We first investigate in this paper further properties of the new concept of (ω, c)-periodic functions; then we apply the results to study the existence of (ω, c)-periodic mild solutions of the fractional differential equations D α t (u(t) − F 1 (t, u(t))) = A(u(t) − F 1 (t, u(t))) + D α−1 t F 2 (t, u(t)), t ∈ R, where 1 < α < 2, A : D(A) ⊆ X → X is a linear densely defined operator of sectorial type on a complex Banach space X, F 1 , F 2 : R×X → X are two (ω, c)-periodic functions satisfying suitable conditions in the second variable. The fractional derivative is understood in the sense of Riemann-Liouville.