Introduction

In this paper we are concerned with the linear functional differential equation

$$\dot{u}(t)=Au(t)+F(t)u_t+f(t)$$ (1)

on a phase space ${\mathcal B}={\mathcal B}((-\infty,0];{\mathbb{X}})$ satisfying some fundamental axioms listed in Section 2.1, where $A$ is the infinitesimal generator of a strongly continuous semigroup $T(t)$ on a Banach space ${\mathbb{X}}$, $u_t$ is an element of ${\mathcal B}$ defined by $u_t(\theta)=u(t+\theta)$ for $\theta\in (-\infty,0]$, $F(t)$ is a bounded linear operator mapping ${\mathcal B}$ into ${\mathbb{X}}$ which depends strongly continuously and periodically on $t$, and $f$ is an ${\mathbb{X}}$-valued bounded and continuous function.

In a recent paper [24], Murakami, Naito and Nguyen have established a variation of constants formula (VCF) in the phase space for Eq. (1). The formula has been then applied to extend a classical theorem of Massera [22] on the existence of periodic solutions of linear ordinary differential equations to almost periodic solutions for Eq. (1).

A key point in [24] is to analyze difference equations associated with Eq. (1), which are derived naturally from the formula.

In this paper we will continue to study the subject, and establish several sharper results on the existence of almost periodic and quasiperiodic solutions for Eq. (1). Our approach employed in this paper is different from the one in [24]. Indeed, we will decompose the variation of constants formula into two parts referred to as the stable part of VCF and the unstable part of VCF (Theorem 3.1), and study each part of VCF to ensure the existence of almost periodic solutions and quasiperiodic solutions for Eq. (1). There are several advantages in our approach. Among them, we point out the following two facts: Roughly speaking, some spectral properties of the function $f$ is inherited to the stable part of VCF (Theorem 3.3). Meanwhile, the unstable part of VCF is reduced to an ordinary differential equations (Theorem 3.5), and via this fact some spectral properties of $f$ is inherited also to the unstable part of VCF (Theorem 3.6).

As an appendix, we refer to the Riesz representation for each element belonging to the dual space of ${\mathcal B}((-\infty];{\mathbb{X}})$ and Favard's type argument to ensure the existence of almost periodic solutions of ordinary differential equations with a discontinuous forcing term. They seem to be unknown in the general situation and will be indispensable in our approach.