Periodic solutions of third-order degenerate differential equations in vector-valued functional spaces

Abstract

We study the well-posedness of the third-order degenerate differential equation $\left( {{P_3}} \right):\alpha {\left( {Mu} \right)^{\prime \prime \prime }}\left( t \right) + {\left( {Mu} \right)^{\prime \prime }}\left( t \right) = \beta Au\left( t \right) + f\left( t \right)$, (t ∈ [0, 2p]) with periodic boundary conditions $Mu\left( 0 \right) = Mu\left( {2\pi } \right),\;Mu'\left( 0 \right) = Mu'\left( {2\pi } \right),\;Mu''\left( 0 \right) = Mu''\left( {2\pi } \right)$, in periodic Lebesgue–Bochner spaces L ^p(T,X), periodic Besov spaces B ^s _p,q (T,X) and periodic Triebel–Lizorkin spaces F ^s _p,q (T,X), where A, B and M are closed linear operators on a Banach space X satisfying D(A) $\cap$ D(B) ⊂ D(M) and α, β, γ ∈ R. Using known operator-valued Fourier multiplier theorems, we completely characterize the well-posedness of (P ₃) in the above three function spaces.