1582 CHAPTER 48. MULTIFUNCTIONS AND THEIR MEASURABILITY

Then, there exists a subsequence {nk}, which may depend on t, such that

limk→∞⟨znk (t) ,unk (t)−u(t)⟩= lim inf

n→∞⟨zn (t) ,un (t)−u(t)⟩< 0. (48.7.24)

Now, condition 3 implies that for all k large enough,

b3∣∣∣∣unk (t)

∣∣∣∣pV −b4 (t)−λ

∣∣unk (t)∣∣2H <

∣∣∣∣znk (t)∣∣∣∣

V ′ ||u(t)||V

≤(

b1∣∣∣∣unk (t)

∣∣∣∣p−1V +b2 (t)

)||u(t)||V ,

therefore,∣∣∣∣unk (t)

∣∣∣∣V and consequently

∣∣∣∣znk (t)∣∣∣∣

V ′ are bounded. This follows from 48.7.22in case λ > 0. Note that

∣∣∣∣znk (t)∣∣∣∣

V ′ is bounded independently of nk because of the assump-tion that A(·, t) is bounded and we just showed that

∣∣∣∣unk (t)∣∣∣∣

V is bounded.Taking a further subsequence if necessary, let unk (t)→ u(t) weakly in U ′ and unk (t)→

ξ weakly in V . Thus, by density considerations, ξ = u(t). Now, 48.7.24 and the limit con-ditions for pseudomonotone operators imply that the liminf condition holds.There existsz∞ ∈ A(u(t) , t) such that

lim infk→∞⟨znk (t) ,unk (t)−u(t)⟩ ≥ ⟨z∞,u(t)−u(t)⟩= 0

> limk→∞⟨znk (t) ,unk (t)−u(t)⟩,

which is a contradiction. This completes the proof of the claim.We continue with the proof of the theorem. It follows from this claim that for every

t /∈ Σ,lim inf

n→∞⟨zn (t) ,un (t)−u(t)⟩ ≥ 0. (48.7.25)

Also, it is assumed thatlim sup

n→∞

⟨zn,un−u⟩V ≤ 0.

Then from the estimates,∫ T

0

(b3 ||un (t)||pV −b4 (t)−λ |un (t)|2H

)dt ≤

∫ T

0∥u(t)∥V

(∥un∥p−1 b1 +b2

)dt

so it is routine to get ∥un∥V is bounded. This follows from the assumptions, in particular48.7.22.

Now, the coercivity condition 3 shows that if y ∈ Lp ([0,T ] ;V ), then

⟨zn (t) ,un (t)− y(t)⟩ ≥ b3 ||un (t)||pV −b4 (t)−λ |un (t)|2H−(

b1 ||un (t)||p−1 +b2 (t))||y(t)||V .

Using p−1 = pp′ , where 1

p +1p′ = 1, the right-hand side of this inequality equals

b3 ||un (t)||pV −b4 (t)−b1 ||un (t)||p/p′ ||y(t)||V −b2 (t) ||y(t)||V −λ |un (t)|2H ,