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A characterization of periodic solutions for time‐fractional differential equations in UMD spaces and applications
Author(s) -
Keyantuo Valentin,
Lizama Carlos
Publication year - 2011
Publication title -
mathematische nachrichten
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.913
H-Index - 50
eISSN - 1522-2616
pISSN - 0025-584X
DOI - 10.1002/mana.200810158
Subject(s) - mathematics , multiplier (economics) , banach space , fractional calculus , lp space , mathematical analysis , pure mathematics , differential equation , characterization (materials science) , physics , optics , economics , macroeconomics
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