16.6.1 Cariste Fixed Point Theorem
As mentioned in [?], the above result can be used to prove a fixed point theorem called
the Cariste fixed point theorem.
Theorem 16.6.3 Let ϕ be lower semicontinuous, proper, and bounded below
on a complete metric space X and let F : X →P
be set valued such that
≠∅ for all x. Also suppose that for each x ∈ X, there exists y ∈ F
Then there exists x0 such that x0 ∈ F
Proof: In the above Ekeland variational principle, let ε = 1 = λ. Then there exists
x0 such that for all y≠x0
for all y≠x0.
From the assumption, there is y ∈ F
Since y≠x0, it follows
a contradiction. Hence x0 ∈ F
It is a funny theorem. It is easy to prove, but you look at it and wonder what it says.
If F is single valued, you would need to have a function ϕ such that for each
and if you have such a ϕ then you can assert there is a fixed point for F. Suppose
F is single valued and d
0 < r <
Of course F
fixed point using easier techniques. However, this also follows from this result.
Then is it true that for each x, there exists y ∈ F
such that the inequality holds for all
Yes, this is certainly so because the right side reduces to
Thus this fixed
point theorem implies the usual Banach fixed point theorem.
The Ekeland variational principle says that when ϕ is lower semicontinuous proper
and bounded below, there exists y such that
In fact this can be proved from the Cariste fixed point theorem. Suppose the EVP does
not hold. This would mean that for all y there exists x≠y such that
Thus, for all x there exists y≠x such that
The inequality is preserved if x = y. Then let
by assumption. This is the hypothesis for the Cariste fixed point theorem. Hence there
exists x0 ∈ F
but this cannot happen because
you can’t have
. Thus the Ekeland variational principle must hold after