Solutions of third order degenerate equations with infinite delay in Banach spaces

We study the well-posedness of the third order degenerate differential equations with infinite delay (P-3) :(Mu)'''(t)+(Lu)''(t)+(Bu)'(t)=Au(t)+ integral(t)(-infinity)a(t-s)Au(s)ds+f(t) on [0,2 pi] in Lebesgue-Bochner spaces L-p(T;X) and periodic Besov spaces B-p,q(s)(T;X), where A, B, L and M are closed linear operators on a Banach space X satisfying D(A)subset of D(B)boolean AND D(L)boolean AND D(M) and a is an element of L-1(R+). Using known operator-valued Fourier multiplier theorems, we give necessary and sufficient conditions for (P-3) to be L-p-well-posed(or B-p,q(s)-well-posed). Concrete examples are also given to support our main abstract results.