76.5. REPLACING Φ WITH σ (u) 2577
=CK2E(∫ t
0
⟨B(ψ
n−1w1−ψn−1w2
),ψn−1w1−ψ
n−1w2⟩(t1)dt1
)≤(CK2)2
E(∫ t
0
∫ t1
0
⟨B(ψ
n−2w1−ψn−2w2
),ψn−2w1−ψ
n−2w2⟩(t2)dt2dt1
)· · · ≤
(CK2)n
E(∫ t
0
∫ t1
0· · ·∫ tn−1
0⟨B(w1−w2) ,w1−w2⟩(tn)dtn · · ·dt2dt1
)=(CK2)n
∫ t
0
∫ t1
0· · ·∫ tn−1
0E (⟨B(w1−w2) ,w1−w2⟩(tn))dtn · · ·dt2dt1
≤(CK2)n
supt
E (⟨B(w1−w2) ,w1−w2⟩(t))T n
n!<
12∥w1−w2∥2
L∞([0,T ],L2(Ω,W ))
provided n is sufficiently large. It follows that
∥ψnw1−ψnw2∥2
L∞([0,T ],L2(Ω,W )) ≤12∥w1−w2∥2
L∞([0,T ],L2(Ω,W ))
for all n sufficiently large. Hence, if one begins with
w ∈ L∞([0,T ] ,L2 (Ω,W )
)∩L2 ([0,T ]×Ω,W ) ,
the sequence of iterates {ψnw}∞
n=1 converges to a fixed point u in L∞([0,T ] ,L2 (Ω,W )
).
This u is automatically in L2 ([0,T ]×Ω,W ) and is progressively measurable since each ofthe iterates is progressively measurable. This proves the following theorem.
Theorem 76.5.2 Suppose f ∈ V ′ is progressively measurable and that (t,ω)→ σ (t,u,ω)is progressively measurable whenever u is. Suppose that B : W →W ′ is a Riesz map.
λB+A(ω) : Vω → V ′ω , λB+A : V → V ′
are monotone hemicontinuous and bounded where
A(ω)u(t)≡ A(t,u(t) ,ω)
and (t,u,ω)→ A(t,u,ω) is progressively measurable. Also suppose for p ≥ 2, the coer-civity, and the boundedness conditions
λ ⟨Bu,u⟩+ ⟨A(t,u,ω) ,u⟩V ≥ δ ∥u∥pV − c(t,ω) (76.5.60)
for all λ large enough.
∥A(t,u,ω)∥V ′ ≤ k∥u∥p−1V + c1/p′ (t,ω) (76.5.61)
where c ∈ L1 ([0,T ]×Ω). Also suppose that
∥σ (t,u,ω)∥W ≤C+C∥u∥W∥σ (t,u,ω)−σ (t, û,ω)∥L2(Q1/2U,W) ≤ K ∥u− û∥W
Then if u0 ∈ L2 (Ω,W ) with u0 F0 measurable, there exists a unique solution u(·,ω) ∈ Vω
with u ∈ V (Lp ([0,T ]×Ω,V ) and progressively measurable) such that for ω off a set ofmeasure zero,
Bu(t,ω)−Bu0 (ω)+∫ t
0A(s,u(s,ω) ,ω)ds =
∫ t
0f ds+B
∫ t
0σ (u)dW.