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Using the method of exponential dichotomies, we establish a new existence and uniqueness theorem for almost automorphic solutions of differential equations with piecewise constant argument of the form where 
$$\begin{eqnarray*}{x}^{\prime } (t)= A(t)x(t)+ B(t)x(\lfloor t\rfloor )+ f(t), \quad t\in \mathbb{R} ,\end{eqnarray*}$$
$\lfloor \cdot \rfloor $ denotes the greatest integer function, and 
$A(t), B(t): \mathbb{R} \rightarrow { \mathbb{R} }^{q\times q} $, 
$f(t): \mathbb{R} \rightarrow { \mathbb{R} }^{q} $ are all almost automorphic.
