Well‐posedness of second order degenerate integro‐differential equations with infinite delay in vector‐valued function spaces

We study the well‐posedness of the second order degenerate differential equations with infinite delay:( M u ) ′ ′( t )= A u ( t ) + ∫ − ∞ t a ( t − s ) A u ( s )d s + ∫ − ∞ t b ( t − s ) B u ( s )d s + f ( t ) , ( 0 ≤ t ≤ 2 π )with periodic boundary conditions( M u ) ( 0 ) = ( M u ) ( 2 π ) , ( M u ) ′ ( 0 ) = ( M u ) ′ ( 2 π ) , where A , B and M are closed linear operators in a Banach space satisfying D ( A ) ∩ D ( B ) ≠ { 0 } , D ( A ) ∩ D ( B ) ⊂ D ( M ) , a , b ∈ L 1 ( R + ) . Using operator‐valued Fourier multiplier techniques, we give necessary and sufficient conditions for the well‐posedness of this problem 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 ) .