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Well‐posedness of degenerate differential equations with fractional derivative in vector‐valued functional spaces
Author(s) -
Bu Shangquan,
Cai Gang
Publication year - 2017
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.201500481
Subject(s) - mathematics , lp space , banach space , function space , degenerate energy levels , fractional calculus , pure mathematics , multiplier (economics) , mathematical analysis , interpolation space , locally convex topological vector space , functional analysis , topological space , physics , quantum mechanics , economics , macroeconomics , biochemistry , chemistry , gene
In this paper, we study the well‐posedness of the degenerate differential equations with fractional derivativeD α ( M u ) ( t ) = A u ( t ) + f ( t ) , ( 0 ≤ t ≤ 2 π )in Lebesgue–Bochner spacesL p ( T ; X ) , periodic Besov spacesB p , q s ( T ; X )and periodic Triebel–Lizorkin spacesF p , q s ( T ; X ) , where A and M are closed linear operators in a complex Banach space X satisfying D ( A ) ⊂ D ( M ) , α > 0 and D α is the fractional derivative in the sense of Weyl. Using known operator‐valued Fourier multiplier results, we completely characterize the well‐posedness of this problem in the above three function spaces by the R ‐bounedness (or the norm boundedness) of the M ‐resolvent of A .