922 CHAPTER 25. NONLINEAR OPERATORS

Theorem 25.8.3 Let V be a reflexive Banach space and let T : V → P (V ′) be pseu-domonotone, bounded and coercive. Then T is onto. More generally, this continues tohold if T is modified bounded pseudomonotone.

Recall the definition of pseudomonotone.

Definition 25.8.4 For X a reflexive Banach space, we say A : X→P (X ′) is pseudomono-tone if the following hold.

1. The set Au is nonempty, closed and convex for all u ∈ X .

2. If F is a finite dimensional subspace of X, u ∈ F, and if U is a weakly open set in V ′

such that Au ⊆U, then there exists a δ > 0 such that if v ∈ Bδ (u)∩F then Av ⊆U.(Weakly upper semicontinuous on finite dimensional subspaces.)

3. If ui→ u weakly in X and u∗i ∈ Aui is such that

lim supi→∞

⟨u∗i ,ui−u⟩ ≤ 0, (25.8.87)

then, for each v ∈ X , there exists u∗ (v) ∈ Au such that

lim infi→∞⟨u∗i ,ui− v⟩ ≥ ⟨u∗(v),u− v⟩. (25.8.88)

Also recall the definition of modified bounded pseudomonotone. It is just the aboveexcept that the limit condition is replaced with the following condition: If ui → u weaklyin X and

lim supi→∞

⟨u∗i ,ui−u⟩ ≤ 0, (25.8.89)

then there exists a subsequence, still denoted as {ui} such that for each v ∈ X , there existsu∗ (v) ∈ Au such that

lim infi→∞⟨u∗i ,ui− v⟩ ≥ ⟨u∗(v),u− v⟩. (25.8.90)

Also recall that this more general limit condition along with the assumption 1 and theassumption that A is bounded is sufficient to obtain condition 2. This was Lemma 25.4.9proved earlier and stated here for convenience.

Lemma 25.8.5 Let A : X →P (X ′) satisfy conditions 1 and 3 above and suppose A isbounded. Also suppose the condition that if xn→ x weakly and

lim supn→∞

⟨zn,xn− x⟩ ≤ 0

implies there exists a subsequence{

xnk

}such that for any y,

lim infn→∞

⟨znk ,xnk − y

⟩≥ ⟨z(y) ,x− y⟩

for z(y) some element of Ax. Then if this weaker condition holds, you have that if U is aweakly open set containing Ax, then Axn ⊆U for all n large enough.