A generalized vector-valued variational principle in Fréchet spaces

In the framework of Fréchet spaces, we give a generalized vector-valued Ekeland.s variational principle, where the perturbation involves the subadditive functions of countable generating semi-norms. By modifying and developing the method of Cammaroto and Chinni, we obtain a density theorem on extremal points of the vector-valued variational principle, which extends and improves the related known results.