In this paper, we establish a new composition theorem for (ω,c)-asymptotically periodic functions. Then, we use the Banach contraction principle to investigate the existence and uniqueness of (ω,c)-asymptotically periodic mild solutions to the fractional integro-differential equation u’(t)=\frac{1}{\Gamma(\alpha-1)}\int_{0}^{t}(t-\tau)^{\alpha-2}Au(\tau)d\tau+F(t,u_t), t≥0 and u_0=\phi \in \mathcal{B}(\mathbb{X}), where \mathcal{B}(\mathbb{X}) is a linear space of functions defined from (-∞,0] \longrightarrow \mathbb{X} and A is a closed but not necessarily bounded linear operator of sectorial type \varpi<0.