Periodic solutions of integro-differential equations in Banach space having Fourier type

The aim of this work is to study the existence of a periodic solutions of integro-differential equations d dt [x(t)− L(x t)] = A[x(t)− L(x t)]+ G(x t)+ t −∞ a(t− s)x(s)ds+ f (t), (0 ≤ t ≤ 2π) with the periodic condition x(0) = x(2π), where a ∈ L 1 (R +). Our approach is based on the M-boundedness of linear operators, Fourier type, B s p,q-multipliers and Besov spaces