Ekeland’s variational principle

Let $(M,d)$ be a complete metric space and let $\psi :M\to (-\mathrm{\infty },+\mathrm{\infty }]$, $\psi \not\equiv +\mathrm{\infty }$, be a lower semicontinuous function which is bounded from below. Then the following hold: For every $\epsilon >0$ and for any $z_0\in M$ there exists $z\in M$ such that

• $\psi (z)\le \psi (z_0)-\epsilon d(z,z_0)$;

• $\psi (x)\ge \psi (z)-\epsilon d(x,z)$, for any $x\in M$.

Title	Ekeland’s variational principle
Canonical name	EkelandsVariationalPrinciple
Date of creation	2013-03-22 15:19:16
Last modified on	2013-03-22 15:19:16
Owner	ncrom (8997)
Last modified by	ncrom (8997)
Numerical id	8
Author	ncrom (8997)
Entry type	Theorem
Classification	msc 49J40