Ekeland’s variational principle

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

• (i)

$\psi(z)\leq\psi(z_{0})-\varepsilon d(z,z_{0})$;

• (ii)

$\psi(x)\geq\psi(z)-\varepsilon d(x,z)$, for any $x\in M$.

Title Ekeland’s variational principle EkelandsVariationalPrinciple 2013-03-22 15:19:16 2013-03-22 15:19:16 ncrom (8997) ncrom (8997) 8 ncrom (8997) Theorem msc 49J40