Periodic solutions of four-order degenerate differential equations with finite delay in vector-valued function spaces

In this paper, we mainly investigate the well-posedness of the four-order degenerate differential equation ($P_4$): $(Mu)''''(t) + \alpha (Lu)'''(t) + (Lu)''(t)$ $=\beta Au(t) + \gamma Bu'(t) + Gu'_t + Fu_t + f(t),\,( t\in [0,\,2\pi ])$ in periodic Lebesgue-Bochner spaces $L<^>p(\mathbb {T}; X)$ and periodic Besov spaces $B_{p,q}<^>s\;(\mathbb {T}; X)$, where $A$, $B$, $L$ and $M$ are closed linear operators on a Banach space $X$ such that $D(A)\cap D(B)\subset D(M)\cap D(L)$ and $\alpha,\,\beta,\,\gamma \in \mathbb {C}$, $G$ and $F$ are bounded linear operators from $L<^>p([-2\pi,\,0];X)$ (respectively $B_{p,q}<^>s([-2\pi,\,0];X)$) into $X$, $u_t(\cdot ) = u(t+\cdot )$ and $u'_t(\cdot ) = u'(t+\cdot )$ are defined on $[-2\pi,\,0]$ for $t\in [0,\, 2\pi ]$. We completely characterize the well-posedness of ($P_4$) in the above two function spaces by using known operator-valued Fourier multiplier theorems.