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# Minimizing by Ekeland’s Variational Principle.

Minimizing by Ekeland’s Variational Principle.

Motivation of Ekeland’s paper “On the variational principle.” JMAA, 47. (1974). If $f$ is a continous function on $\mathbb{R}^n$ or Banach space $X$ or complete Riemannian manifold $\overline{M}$ and $f$ has not minimum (of course $f$ is bounded below) then we can choose a sequence $(x_n)$ such that $f(x_n)$ is “near” to the infimum of $f$.

We provide following Ekeland’s theorem with $X$ is complete metric space

Theorem (Ekeland’s Variational Principle).  Let $X$ be a complete metric space and $F: X \to \mathbb{R} \cup \{+\infty\}$ a lower semi continous function, $\ne +\infty$, bounded from below. For every point $u \in X$ satisfying $\inf F \le F(u) \le \inf F + \epsilon$ and every $\lambda > 0$, there exists some point $v \in X$ such that

$F(v) \le F(u)$,

$d(u,v) \le \lambda$,

$\forall w \ne v, F(w) > F(v) - (\epsilon/\lambda)d(v,w)$.

In part 5 of the paper, the author states an interesting result on complete Riemannian manifold

Proposition. Let $F$ be a $C^1$ function on a complete Riemannian manifold $\overline{M}$. If $F$ bounded from below, then, for every $\epsilon \ge 0$, there exists some point $p_\epsilon \in \overline{M}$ such that

$F(p_\epsilon) \le \inf F + \epsilon^2$,

$\|gradF(p_\epsilon)\|_{p_\epsilon} \le \epsilon$.

Conclusion. $p_\epsilon$ is the sequence of minimizer.