Stepanov-Almost Periodic Solutions of Non-autonomous Neutral Functional Differential Equations with Functional Delay

In this paper, we study the existence and uniqueness of Stepanov-almost periodic mild solution to the non-autonomous neutral functional differential equation

$$\begin{aligned} \frac{{\hbox {d}}}{\hbox {d}t}u(t)-F(t,u(t-\alpha (t)))]= & {} A(t)u(t)-F(t,u(t-\alpha (t)))]\\&+\,G(t,u(t),u(t-\alpha (t))),\quad t\in \mathbb {R}, \end{aligned}$$

in a Banach space \(\mathbb {X},\) where the family of linear operators A(t) satisfies the ‘Acquistapace–Terreni’ conditions, the evolution family generated by \(A(t),t\in \mathbb {R},\) is exponentially stable, \((\gamma -A(\cdot ))^{-1}\) and \(\alpha (\cdot )\) are almost periodic, and F and G are Stepanov-almost periodic continuous functions.