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| The purpose of this paper is to study the existence and uniqueness of periodic solutions to the second order degenerate differential equation [(P_2): (Mu)''(t)=Au(t)+f(t), (0leq tleq 2pi)]with periodic boundary conditions $ Mu(0)=Mu(2pi),(Mu)'(0)=(Mu)'(2pi)$, in periodic 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 two closed linear operators in a Banach space satisfying $D(A)subset D(M)$. We use operator-valued Fourier multiplier techniques to obtain necessary and sufficient conditions to guarantee the existence and uniqueness of $(P_2)$. |
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| Keywords:Operator-valued Fourier multipliers; Degenerate differentialequation; Maximal regularity; Besov spaces; Triebel-Lizorkinspaces |
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