A Characterization of Periodic Solutions for Time-Fractional Differential Equations in Umd Spaces and Applications

We study the fractional differential equation D?u(t) + BD?u(t) + Au(t) = f(t), 0 ? t ? 2? (0 ? ? < ? ? 2) in periodic Lebesgue spaces Lp(0, 2?; 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)?((ik)? + (ik)?B + A)-1}k?Z and {(ik)?B((ik)? + (ik)?B + A)-1}k?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. © 2011 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim.