Well‐posedness of fractional degenerate differential equations with finite delay on vector‐valued functional spaces

We study the well‐posedness of the fractional degenerate differential equations with finite delay( P α ) : D α ( M u ) ( t ) = A u ( t ) + F u t + f ( t ) , ( 0 ≤ t ≤ 2 π , α > 0 )on 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 on a Banach space X satisfying D ( A ) ⊂ D ( M ) , F is a bounded linear operator fromL p ( [ − 2 π , 0 ] ; X )(resp.B p , q s ( [ − 2 π , 0 ] ; X )andF p , q s ( [ − 2 π , 0 ] ; X ) ) into X , where u t is given byu t ( s ) = u ( t + s )when s ∈ [ − 2 π , 0 ] and t ∈ [ 0 , 2 π ] . Using known operator‐valued Fourier multiplier theorems, we give necessary or sufficient conditions for the well‐posedness of ( P α ) in the above three function spaces.