Abstract: Let $X$ be a complex Banach space and let $B$ and $C$ be two closed linear operators on $X$ satisfying the condition $D(B)\subset D(C)$, and let $d\in L^1(\mathbb{R}_+)$ and $0 \leq \beta < \alpha\leq 2$. We characterize the well-posedness of the fractional integro-differential equations $D^\alpha u(t) + CD^\beta u(t)$ $= Bu(t) + \int_{-\infty}^t d(t-s)Bu(s){\rm d}s + f(t),\ (0\leq t\leq 2\pi)$ on periodic Lebesgue-Bochner spaces $L^p(\mathbb{T}; X)$ and periodic Besov spaces $B_{p,q}^s(\mathbb{T}; X)$.

Key words: Lebesgue-Bochner spaces, fractional integro-differential equations, multiplier, well-posedness