
Periodic solutions of second order degenerate differential equations with delay in Banach spaces
We give necessary and sufficient
conditions of the $L^p$wellposedness (resp. $B_{p,q}^s$wellposedness) for the second order degenerate
differential equation with finite delays:
$(Mu)''(t)+Bu'(t)+Au(t)=Gu'_t+Fu_t+f(t),(t\in [0,2\pi])$ with periodic
boundary conditions $(Mu)(0)=(Mu)(2\pi)$, $(Mu)'(0)=(Mu)'(2\pi)$, where
$A, B, M$ are closed linear operators on a complex Banach space $X$ satisfying
$D(A)\cap D(B)\subset D(M)$, $F$ and $G$ are bounded linear operators from
$L^p([2\pi,0];X)$ (resp. $B_{p,q}^s([2\pi,0];X)$) into $X$.
Keywords:second order degenerate differential equation, Fourier multiplier theorem, wellposedness, LebesgueBochner space, Besov space Categories:34G10, 34K30, 43A15, 47D06 