18.6. A VARIATIONAL PRINCIPLE OF EKELAND 507

The variational principle of Ekeland is the following theorem [55]. You start with anapproximate minimizer x0. It says there is yλ fairly close to x0 such that if you subtract a“cone” from the value of φ at yλ , then the resulting function is less than φ (x) for all x ̸= yλ .This cone is like a supporting plane for a convex function but pertains to functions whichare certainly not convex.

x0

yλ

Theorem 18.6.2 Let X be a complete metric space and let φ : X → (−∞,∞] be proper,lower semicontinuous and bounded below. Let x0 be such that

φ (x0)≤ infx∈X

φ (x)+ ε

Then for every λ > 0 there exists a yλ such that

1. φ (yλ )≤ φ (x0)

2. d (yλ ,x0)≤ λ

3. φ (yλ )− ε

λd (x,yλ )< φ (x) for all x ̸= yλ

To motivate the proof, see the following picture which illustrates the first two steps.The Siwill be sets in X but are denoted symbolically by labeling them in X× (−∞,∞].

x1

x2

S1 S1

Then the end result of this iteration would be a picture like the following.

yλ