Nonexistence of periodic solutions and asymptotically periodic solutions for fractional differential equations

► Nonexistence of periodic solutions to fractional differential equations is firstly shown. ► Existence and uniqueness of S(Sv)-asymptotically T-periodic solutions to fractional differential equations are presented. ► More interesting results are extended to a subset of a Fréchet space. Using the final value theorem of Laplace transform, it is firstly shown that nonhomogeneous fractional Cauchy problem does not have nonzero periodic solution. Secondly, two basic existence and uniqueness results for asymptotically periodic solution of semilinear fractional Cauchy problem in an asymptotically periodic functions space. Furthermore, existence and uniqueness results are extended to a closed, nonempty and convex set which is a subset of a Fréchet space. Some examples are given to illustrate the results.