Asymptotic homogenization for delay-differential equations and a question of analyticity

We consider a class of nonautonomous delay-differential equations in which the time-varying coefficients have an oscillatory character, with zero mean value, and whose frequency approaches \begin{document}$ +\infty $\end{document} as \begin{document}$ t\to\pm\infty $\end{document} . Typical simple examples are \begin{document}$ x'(t) = \sin (t^q)x(t-1) \qquad\text{and}\qquad x'(t) = e^{it^q}x(t-1), \;\;\;\;\;\;\;\;{(*)} $\end{document} where \begin{document}$ q\ge 2 $\end{document} is an integer. Under various conditions, we show the existence of a unique solution with any prescribed finite limit \begin{document}$ \lim\limits_{t\to-\infty}x(t) = x_- $\end{document} at \begin{document}$ -\infty $\end{document} . We also show, under appropriate conditions, that any solution of an initial value problem has a finite limit \begin{document}$ \lim\limits_{t\to+\infty}x(t) = x_+ $\end{document} at \begin{document}$ +\infty $\end{document} , and thus we establish the existence of a class of heteroclinic solutions. We term this limiting phenomenon, and thus the existence of such solutions, "asymptotic homogenization." Note that in general, proving the existence of a bounded solution of a given delay-differential equation on a semi-infinite interval \begin{document}$ (-\infty,-T] $\end{document} is often highly nontrivial. Our original interest in such solutions stems from questions concerning their smoothness. In particular, any solution \begin{document}$ x: \mathbb{R}\to \mathbb{C} $\end{document} of one of the equations in \begin{document}$ (*) $\end{document} with limits \begin{document}$ x_\pm $\end{document} at \begin{document}$ \pm\infty $\end{document} is \begin{document}$ C^\infty $\end{document} , but it is unknown if such solutions are analytic. Nevertheless, one does know that any such solution of the second equation in \begin{document}$ (*) $\end{document} can be extended to the lower half plane \begin{document}$ \{z\in \mathbb{C}\:|\: \mathop{{{\rm{Im}}}} z as an analytic function.