Almost automorphic and pseudo almost periodic solutions to damped Euler–Lagrange equations

This paper investigates the existence of almost automorphic, asymptotically almost automorphic, pseudo almost automorphic, and pseudo almost periodic solutions to damped Euler–Lagrange equations. The analysis relies on a variational framework and the continuity properties of Nemytskii operators. Both autonomous and non-autonomous cases, including external forcing terms, are addressed. A central result shows that the set of forcing functions leading to almost automorphic solutions is dense with respect to a suitable norm. The main contribution of this work lies in extending classical existence results for almost periodic solutions to more general classes of functions, thereby emphasizing the robustness and generality of such behaviors in variational systems.