In this work, we present a criterion for the existence of an exponential dichotomy over all \(\mathbb {R}\) for delayed systems of the form \(\begin{aligned} x'(t)=L(t)x_t=\sum \limits _{j=0}^NA_j(t)x(t-r_j)+\int _{-r}^0A(t,\theta )x(t+\theta )d\theta . \end{aligned}\) Specifically, we study systems where \(L_{\pm }=\lim _{t\rightarrow \pm \infty }L(t)\) are both autonomous and hyperbolic. To establish our criterion, we use a novel choice of weighted Sobolev spaces, namely \(L^p(\mathbb {R},\mu )\) and \(W^{1,p}(\mathbb {R},\mu )\) . The differential operators analyzed become Fredholm operators in these spaces, and we make extensive use of this property. In the literature, the existence of an exponential dichotomy over all \(\mathbb {R}\) has been a key point for example in the proofs of non-autonomous versions of Hartman-Grobman’s Theorem.
