We give necessary and sufficient conditions of the
${L}^{p}$wellposedness (respectively,
${B}_{p,q}^{s}$wellposedness)
for the secondorder degenerate differential equation with finite delay:
${\left(M{u}^{\prime}\right)}^{\prime}\left(t\right)+\alpha {u}^{\prime}\left(t\right)=Au\left(t\right)+G{u}_{t}^{\prime}+F{u}_{t}+f\left(t\right)$,
$\left(t\in \left[0,2\pi \right]\right)$ with periodic
boundary conditions
$u\left(0\right)=u\left(2\pi \right)$,
$\left(M{u}^{\prime}\right)\left(0\right)=\left(M{u}^{\prime}\right)\left(2\pi \right)$, where
$A$ and
$M$ are closed linear operators
on a Banach space
$X$
satisfying
$D\left(A\right)\subset D\left(M\right)$,
and
$F$ and
$G$ are bounded linear
operators from
${L}^{p}\left(\left[2\pi ,0\right];X\right)$
(respectively,
${B}_{p,q}^{s}\left(\left[2\pi ,0\right];X\right)$)
into
$X$.
