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.

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