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 $ +\infty $ as $ t\to\pm\infty $. Typical simple examples are

$ x'(t) = \sin (t^q)x(t-1) \qquad\text{and}\qquad x'(t) = e^{it^q}x(t-1), \;\;\;\;\;\;\;\;{(*)} $

where $ q\ge 2 $ is an integer. Under various conditions, we show the existence of a unique solution with any prescribed finite limit $ \lim\limits_{t\to-\infty}x(t) = x_- $ at $ -\infty $. We also show, under appropriate conditions, that any solution of an initial value problem has a finite limit $ \lim\limits_{t\to+\infty}x(t) = x_+ $ at $ +\infty $, 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 $ (-\infty,-T] $ is often highly nontrivial.

Our original interest in such solutions stems from questions concerning their smoothness. In particular, any solution $ x: \mathbb{R}\to \mathbb{C} $ of one of the equations in $ (*) $ with limits $ x_\pm $ at $ \pm\infty $ is $ C^\infty $, but it is unknown if such solutions are analytic. Nevertheless, one does know that any such solution of the second equation in $ (*) $ can be extended to the lower half plane $ \{z\in \mathbb{C}\:|\: \mathop{{{\rm{Im}}}} z<0\} $ as an analytic function.