Ekeland's variational principle

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 cannot 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.

Ekeland's principle has been shown by F. Sullivan to be equivalent to completeness of metric spaces.

Ekeland was associated with the Paris Dauphine University when he proposed this theorem.

Let (X, d) be a complete metric space, and let F: X → R ∪ {+∞} be a lower semicontinuous functional on X that is bounded below and not identically equal to +∞. Fix ε > 0 and a point u ∈ X such that

Then there exists a point v ∈ X such that

and, for all w ≠ v,

This theorem has been shown by F. Sullivan to be equivalent to completeness for metric spaces.

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