An Abstract Variational Theorem

Let

be a Banach space and

f\colon X \to \mathbb{R} \cup \{\infty\} f  ⁣ : X → R ∪ { ∞ }

be a proper function. Then the Fenchel conjugate of

f f

is the function

f^*\colon X^* \to \mathbb{R} \cup \{\infty\} f ∗  ⁣ : X ∗ → R ∪ { ∞ }

defined by,

f^*(x^*):= \sup\{(x^*-f)(x):x \in X\}. f ∗ ( x ∗ ) : = sup ⁡ { ( x ∗ − f ) ( x ) : x ∈ X } .

In this article we will prove a theorem more general than the following. Theorem: Let

f\colon X \to \mathbb{R} \cup \{\infty\} f  ⁣ : X → R ∪ { ∞ }

be a proper function on a Banach space

(X,\|\cdot\|) ( X , ∥ ⋅ ∥ )

. If there is a nonempty open subset

A A

of

\mathrm{Dom}(f^*) D o m ( f ∗ )

such that

\mathrm{argmax}(x^*-f) \not= \varnothing a r g m a x ( x ∗ − f ) ≠ ∅

for each

x^* \in A x ∗ ∈ A

, then there is a dense and

G_\delta G δ

subset

R R

of

A A

such that

(x^*-f) \colon X \to \mathbb{R} \cup \{-\infty\} ( x ∗ − f )  ⁣ : X → R ∪ { − ∞ }

has a strong maximum for each

x^* \in R x ∗ ∈ R

. In addition, if

0 \in A 0 ∈ A

and

0<\varepsilon 0 < ε

then there is an

x^* \in X^* x ∗ ∈ X ∗

with

\|x^*\| < \varepsilon ∥ x ∗ ∥ < ε

such that

(x^* -f) \colon X \to \mathbb{R} \cup \{-\infty\} ( x ∗ − f )  ⁣ : X → R ∪ { − ∞ }

has a strong maximum.