25.8. PERTURBATION THEOREMS 929

Then one can conclude thatlim sup

n→∞

⟨u∗n,un−u⟩ ≤ 0

Proof: Let α ≡ limsupn→∞ ⟨u∗n,un⟩ . It is a finite number because these sequences arebounded. Then using the weak convergence,

0 ≥ lim supm→∞

(lim sup

n→∞

⟨u∗n−u∗m,un−um⟩)

= lim supm→∞

(lim sup

n→∞

(⟨u∗n,un⟩+ ⟨u∗m,um⟩−⟨u∗n,um⟩−⟨u∗m,un⟩))

= lim supm→∞

(α + ⟨u∗m,um⟩−⟨u∗,um⟩−⟨u∗m,u⟩)

= (α +α−⟨u∗,u⟩−⟨u∗,u⟩) = 2α−2⟨u∗,u⟩

Nowlim sup

n→∞

⟨u∗n,un−u⟩= α−⟨u∗,u⟩ ≤ 0.

To begin with, consider the approximate problem which is to determine whether L+A+Bλ is onto. Here Bλ x = −λ

−1F (xλ − x) where 0 ∈ F (xλ − x)+ λBx. In the nota-tion given above, Bλ x = −λ

−1F (Jλ x− x). Then by Theorem 25.7.36, Bλ is monotone,demicontinuous, and bounded. In addition, we assume 0 ∈ D(B) . Then

⟨Bλ x,x⟩ ≥ ⟨Bλ 0,x⟩ ≥ −|B(0)|∥x∥ (25.8.94)

Lemma 25.8.12 Let A be pseudomonotone, bounded and coercive and let 0 ∈D(B). Thenif y∗ ∈ X ′, there exists a solution xλ to

y∗ ∈ Lxλ +Axλ +Bλ xλ

Proof: From the inequality 25.8.94, A+Bλ is coercive. It is also bounded and pseu-domonotone. It is pseudomonotone from Theorem 25.7.27. Therefore, there exists a solu-tion xλ by Theorem 25.8.8.

Acting on xλ and using the inequality 25.8.94, it follows that these solutions xλ lie in abounded set. The details follow. Letting z∗

λ∈ Axλ be such that equality holds in the above

inclusion,y∗ = Lxλ + z∗

λ+Bλ xλ (25.8.95)

∥y∗∥ ≥ ⟨y∗,xλ ⟩∥xλ∥

=⟨Lxλ ,xλ ⟩+

⟨z∗

λ,xλ

⟩+ ⟨Bλ xλ ,xλ ⟩

∥xλ∥

≥⟨Lxλ ,xλ ⟩+

⟨z∗

λ,xλ

⟩−|B(0)|∥xλ∥

∥xλ∥

≥⟨z∗

λ,xλ

⟩∥xλ∥

− |B(0)|