Maximal Regularity for Degenerate Differential Equations with Infinite Delay in Periodic Vector-Valued Function Spaces

Let A and M be closed linear operators defined on a complex Banach space X and let a â?? L 1(â"+) 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 with initial condition Mu(0) = Mu(2?), 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. Copyright © Edinburgh Mathematical Society 2013.