25.4. SET-VALUED MAPS, PSEUDOMONOTONE OPERATORS 849

Lemma 25.4.6 Suppose in addition to 25.4.11 and 25.4.12, (compact convex valued andupper semicontinuous) A is coercive,

lim|x|→∞

inf{

Re(y,x)|x|

: y ∈ Ax}= ∞.

Then A is onto.

Proof: Let y ∈ Cn and let Kr ≡ B(0,r). By Lemma 25.4.5 there exists xr ∈ Kr andwr ∈ Axr such that

Re(y−wr,z−xr)≤ 0 (25.4.13)

for all z ∈ Kr. Letting z = 0,

Re(wr,xr)≤ Re(y,xr).

Therefore,

inf{

Re(w,xr)

|xr|: w ∈ Axr

}≤ |y| .

It follows from the assumption of coercivity that |xr| is bounded independent of r. There-fore, picking r strictly larger than this bound, 25.4.13 implies

Re(y−wr,v)≤ 0

for all v in some open ball containing 0. Therefore, for all v in this ball

Re(y−wr,v) = 0

and hence this holds for all v ∈ Cn and so y = wr ∈ Axr. This proves the lemma.

Lemma 25.4.7 Let F be a finite dimensional Banach space of dimension n, and let T be amapping from F to P (F ′) such that 25.4.11 and 25.4.12 both hold for F ′ in place of Cn.Then if T is also coercive,

lim||u||→∞

inf{

Rey∗ (u)||u||

: y∗∈ T u}= ∞, (25.4.14)

it follows T is onto.

Proof: Let |·| be an equivalent norm for F such that there is an isometry of Cn and F,θ .Now define A : Cn→P (Cn) by Ax≡ θ

∗T θx.

P (F ′) θ∗→ Cn

T ↑ ◦ ↑ A

F θ← Cn

Thus y ∈ Ax means that there exists z∗ ∈ T θx such that

(w,y)Cn = z∗ (θw)