Minimizing by Ekeland’s Variational Principle.
Motivation of Ekeland’s paper “On the variational principle.” JMAA, 47. (1974). If is a continous function on or Banach space or complete Riemannian manifold and has not minimum (of course is bounded below) then we can choose a sequence such that is “near” to the infimum of .
We provide following Ekeland’s theorem with is complete metric space
Theorem (Ekeland’s Variational Principle). Let be a complete metric space and a lower semi continous function, , bounded from below. For every point satisfying and every , there exists some point such that
In part 5 of the paper, the author states an interesting result on complete Riemannian manifold
Proposition. Let be a function on a complete Riemannian manifold . If bounded from below, then, for every , there exists some point such that
Conclusion. is the sequence of minimizer.