Periodic Solutions of Third-order Differential Equations with Finite Delay in Vector-valued Functional Spaces

Abstract

In this paper, we study the well-posedness of the third-order differential equation with finite delay (P3): αu‴ (t) + u″(t) = Au(t) + Bu′ (t) + Fut + f(t)(t ∈ $$\mathbb{T}$$ := [0, 2π]) with periodic boundary conditions u(0) = u(2π), u′(0) = u′(2π), u″ (0) = u″(2π), in periodic Lebesgue–Bochner spaces Lp( $$\mathbb{T}$$ ;X) and periodic Besov spaces Bp,qs( $$\mathbb{T}$$ ;X), where A and B are closed linear operators on a Banach space X satisfying D(A) ∩ D(B) ≠{0}, α ≠ 0 is a fixed constant and F is a bounded linear operator from Lp([−2π, 0];X) (resp. Bp,qs([−2π, 0];X)) into X, ut is given by ut(s) = u(t + s) when s ∈ [−2π, 0]. Necessary and sufficient conditions for the Lp-well-posedness (resp. Lp-well-posedness) of (P3) are given in the above two function spaces. We also give concrete examples that our abstract results may be applied.