1570 CHAPTER 48. MULTIFUNCTIONS AND THEIR MEASURABILITY

By Theorem 48.5.2,

wB (ω) ∈ B(u(ω) ,ω) ,wC (ω) ∈C (u(ω) ,ω) .

Also, there is a subsequence, still denoted with n(ω) such that

lim infn(ω)→∞

⟨wB

n(ω) (ω)+wCn(ω) (ω) ,un(ω) (ω)−u(ω)

⟩≥ ⟨w(u(ω)) ,u(ω)−u(ω)⟩= 0

for some w(u(ω)) ∈ B(u(ω) ,ω)+C (u(ω) ,ω) because the sum of pseudomonotone op-erators is pseudomonotone. Thus for this subsequence, since

lim supn(ω)→∞

⟨wB

n(ω) (ω)+wCn(ω) (ω) ,un(ω) (ω)−u(ω)

⟩≤ 0≤ lim inf

n(ω)→∞

⟨wB

n(ω) (ω)+wCn(ω) (ω) ,un(ω) (ω)−u(ω)

⟩,

it follows that

limn(ω)→∞

⟨wB

n(ω) (ω)+wCn(ω) (ω) ,un(ω) (ω)−u(ω)

⟩= 0.

We will consider this subsequence or a further subsequence. Then since the limsup condi-tion holds for this subsequence, there exists for any v ∈V,

wB (v) ∈ B(u(ω) ,ω) ,wC (v) ∈C (u(ω) ,ω)

such that

⟨wB (ω)+wC (ω) ,u(ω)− v⟩

≥ lim infn(ω)→∞

 ⟨wB

n(ω) (ω)+wCn(ω) (ω) ,un(ω) (ω)−u(ω)

⟩+⟨

wBn(ω) (ω)+wC

n(ω) (ω) ,u(ω)− v⟩ 

= lim infn(ω)→∞

⟨wB

n(ω) (ω)+wCn(ω) (ω) ,un(ω) (ω)− v

⟩≥ ⟨wB (v)+wC (v) ,u(ω)− v⟩

Finally, let v ∈ K (ω) . Then it follows that there exists a sequence {v̂n} such that v̂n ∈Kn (ω) which converges strongly to v. Thus⟨

wBn (ω)+wC

n (ω) ,un (ω)− v̂n⟩≤ ⟨ f (ω) ,un (ω)− v̂n⟩

Then⟨wB (ω)+wC (ω) ,u(ω)− v⟩=

lim supn(ω)→∞

→0⟨

wBn(ω) (ω)+wC

n(ω) (ω) ,un(ω) (ω)−u(ω)⟩

+⟨

wBn(ω) (ω)+wC

n(ω) (ω) ,u(ω)− v⟩

