On the Bohr transform of almost-periodic solutions for some differential equations in abstract spaces

Abstract

We consider abstract differential equations of the form u′(t)=Au(t)+f(t) or u″(t)=Au(t)+f(t) in Banach spaces X, where f(⋅), ℝ→X is almost-periodic, while A is a linear operator, 𝒟(A)⊂X→X. If the solution u(⋅) is likewise almost-periodic, ℝ→X, we establish connections between their Bohr-transforms, uˆ(λ) and fˆ(λ).