Let A and M be closed linear operators defined on a complex Banach space X and let a ∈ L1(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
![](//static.cambridge.org/content/id/urn%3Acambridge.org%3Aid%3Aarticle%3AS0013091513000606/resource/name/S0013091513000606_eqnU1.gif?pub-status=live)
with initial condition Mu(0) = Mu(2π), 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.