25.8. PERTURBATION THEOREMS 923

Definition 25.8.6 Now let L : D(L) ⊆ V → V ′ such that L is linear, monotone, D(L) isdense in V , L is closed, and L∗ is monotone. Let A : V →P (V ′) be a bounded operator.Then A is called L pseudomonotone if Av is closed and convex in V ′ and for any sequence{un} ⊆ D(L) such that un→ u weakly in V and Lun→ Lu weakly in V ′, and for z∗n ∈ Aun,

lim supn→∞

⟨z∗n,un−u⟩ ≤ 0

then for every v ∈V, there exists z∗ (v) ∈ Au such that

lim infn→∞⟨z∗n,un− v⟩ ≥ ⟨z∗ (v) ,u− v⟩

It is called L modified bounded pseudomonotone if the above liminf condition holds forsome subsequence whenever un→ u weakly and Lun→ Lu weakly and

lim supn→∞

⟨z∗n,un−u⟩ ≤ 0

Lemma 25.8.7 Suppose X is the Banach space

X = D(L) , ∥u∥X ≡ ∥u∥V +∥Lu∥V ′

where L is as described in the above definition. Also assume that A is bounded. Then if Ais L pseudomonotone, it follows that A is pseudomonotone as a map from X to P (X ′). IfA is L modified bounded pseudomonotone, then A is modified bounded pseudomonotone asa map from X to P (X ′).

Proof: Is A bounded? Of course, because the norm of X is stronger than the norm onV . Is Au convex and closed? This also follows because X ⊆V . It is clear that Au is convex.If {zn} ⊆ Au and zn→ z in X ′, then does it follow that z ∈ Au? Since A is bounded, thereis a further subsequence which converges weakly to w in V ′. However, Au is convex andclosed so it is weakly closed. Hence w ∈ Au and also w = z. It only remains to verify thepseudomonotone limit condition. Suppose then that un→ u weakly in X and for z∗n ∈ Aun,

lim supn→∞

⟨z∗n,un−u⟩ ≤ 0

Then it follows that Lun→ Lu weakly in V ′ and un→ u weakly in V so u ∈ X . Hence theassumption that A is L pseudomonotone implies that for every v ∈ V, and for every v ∈ X ,there exists z∗ (v) ∈ Au⊆V ′ ⊆ X ′ such that

lim infn→∞⟨z∗n,un− v⟩ ≥ ⟨z∗ (v) ,u− v⟩

The last claim goes the same way. You just have to take a subsequence.Then we have the following major surjectivity result. In this theorem, we will assume

for simplicity that all spaces are real spaces. Versions of this appear to be due to Brezis [23]and Lions [91]. Of course the theorem holds for complex spaces as well. You just need touse Re⟨ ⟩ instead of ⟨ ⟩ .