868 CHAPTER 25. NONLINEAR OPERATORS

In particular, this holds for z = x and this implies limsupx∗n (x− xn) ≤ 0 which along with25.6.35 yields

limn→∞

x∗n (xn− x) = 0 (25.6.37)

.Now let z be arbitrary. There exists a subsequence, nk, depending on z such that

limk→∞

x∗nk

(xnk − z

)= liminfx∗nk

(xnk − z

).

Now from Lemma 25.6.2 and its proof, the ||x∗n|| are all bounded by Lipx ( f ) whenever n islarge enough. Therefore, there is a further subsequence, still denoted by nk such that

x∗nkconverges weakly to x∗ (z) .

We need to verify that x∗ (z)∈ ∂ f (x) . To do so, let y be arbitrary. Then from the definition,

x∗n (y− xn)≤ f 0 (xn,y− xn) . (25.6.38)

From 25.6.37, we can take the limsup of both sides and obtain, using 25.6.36

x∗ (z)(y− x)≤ limsup f 0 (xn,y− xn)≤ f 0 (x,y− x) .

Since y is arbitrary, this shows x∗ (z) ∈ ∂ f (x) and proves the theorem.

25.7 Maximal Monotone OperatorsHere it is assumed that the spaces are all real spaces to simplify the presentation.

Definition 25.7.1 Let A : D(A)⊆ X →P (X) be a set valued map. It is said to be mono-tone if whenever yi ∈ Axi,

⟨y1− y2,x1− x2⟩ ≥ 0

Denote by G (A) the graph of A consisting of all pairs (x,y) where y∈ Ax. Such a monotoneoperator is said to be maximal monotone if

F +A

is onto where F is the duality map with p = 2.

Actually, it is more usual to say that the graph is maximal monotone if the graph ismonotone and there is no monotone graph which properly contains the given graph. How-ever, the two conditions are equivalent and I am more used to using the version in the abovedefinition.

There is a fundamental result about these which is given next.

Theorem 25.7.2 Let X, X ′ be reflexive and have strictly convex norms. Let A be a mono-tone set valued map as just described. Then if λF+A is onto for some λ > 0, then whenever

⟨y− z,x−u⟩ ≥ 0 for all [x,y] ∈ G (A)

it follows that z ∈ Au and u ∈ D(A). That is, the graph is maximal.