In this paper, we study well-posedness for the following third-order in time equation with delay $$\left( {0.1} \right)\;\alpha \left( {Mu'} \right)''\left( t \right) + \left( {Nu'} \right)'\left( t \right) = \beta Au\left( t \right) + \gamma Bu{'(t)} + Gu{'_t}+F{u_t} + f\left( t \right),\;t \in \left[ {0,2\pi } \right]$$ where α, β, γ are real numbers, A and B are linear operators defined on a Banach space X with domains D(A) and D(B) such that $$D(A)\cap{D(B)}\subset{D(M)}\cap{D(N)};$$ u(t)is the state function taking values in X and ut: (−∞, 0] → X defined as ut(θ) = u(t+θ) for θ < 0 belongs to an appropriate phase space where F and G are bounded linear operators. Using operator-valued Fourier multiplier techniques we provide optimal conditions for well-posedness of equation (0.1) 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 $$F^s_{p,q}(\mathbb{T}, X)$$ . A novel application to an inverse problem is given.