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

(i)
$\psi (z)\le \psi ({z}_{0})\epsilon d(z,{z}_{0})$;

(ii)
$\psi (x)\ge \psi (z)\epsilon d(x,z)$, for any $x\in M$.
Title  Ekeland’s variational principle 

Canonical name  EkelandsVariationalPrinciple 
Date of creation  20130322 15:19:16 
Last modified on  20130322 15:19:16 
Owner  ncrom (8997) 
Last modified by  ncrom (8997) 
Numerical id  8 
Author  ncrom (8997) 
Entry type  Theorem 
Classification  msc 49J40 