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 $x\in M$.