Periodic solutions of a fractional neutral equation with finite delay

In this paper, we prove the maximal regularity property of an abstract fractional differential equation with finite delay on periodic Besov and Triebel–Lizorkin spaces and use these results to guarantee the existence and uniqueness of periodic solution of a neutral fractional differential equation with finite delay. The main tool used to achieve our goal is an operator-valued version of Miklhin’s Fourier multiplier theorem and fixed-point argument.