Exponential dichotomy for a class of asymptotically autonomous delayed differential equations

In this work we give a criterion to have an exponential dichotomy over all $\mathbb{R}$ for delayed systems $x^{\mathrm{\prime }}(t)=L(t)x_t$, where $L_{\pm }={\lim }_{t\to \pm \mathrm{\infty }}L(t)$, and the systems $x^{\mathrm{\prime }}(t)=L_{\pm }{x}_t$ are autonomous and hyperbolic. The proof is based in a suitable choice of weighted Sobolev spaces $L^p(\mathbb{R},\mu )$ and $W^{1,p}(\mathbb{R},\mu )$, where the involves differential operators become Fredholm’s operators and this property are strongly exploited.