76.5. REPLACING Φ WITH σ (u) 2573
≤
∫Ω
sups∈[0,t] ⟨B(u1−u2) ,u1−u2⟩1/2 ·
2ĈeλT(∫ t
0 K2 ∥w1−w2∥2W dt
)1/2dP
≤ E
(12
sups∈[0,t]
⟨B(u1−u2) ,u1−u2⟩(s)
)
+ĈeλT E(∫ t
0K2 ∥w1−w2∥2
W dt)
It follows that, after adjusting constants as needed, one gets an inequality of the followingform.
12
E
(sup
s∈[0,t]⟨B(u1−u2) ,u1−u2⟩(s)
)+4
∫Ω
∫ t
0∥u1−u2∥2
W dsdP
≤ ĈeλT E(∫ t
0K2 ∥w1−w2∥2
W dt)
This holds for every t ≤ T and so, from the estimate on the size of T, it follows that∫ T
0
∫Ω
∥u1−u2∥2W dsdP≤ 3
4
∫ T
0
∫Ω
∥w1−w2∥2W dtdP
Therefore, there is a unique fixed point to this mapping which takes w∈W to u the solutionto the integral equation. We denote it as u. Thus u is progressively measurable and for ω
off a set of measure zero, we have a solution to the integral equation
Bu(t,ω)−Bu0 (ω)+∫ t
0A(s,u(s,ω) ,ω)ds
=∫ t
0f ds+B
∫ t
0σ (u)dW, t ∈ [0,T ]
Now the same argument can be repeated for the succession of intervals mentioned above.However, you need to be careful that at T, you have Bu(T,ω) = B(u(T,ω)) for ω off aset of measure zero. If this is not so, you locate T ′ close to T for which it is so as inLemma 73.3.1 and use this T ′ instead, but these are mainly technical issues. This provesthe following existence and uniqueness theorem.
Theorem 76.5.1 Suppose f ∈ V ′ is progressively measurable and that
(t,ω)→ σ (t,ω,u(t,ω))
is progressively measurable whenever u is. Suppose that
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)