900 CHAPTER 25. NONLINEAR OPERATORS

where p > 1 and an→ 0. Then

lim supλ→0

ypλ≤ lim sup

λ→0yp−1

λan

and so,lim sup

λ→0yλ ≤ an

Hencelim sup

λ→0∥Jλ x− x∥ ≤ ∥xn− x∥

Since xn is arbitrary, it follows that for every ε > 0,

lim supλ→0∥Jλ x− x∥ ≤ ε

and so in fact, limsupλ→∞ ∥Jλ x− x∥= 0.Now here is an interesting corollary.

Corollary 25.7.37 Let A be maximal monotone. A : X → X ′ where X is a strictly convexreflexive Banach space. Then D(A) is convex.

Proof: It is known that Jλ : X → D(A) for any λ . Also, if x ∈ conv(D(A)), then it wasshown that Jλ x→ x. Clearly

conv(D(A))⊇ D(A)

Now if x is in the set on the left, Jλ x→ x and so in fact, since Jλ x ∈ D(A) , it must bethe case that x ∈ D(A). Thus the two sets are the same and so in fact, D(A) is closed andconvex.

Note that this implies that A(D(A)) is also convex. This is because A−1 describedabove, is maximal monotone with domain A(D(A)).

Next is a useful generalization of some of the earlier material used to establish theabove results on approximation. It will include the general case of F a duality map forp > 1.

Proposition 25.7.38 Suppose A : X →P (X ′) where X is a reflexive Banach space withstrictly convex norm. Suppose also that A is maximal monotone. Then if λ n → 0 and ifxn→ x weakly, Aλ n xn→ x∗ weakly, and

lim supn,m→∞

⟨Aλ nxn−Aλ mxm,xn− xm

⟩≤ 0

Thenlim

n,m→∞

⟨Aλ nxn−Aλ mxm,xn− xm

⟩= 0,

[x,x∗] ∈ G (A), and⟨Aλ nxn,xn

⟩→ ⟨x∗,x⟩.