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In this paper, we establish some new properties of-periodic and asymptotically-periodic functions, then we apply them to study the existence and uniqueness of mild solutions of these types to the following semilinear fractional differential equations:(1)and(2)wherestands for the Caputo derivative and  A  is a linear densely defined operator of sectorial type on a complex Banach spaceand the functionis-periodic or asymptotically-periodic with respect to the first variable. Our results are obtained using the Leray–Schauder alternative theorem, the Banach fixed point principle and the Schauder theorem. Then we illustrate our main results with an application to fractional diffusion-wave equations.

(ω, c )-periodic and asymptotically (ω, c )-periodic mild solutions to fractional Cauchy problems