We study the fractional differential equation (*) D-alpha u(t) + BD beta u(t) + Au(t) = f(t), 0 <= t <= 2 pi (0 <= beta < alpha <= 2) in periodic Lebesgue spaces L-p (0, 2 pi; X) where X is a Banach space. Using functional calculus and operator valued Fourier multiplier theorems, we characterize, in UMD spaces, the well posedness of (*) in terms of R-boundedness of the sets {(ik)(alpha) ((ik)(alpha) + (ik)(beta) B + A)(-1)}k is an element of Z and {(ik)(beta) B((ik)(alpha) + (ik)(beta) B + A)(-1)}k is an element of Z. Applications to the fractional problems with periodic boundary condition, which includes the time diffusion and fractional wave equations, as well as an abstract version of the Basset-Boussinesq-Oseen equation are treated. (C) 2011 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim