Well-posedness of degenerate fractional integro-differential equations in vector-valued functional spaces

We study the well-posedness of the fractional degenerate integro-differential equations (P-alpha) : D-alpha (Mu)(t) = Au(t) + integral(t )(-infinity)a(t - s)Au(s) ds + integral(t)(-infinity) b(t - s)Bu(s)ds + f(t), t is an element of T := [0, 2 pi]), in Lebesgue-Bochner spaces L-p(T; X) and Besov spaces B-p,q(s) (T; X), where A, B and M are closed linear operators on a Banach space X satisfying D(A) boolean AND D(B) subset of D(M), D(A) boolean AND D(B) not equal (0), alpha > 0 and a, b is an element of L-1 (R+). We completely characterize the well-posedness of (P-alpha) in the above vector-valued function spaces on T by using operator-valued Fourier multiplier. We also give an example that our abstract results may be applied.