Solutions of second order degenerate integro-differential equations in vector-valued function spaces

Abstract

We study the well-posedness of the second order degenerate integro-differential equations (P ₂): (Mu)″(t) + α(Mu)′(t) = Au(t) + Σ ^t_−∞ a(t − s)Au(s)ds + f(t), 0 ⩽ t ⩽ 2π, with periodic boundary conditions Mu(0) = Mu(2π), (Mu)′(0) = (Mu)′(2π), in periodic Lebesgue-Bochner spaces L ^p($\mathbb{T}$,X), periodic Besov spaces B ^s _p,q ($\mathbb{T}$,X) and periodic Triebel-Lizorkin spaces F ^s _p,q ($\mathbb{T}$,X), where A and M are closed linear operators on a Banach space X satisfying D(A) ⊂ D(M), a ∈ L ¹(ℝ⁺) and α is a scalar number. Using known operatorvalued Fourier multiplier theorems, we completely characterize the well-posedness of (P ₂) in the above three function spaces.