1370 CHAPTER 42. MAXIMAL MONOTONE OPERATORS, HILBERT SPACE

Proof: From the above, if u ∈ D(A) and y ∈ Au, then∣∣∣∣ 1λ

u− 1λ

Jλ u∣∣∣∣≤ |y|

Hence Jλ u→ u. Now for x arbitrary,

|Jλ x− x| ≤ |Jλ x− Jλ u|+ |Jλ u−u|+ |u− x|< 2ε + |Jλ u−u|

where the last term converges to 0 as λ → 0. Since ε is arbitrary, this shows the proposition.Thus in the case where D(A) is dense, if you have

x ∈ εAxε + xε

so that xε = Jε x, then |x− xε | → 0.The next lemma gives a way to determine whether a pair [x,y] is in the graph of A

defined as{[x,y] : y ∈ Ax} ≡ G (A)

Here I am writing [·, ·] rather than (·, ·) to avoid confusion with the inner product. It is theconclusion of this lemma which accounts for the use of the term “maximal”. It essentiallysays there is no larger monotone graph which includes the one for A.

Lemma 42.1.5 Suppose (y1− y,x1− x)≥ 0 for all [x,y]∈G (A) where A is maximal mono-tone. Then x1 ∈ D(A) and y1 ∈ Ax1. Also if [xk,yk] ∈ G (A) and xk → x,yk ⇀ y where thehalf arrow denotes weak convergence, then [x,y] ∈ G (A).

Proof: I want to show y1 ∈ Ax1 or in other words I want to show

x1 +λy1 ∈ x1 +λAx1

or in other wordsJλ (x1 +λy1) = x1.

This is the motivation for the following argument.From Lemma 42.1.3 Aλ (x1 +λy1) ∈ AJλ (x1 +λy1) and so by the above assumption

0≤ (y1−Aλ (x1 +λy1) ,x1− Jλ (x1 +λy1))

=

(y1−

(1λ(x1 +λy1)−

1λ

Jλ (x1 +λy1)

),x1− Jλ (x1 +λy1)

)=

((− 1

λx1 +

1λ

Jλ (x1 +λy1)

),x1− Jλ (x1 +λy1)

)= − 1

λ(x1− Jλ (x1 +λy1) ,x1− Jλ (x1 +λy1))

which requiresx1 = Jλ (x1 +λy1)