896 CHAPTER 25. NONLINEAR OPERATORS

Definition 25.7.35 Let xλ just defined

0 ∈ F (xλ − x)+λAxλ

be denoted by Jλ x and define also

Aλ (x) =−λ−(p−1)F (xλ − x) =−λ

−(p−1)F (Jλ x− x) .

This is for F a duality map with p > 1. Thus for the usual duality map, you would have

Aλ (x) =−λ−1F (Jλ x− x)

Recall how this xλ is defined. In general,

0 ∈ F (Jλ x− x)+λp−1Axλ

Thus, from the definition,Aλ (x) ∈ A(Jλ x)

Formally, and to help remember what is going on, you are looking at a generalizationof

Aλ x =A

1+λAx =

1λ

(x− (I +λA)−1 x

)This is in the case where F = I to keep things simpler. You have 0 = xλ −x+λAxλ and soformally xλ = (I +λA)−1 x. Thus you are looking at 1

λ(x− xλ ) =

1λ

(x− (I +λA)−1 x

)=

Aλ x. In fact, this is exactly what you do when you are in a single Hilbert space. This is justa generalization to mappings between Banach spaces and their duals.

Then there are some things which can be said about these operators. It is presented forthe general duality map for p > 1.

Theorem 25.7.36 The following hold. Here X is a reflexive Banach space with strictlyconvex norm. A : D(A)→P (X ′) is maximal monotone. Then

1. Jλ and Aλ are bounded single valued operators defined on X . Bounded means theytake bounded sets to bounded sets. Also Aλ is a monotone operator.

2. Aλ ,Jλ are demicontinuous. That is, strongly convergent sequences are mapped toweakly convergent sequences.

3. For every x ∈ D(A) ,

∥Aλ (x)∥ ≤ |Ax| ≡ inf{∥y∗∥ : y∗ ∈ Ax} .

For every x ∈ conv(D(A)), it follows that limλ→0 Jλ (x) = x. The new symbol meansthe closure of the convex hull. It is the closure of the set of all convex combinationsof points of D(A).