$C^1$ invariant and inertial manifolds for non-autonomous dynamical systems

Abstract: Recent article by Carvalho et al. [J. Differential Equations 416 (2025)] presents in a~unified framework the Lyapunov--Perron construction of invariant and inertial manifolds together with the saddle point property. We extend their results on the existence of non-autonomous inertial manifolds by considering the gap condition which is known to be sharp. Moreover, we show that the constructed inertial manifolds have $C^1$ regularity.