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Abstract

Using known results on operator-valued Fourier multipliers on vector-valued function spaces, we give necessary or sufficient conditions for the well-posedness of the second order degenerate equations ($P_2$): $\frac {d}{dt}(Mu')(t) =Au(t)+f(t)$ $(0\leq t\leq 2\pi )$ with periodic boundary conditions $ u(0)=u(2\pi )$, $(Mu')(0)=(Mu')(2\pi )$, in Lebesgue–Bochner spaces $L^p({\mathbb T}, X)$, periodic Besov spaces $B_{p,q}^s({\mathbb T}, X)$ and periodic Triebel–Lizorkin spaces $F_{p,q}^s({\mathbb T}, X)$, where $A$ and $M$ are closed operators in a Banach space $X$ satisfying $D(A)\subset D(M)$. Our results generalize the previous results of W. Arendt and S. Q. Bu when $M= I_X$.