A characterization of periodic solutions for time‐fractional differential equations in UMD spaces and applications |
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| Authors: | Valentin Keyantuo Carlos Lizama |
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| Affiliation: | 1. University of Puerto Rico, Department of Mathematics, Faculty of Natural Sciences, P.O. Box 70377, San Juan PR 00936‐8377, USA, Phone: +1?787?764?0000 (Extension 4692), Fax: (787) 281‐0651;2. Universidad de Santiago de Chile, Departamento de Matemática y Ciencia de la Computación, Facultad de Ciencia, Casilla 307‐Correo 2, Santiago, Chile |
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| 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 |
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| Keywords: | Well posedness fractional problems UMD spaces R‐boundedness MSC (2010) 35B10 42A45 35B65 44A45 |
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