862 CHAPTER 25. NONLINEAR OPERATORS

Then, using this subsequence,

0≥ lim supn→∞

⟨zn +wn,un−u⟩ ≥ δ + lim supn→∞

⟨wn,un−u⟩ ≥ δ

which is a contradiction. Thus the liminf condition must hold for some subsequence.The following is mostly in [99].

Theorem 25.5.4 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, the same holds ifT is modified or generalized bounded pseudomonotone and coercive.

Proof: The proof is for modified bounded pseudomonotone since this is more general.Let F be the set of finite dimensional subspaces of V and let F ∈F . Then define TF as

TF ≡ i∗F TiF

where here iF is the identity map from F to V. Then TF satisfies the conditions of Lemma25.4.7 thanks to Lemma 25.4.9 or Lemma 25.4.14 and so TF is onto P (F ′). Let w∗ ∈V ′.Then since TF is onto, there exists uF ∈ F such that

i∗F w∗ ∈ i∗F TiF uF .

Thus for each finite dimensional subspace F , there exists uF ∈ F such that for all v ∈ F ,

⟨w∗,v⟩= ⟨u∗F ,v⟩ , u∗F ∈ TuF . (25.5.29)

Replacing v with uF , in 25.5.29,

⟨u∗F ,uF⟩||uF ||

=⟨w∗,uF⟩||uF ||

≤ ||w∗||.

Therefore, the assumption that T is coercive implies {uF : F ∈F} is bounded in V . Nowdefine

WF ≡ ∪{

uF ′ : F ′ ⊇ F}.

Then WF is bounded and if WF ≡ weak closure of WF , then{WF : F ∈F

}is a collection of nonempty weakly compact (since V is reflexive and the uF were justshown bounded) sets having the finite intersection property because WF ̸= /0 for each F .(If Fi, i = 1, · · · ,n are finite dimensional subspaces, let F be a finite dimensional subspacewhich contains all of these. Then WF ̸= /0 and WF ⊆ ∩n

i=1WFi .) Thus there exists

u ∈ ∩{

WF : F ∈F}.

I will show w∗ ∈ Tu. If w∗ /∈ Tu, a closed convex set, there exists v ∈V such that

Re⟨w∗,u− v⟩< Re⟨u∗,u− v⟩ (25.5.30)