Solutions of second order degenerate integro-differential equations in vector-valued function spaces

We study the well-posedness of the second order degenerate integro-differential equations (P (2)): (Mu)aEuro(3)(t) + alpha(Mu)'(t) = Au(t) + I pound (-a) (t) a(t - s)Au(s)ds + f(t), 0 a (c) 1/2 t a (c) 1/2 2 pi, with periodic boundary conditions Mu(0) = Mu(2 pi), (Mu)'(0) = (Mu)'(2 pi), in periodic Lebesgue-Bochner spaces L (p) (,X), periodic Besov spaces B (p,q) (s) (,X) and periodic Triebel-Lizorkin spaces F (p,q) (s) (,X), where A and M are closed linear operators on a Banach space X satisfying D(A) aS, D(M), a a L (1)(a"e(+)) and alpha is a scalar number. Using known operatorvalued Fourier multiplier theorems, we completely characterize the well-posedness of (P (2)) in the above three function spaces.