A Generalized Palais-Smale Condition in the Fr\'echet space setting

We extend the Palais-Smale condition to Keller’s $C_c^1$-functionals on Fr'{e}chet spaces. Using this condition together with Ekeland’s variational principle, we obtain some results regarding the existence of minima. In this setting, we prove that the Palais-Smale condition for functionals bounded below implies the coercivity.