18.6. A VARIATIONAL PRINCIPLE OF EKELAND 511
The inequality is preserved if x = y. Then let
F (x)≡ {y ̸= x : φ (x)−d (x,y)≥ φ (y)} ̸= /0
by assumption. This is the hypothesis for the Cariste fixed point theorem. Hence thereexists x0 ∈F (x0) = {y ̸= x0 : φ (x0)−d (x0,y)≥ φ (y)} but this cannot happen because youcan’t have x0 ̸= x0. Thus the Ekeland variational principle must hold after all.
18.6.2 A Density ResultThere are several applications of the Ekeland variational principle. For more of them, see[55]. One of these is to show that there is a point where φ
′ is small assuming φ is boundedbelow, lower semicontinuous, and Gateaux differentiable. Here⟨
φ′ (x) ,v
⟩≡ lim
h→0
φ (x+hv)−φ (x)h
, φ′ (x) ∈ X ′
It is sort of an approximate critical point at a point which causes φ to be near the infimum.
Theorem 18.6.4 Let X be a Banach space and φ : X → R be Gateaux differentiable,bounded from below, and lower semicontinuous. Then for every ε > 0 there exists x ∈ Xsuch that
φ (xε)≤ infx∈X
φ (x)+ ε and∥∥φ′ (xε)
∥∥X ′ ≤ ε
Proof: From the Ekeland variational principle with λ = 1, there exists xε such that
φ (xε)≤ φ (x0)≤ infx∈X
φ (x)+ ε
and for all x,φ (xε)< φ (x)+ ε ∥x− xε∥
Then letting x = xε +hv where ∥v∥= 1,
φ (xε +hv)−φ (xε)>−ε |h|
Let h < 0. Then divide by it
φ (xε +hv)−φ (xε)
h< ε
Passing to a limit as h→ 0 yields ⟨φ′ (xε) ,v
⟩≤ ε
Now v was arbitrary with norm 1 and so
sup∥v∥=1
⟨φ′ (xε) ,v
⟩=∥∥φ′ (xε)
∥∥≤ ε
There is another very interesting application of the Ekeland variational principle [55].