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The aim of this work is to study the existence of a periodic solutions ofintegro-differential equations d dt [x(t)-- L(x t)] = A[x(t)-- L(x t)]+ G(x t)+t --$\infty$ a(t-- s)x(s)ds+ f (t), (0 $\le$ t $\le$ 2$\pi$) with the periodiccondition x(0) = x(2$\pi$), where a $\in$ L 1 (R +). Our approach is based onthe M-boundedness of linear operators, Fourier type, B s p,q-multipliers andBesov spaces.

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