Well‐posedness of degenerate differential equations with fractional derivative in vector‐valued functional spaces

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 .