886 CHAPTER 25. NONLINEAR OPERATORS

Definition 25.7.22 A set valued operator B is quasi-bounded if whenever x ∈ D(B) andx∗ ∈ Bx are such that

|⟨x∗,x⟩| , ∥x∥ ≤M,

it follows that ∥x∗∥ ≤ KM . Bounded would mean that if ∥x∥ ≤M, then ∥x∗∥ ≤ KM . Hereyou only know this if there is another condition.

By Proposition 25.7.23 an example of a quasi-bounded operator is a maximal monotoneoperator G for which 0 ∈ int(D(G)).

Then there is a useful result which gives examples of quasi-bounded operators [25].

Proposition 25.7.23 Let A : D(A)⊆ X →P (X ′) be maximal monotone and suppose 0 ∈int(D(A)) . Then A is quasi-bounded.

Proof: From local boundedness, Theorem 25.7.20, there exists δ ,C > 0 such that

sup{∥x∗∥ : x∗ ∈ A(x) for ∥x∥ ≤ δ}<C

Now suppose that ∥x∥ , |⟨x∗,x⟩| ≤M. Then letting ∥y∥ ≤ δ ,y∗ ∈ Ay,

0≤ ⟨x∗− y∗,x− y⟩= ⟨x∗,x⟩−⟨x∗,y⟩−⟨y∗,x⟩+ ⟨y∗,y⟩

and so for ∥y∥ ≤ δ ,

⟨x∗,y⟩ ≤ ⟨x∗,x⟩−⟨y∗,x⟩+ ⟨y∗,y⟩ ≤M+MC+Cδ

Hence, ∥x∗∥ ≤M+MC+Cδ ≡ KM .This is actually quite a restrictive requirement and leaves out a lot which would be

interesting.

Definition 25.7.24 Let V be a Reflexive Banach space. We say T : V →P (V ′) is pseu-domonotone if the following conditions hold.

Tu is closed, nonempty, convex. (25.7.51)

If F is a finite dimensional subspace of V , then if u ∈ F and W ⊇ Tu for W a weakly openset in V ′, then there exists δ > 0 such that

v ∈ B(u,δ )∩F implies T v⊆W. (25.7.52)

If uk ⇀ u and if u∗k ∈ Tuk is such that

lim supk→∞

u∗k (uk−u)≤ 0,

then for all v ∈V , there exists u∗ (v) ∈ Tu such that

lim infk→∞

u∗k (uk− v)≥ u∗ (v)(u− v). (25.7.53)

Then here is an interesting result [39].