76.3. THE EXISTENCE OF APPROXIMATE SOLUTIONS 2553

Thus ∥∥∥∥∫ t∧τn

0ΦdW

∥∥∥∥E≤ 2n

Then you could pick u0 ∈ Lp (Ω,V ) ,u0 being F0 measurable, and let

q(t,ω) =∫ t∧τn

0ΦdW.

The result is clearly in V and is continuous in t. Therefore, from Corollary 76.3.7, thereexists a unique solution u ∈ V to the initial value problem(

Bu−B∫ t∧τn

0ΦdW

)′(·,ω)+A(ω)(u(·,ω)) = f (·,ω) , Bu(0) = Bu0

Integrating, one obtains a unique solution un ∈ V to the integral equation

Bun (t,ω)−Bu0 (ω)+∫ t

0A(s,un,ω)ds =

∫ t

0f (s,ω)ds+B

∫ t∧τn

0ΦdW

This holds in V ′ω and is so for all ω off a set of measure zero Nn. Let N = ∪nNn. Forω /∈ N, t →

∫ t0 ΦdW is continuous and so for all n large enough, τn = ∞. Thus for a fixed

ω, it follows that for all n large enough τn = ∞ and so one obtains

Bun (t,ω)−Bu0 (ω)+∫ t

0A(s,un,ω)ds =

∫ t

0f (s,ω)ds+B

∫ t

0ΦdW

Then for k some other index sufficiently large, the same holds for uk. By the uniquenessassumption 76.3.17, uk (t,ω) = un (t,ω) and so it follows that limn→∞ un (t,ω) exists be-cause for each ω off a set of measure zero, there is eventually no change in un. Definingu(t,ω) ≡ limn→∞ un (t,ω) ≡ un (t,ω) for all n large enough, it follows that u is progres-sively measurable since it is the pointwise limit of progressively measurable functions and

Bu(t,ω)−Bu0 (ω)+∫ t

0A(s,u,ω)ds =

∫ t

0f (s,ω)ds+B

∫ t

0ΦdW

This has shown the following lemma.

Lemma 76.3.8 Let (t,u,ω)→ A(t,u,ω) be progressively measurable into V ′ and supposefor some λ ,

λB+A(ω) : Vω → V ′ω ,

λB+A : V → V ′

are both monotone bounded and hemicontinuous. Also suppose the two estimates givingboundedness and coercivity 76.3.23 - 76.3.24 of Corollary 76.3.7 above. Here V,W areas described above V ⊆W,W ′ ⊆V ′, W is a separable Hilbert space and V is a separablereflexive Banach space. B : W →W ′ is nonnegative and self adjoint. Let f ∈ V ′ and letu0 ∈ Lp (Ω,V ) where u0 is F0 measurable. Then if Φ ∈ L2

([0,T ]×Ω,L2

(Q1/2U,E

)), Φ