A characterization of periodic solutions for time‐fractional differential equations in UMD spaces and applications

We study the fractional differential equation (*) D α u ( t ) + BD β u ( t ) + Au ( t ) = f ( t ),  0 ⩽ t ⩽ 2π (0 ⩽ β < α ⩽ 2) in periodic Lebesgue spaces L p (0,  2π; 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 ) α (( ik ) α + ( ik ) β B + A ) −1 } k ∈ Z and {( ik ) β B (( ik ) α + ( ik ) β 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