# Ngô Quốc Anh

## May 19, 2011

### The Ekeland variational principle

Filed under: PDEs — Ngô Quốc Anh @ 14:43

In mathematical analysis, Ekeland’s variational principle, discovered by Ivar Ekeland, is a theorem that asserts that there exists nearly optimal solutions to some optimization problems. Ekeland’s variational principle can be used when the lower level set of a minimization problems is not compact, so that the Bolzano–Weierstrass theorem can not be applied. Ekeland’s principle relies on the completeness of the metric space. Ekeland’s principle leads to a quick proof of the Caristi fixed point theorem.

Theorem (Ekeland’s variational principle). Let $(X, d)$ be a complete metric space, and let $F: X \to \mathbb R\cup \{+\infty\}$ be a lower semicontinuous functional on $X$ that is bounded below and not identically equal to $+\infty$. Fix $\varepsilon > 0$ and a point $u\in X$ such that

$F(u) \leq \varepsilon + \inf_{x \in X} F(x).$

Then there exists a point $v\in X$ such that

1. $F(v) \leq F(u)$,
2. $d(u, v) \leq 1$,
3. and for all $w \ne v$, $F(w) > F(v) - \varepsilon d(v, w)$.

Source: Wiki.