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

Abstract

We study the fractional differential equation (*) Dαu(t) + BDβu(t) + Au(t) = f(t), 0 ⩽ t ⩽ 2π (0 ⩽ β < α ⩽ 2) in periodic Lebesgue spaces Lp(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)βB + A)−1}k∈Z and {(ik)βB((ik)α + (ik)βB + 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