WELL-POSEDNESS OF SECOND ORDER DEGENERATE INTEGRO-DIFFERENTIAL EQUATIONS IN VECTOR-VALUED FUNCTION SPACES

We consider the well-posedness of the second order degenerate integrodifferential equations (P-2): (Mu')'(t) + alpha u'(t) = Au(t) + integral(t)(-infinity) a(t - s)Au(s)ds + f(t), (0 < t <= 2 pi) with periodic boundary conditions u(0) = u(2 pi), (Mu')(0) = (Mu')(2 pi), in periodic Lebesgue-Bochner spaces L-p(T, X), periodic Besov spaces B-p,q(s) (T, X) and periodic Triebel-Lizorkin spaces F-p,q(s)(T, X), where A and M are closed linear operators on a Banach space X satisfying D(A) subset of D(M), a is an element of L-1 (R+) and alpha is a scalar number. Using known operator-valued Fourier multiplier theorems, we give necessary and sufficient conditions for the well-posedness of (P-2) in the above three function spaces.