25.7. MAXIMAL MONOTONE OPERATORS 881

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.

Theorem 25.7.13 Let X be a strictly convex reflexive Banach space. Suppose the graphof A : X →P (X) is maximal monotone in the sense that it is monotone and no monotonegraph can properly contain the graph of A. Then for all λ > 0,λF +A is onto. Conversely,if for some λ > 0,λF +A is onto, then the graph of A is maximal with respect to beingmonotone.

Proof: In Theorem 25.7.9, let Bx≡ λF (x)−y0. Then from the properties of the dualitymap, Theorem 25.2.3 above, it follows that B satisfies the necessary conditions to use theresult of Corollary 25.7.11 with K = X . This B is monotone hemicontinuous, and coercive.Thus there exists x such that for all [u,v] ∈ G (A) ,

⟨λF (x)− y0 + v,u− x⟩X ′,X ≥ 0⟨v− (y0−λF (x)) ,u− x⟩X ′,X ≥ 0

By maximality of the graph, it follows that x ∈ D(A) and

y0−λF (x) ∈ A(x) , y0 = λF (x)+A(x)

so λF +A is onto as claimed. The converse was proved in Theorem 25.7.2.Note that this theorem holds if F is a duality map for p > 1. That is, ⟨Fx,x⟩ =

∥x∥p ,∥Fx∥= ∥x∥p−1.Suppose A : X →P (X) is maximal monotone. Then let z ∈ X and define a new map-

ping  as follows.

D(Â)≡ {x : x− x0 ∈ D(A)} , Â(x)≡ A(x− x0)

Proposition 25.7.14 Let A,  be as just defined. Then  is also maximal monotone.

Proof: From Theorem 25.7.13 it suffices to show that graph of  is monotone and ismaximal. Suppose then that x∗i ∈ Âxi. Then

⟨x∗1− x∗2,x1− x2⟩= ⟨x∗1− x∗2,x1− x0− (x2− x0)⟩

by definition, x∗i ∈ A(xi− x0) and so the above is ≥ 0. Next suppose for all [x,x∗] ∈ G(Â),

⟨x∗− z∗,x− z⟩ ≥ 0

Does it follow that [z,z∗] ∈ G(Â)? The above says that

⟨x∗− z∗,x− x0− (z− x0)⟩ ≥ 0

whenever x−x0 ∈D(A) and x∗ ∈ A(x− x0) . Hence, since A is given to be maximal mono-tone, z− x0 ∈ D(A) and z∗ ∈ A(z− x0) which says that z∗ ∈ Â(z). Thus  is maximalmonotone by the Theorem 25.7.13.