Remove the in-time continuity assumptions from the lifting-based invariant-manifold construction

Establish whether the in-time continuity assumptions imposed on the nonautonomous vector field can be removed by circumventing the lifting procedure and reformulating the parameterization-method proof directly for the nonautonomous setting.

Background

The paper constructs invariant manifolds for nonautonomous systems by lifting the dynamics to an evolution flow on a function space. The authors note that the required continuity assumptions in time may arise from this lifting strategy rather than from the underlying invariant-manifold problem itself.

A direct proof formulated in the nonautonomous setting could potentially establish the same invariant-manifold conclusions under weaker hypotheses. The paper does not pursue this reformulation, leaving the necessity of the in-time continuity assumptions unresolved.