Ponce, R (Ponce, Rodrigo)[ 2 ] Univ Talca, Inst Matemat & Fis, Talca, ChileLet A and M be closed linear operators defined on a complex Banach space X and let a is an element of L-1(R+) be a scalar kernel. We use operator-valued Fourier multipliers techniques to obtain necessary and sufficient conditions to guarantee the existence and uniqueness of periodic solutions to the equation d/dt (Mu(t)) = Au(t) + integral(t)(-infinity) a(t - s)Au(s)ds + f(t), t > 0, with initial condition Mu(0) = Mu(2 pi), solely in terms of spectral properties of the data. Our results are obtained in the scales of periodic Besov, Triebel-Lizorkin and Lebesgue vector-valued function spaces.