1584 CHAPTER 48. MULTIFUNCTIONS AND THEIR MEASURABILITY

We will obtain a contradiction to this. In what follows we continue to use the subsequencejust described which satisfies the above inequality.

The estimate 48.7.26 implies,

0≤ ⟨zn (t) ,un (t)−u(t)⟩− ≤ c(t)+C ||u(t)||pV , (48.7.30)

where c is a function in L1 (0,T ). Thanks to (48.7.25),

lim infn→∞⟨zn (t) ,un (t)−u(t)⟩ ≥ 0,

and, therefore, the following pointwise limit exists,

limn→∞⟨zn (t) ,un (t)−u(t)⟩− = 0,

and so we may apply the dominated convergence theorem using (48.7.30) and conclude

limn→∞

∫ T

0⟨zn (t) ,un (t)−u(t)⟩−dt =

∫ T

0limn→∞⟨zn (t) ,un (t)−u(t)⟩−dt = 0

Now, it follows from (48.7.27) and the above equation, that

limn→∞

∫ T

0⟨zn (t) ,un (t)−u(t)⟩+dt

= limn→∞

∫ T

0⟨zn (t) ,un (t)−u(t)⟩+ ⟨zn (t) ,un (t)−u(t)⟩−dt

= limn→∞⟨zn,un−u⟩V ′,V = 0.

Therefore, both∫ T

0 ⟨zn (t) ,un (t)−u(t)⟩+dt and∫ T

0 ⟨zn (t) ,un (t)−u(t)⟩−dt converge to 0,thus,

limn→∞

∫ T

0|⟨zn (t) ,un (t)−u(t)⟩|dt = 0 (48.7.31)

limn→∞⟨zn,un−u⟩V ′,V = 0

From the above, it follows that there exists a further subsequence {nk} not depending on tsuch that ∣∣⟨znk (t) ,unk (t)−u(t)⟩

∣∣→ 0 a.e. t. (48.7.32)

Therefore, by the pseudomonotone limit condition for A there exists wt ∈ A(u(t) , t)such that for a.e. t,

α (t) ≡ lim infk→∞⟨znk (t) ,unk (t)− y(t)⟩

= lim infk→∞⟨znk (t) ,u(t)− y(t)⟩ ≥ ⟨wt ,u(t)− y(t)⟩.

Then on the exceptional set, let α (t)≡ ∞, and consider the set

F (t)≡ {w ∈ A(u(t) , t) : ⟨w,u(t)− y(t)⟩ ≤ α (t)} ,