Exponential ordering for nonautonomous neutral functional differential equations

The paper’s main theorem (Theorem 5.6) proves that under (F1)–(F6), (A1), and in Case III also (D1)–(D7), if the forward orbit is relatively compact and uniformly stable for the exponential order in bounded sets (with Lipschitz initial data in Cases II–III), then the omega-limit set is a 1-cover of the base and the trajectory converges fiberwise to a continuous equilibrium; this uses a semicontinuous super-equilibrium a, an induced equilibrium c, and a dynamical L-argument that relies on (F6) and uniform stability to conclude K equals the graph of c and convergence holds . The model’s solution is aligned in scope and conclusions: it establishes monotonicity (matching Theorem 3.5) , uses backward extensions of points in the omega-limit set (flow extension) , and then derives a 1-cover and convergence. Its proof strategy differs: it invokes a strong order-preserving (SOP) mechanism on K via (F6) and a nonordering principle (à la Smith–Thieme) instead of the paper’s super-equilibrium approach. Technically, one caveat is that the exponential-order cones in BU have empty interior, so appeals to a generic ‘≪’-type interior-based SOP/nonordering principle require careful adaptation (or restriction to Lipschitz subspaces where eventual strong monotonicity is available); the paper avoids this by a tailored argument using (F5)–(F6) and the L-set continuity method. With that nuance noted, the model’s outline matches the paper’s result and is correct in substance though it omits several technical details that the paper supplies. Key ingredients used by both include the exponential-order framework and D̂-transform in the neutral case and the precise monotonicity condition (F4) and its strengthening (F6) .