be a real
function defined on a Banach space
. In the first part of the paper there are presented some applications of Ekeland’s Principle in obtaining critical points of functions
which satisfy the Palais-Smale property. The main result is given by Theorem 1, proved in this part by using Ekeland’s Principle. In the second part is presented a general deformation theorem (Theorem 3). Next a new proof of a recent theorem of Ghoussoub (Theorem 2) is given by using deformation Theorem 3. In the third part, the authors apply Theorem 2 to functions
which are bounded below and satisfy the Palais-Smale property. Finally, in the Appendix, the authors give a new proof of Theorem 1 based on deformation Theorem 3.