable. By that result, sharp stability conditions are derived for vector functional differential equations 'close' to autonomous ones and for equations with small delays.