In this paper, we consider an equivalence relation on the space $AP(\mathbb {R},X)$ of almost periodic functions with values in a prefixed Banach space X. In this context, it is known that the normality or Bochner-type property, which characterizes these functions, is based on the relative compactness of the family of translates. Now, we prove that every equivalence class is sequentially compact and the family of translates of a function belonging to this subspace is dense in its own class, i.e., the condition of almost periodicity of a function $f\in AP(\mathbb {R},X)$ yields that every sequence of translates of f has a subsequence that converges to a function equivalent to f. This extends previous work by the same authors on the case of numerical almost periodic functions.

This paper deals with feedback stabilization of second order equations ofthe form

ytt + A0y + u (t) B0y (t) = 0, t ∈ [0, +∞[,

where A0 is a densely defined positive selfadjoint linear operator on areal Hilbert space H, with compact inverse and B0 is a linear map in diagonal form. It isproved here that the classical sufficient ad-condition of Jurdjevic-Quinn andBall-Slemrod with the feedback control u = ⟨yt, B0y⟩Himplies thestrong stabilization. This result is derived from a general compactnesstheorem for semigroup with compact resolvent and solves several open problems.

A bounded continuous function is said to be S-asymptotically ω-periodic if . This paper is devoted to study the existence and qualitative properties of S-asymptotically ω-periodic mild solutions for some classes of abstract neutral functional differential equations with infinite delay. Furthermore, applications to partial differential equations are given.