850 CHAPTER 25. NONLINEAR OPERATORS

for all w ∈ Cn. Then A satisfies the conditions of Lemma 25.4.6 and so A is onto. Conse-quently T is also onto.

With these lemmas, it is possible to prove a very useful result about a class of mappingswhich map a reflexive Banach space to the power set of its dual space. For more theoremsabout these mappings and their applications, see [99]. In the discussion below, we will usethe symbol, ⇀, to denote weak convergence.

Definition 25.4.8 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. (25.4.15)

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. (25.4.16)

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

lim supk→∞

Reu∗k (uk−u)≤ 0,

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

lim infk→∞

Reu∗k (uk− v)≥ Reu∗ (v)(u− v). (25.4.17)

We say T is coercive if

lim||v||→∞

inf{

Rez∗ (v)||v||

: z∗ ∈ T v}= ∞. (25.4.18)

In the case that T takes bounded sets to bounded sets so it is a bounded set valuedoperator, it turns out you don’t have to consider the second of the above conditions aboutthe upper semicontinuity. It follows from the other conditions. It is convenient to use thenotation

⟨u∗,v⟩ ≡ u∗ (v) ,u∗ ∈V ′,v ∈V.

and this will be used interchangeably with the earlier notation from now on.The next lemma has to do with upper semicontinuity being obtained from simpler con-

ditions.

Lemma 25.4.9 Let T : X→P (X ′) satisfy conditions 25.4.15 and 25.4.17 above and sup-pose T is bounded (T x for x in a bounded set is bounded). Then if xn→ x in X , and if Uis a weakly open set containing T x, then T xn ⊆U for all n large enough. If fact the limitcondition 25.4.17 can be weakened to the following more general condition: If uk ⇀ u, and

lim supk→∞

Reu∗k (uk−u)≤ 0, (**)