2620 CHAPTER 77. STOCHASTIC INCLUSIONS

Thus, by monotonicity,⟨kn (ω)(Bun)

′ ,un−u⟩V ′,V =

⟨k (ω)(Bun)

′ ,un−u⟩V ′,V

+⟨(kn (ω)− k (ω))(Bun)

′ ,un−u⟩V ′,V

≥⟨k (ω)(Bu)′ ,un−u

⟩V ′,V +

⟨(kn (ω)− k (ω))(Bun)

′ ,un−u⟩V ′,V

The last term in the above expression converges to 0 due to the convergence of kn (ω) tok (ω). Thus ⟨

(B(ω)u)′ ,un−u⟩V ′,V + ⟨Fun,un−u⟩V ′,V ≤ ⟨ fn,un−u⟩V ′,V

and solim sup

n→∞

⟨Fun,un−u⟩V ′,V ≤ 0

Then as before, one can conclude that Fu = ξ . Then passing to the limit gives the desiredsolution to the equation, this for each ω . However, by uniqueness, it follows that if ū isthe solution to the evolution equation of Lemma 77.4.1, then for each ω,u = ū in V . Alsothis u just obtained is measurable into V thanks to the Pettis theorem. Therefore, ū canbe modified on a set of measure zero for each fixed ω to equal u a function measurableinto V . Hence there exists a solution to the evolution equation of this lemma u which ismeasurable into V . By the Lemma 77.4.2, it follows that there is a representative for uwhich is product measurable into V .

77.5 The Main ResultThe main result is an existence theorem for product measurable solutions to the system

(B(ω)u(·,ω))′+u∗ (·,ω) = f (·,ω) in V ′

B(ω)u(0,ω) = B(ω)u0 (ω) (77.5.22)

where u∗ (·,ω) ∈ A(u(·,ω) ,ω). It is Theorem 77.5.6 below. First are some assumptions.Here I will denote a subinterval of [0,T ] , of the form I =

[0, T̂], T̂ ≤ T , andVI ≡

Lp (I,V ) with similar things defined analogously. We assume only that p > 1.

Definition 77.5.1 For X a reflexive Banach space, we say A : X→P (X ′) is pseudomono-tone and bounded if the following hold.

1. The set Au is nonempty, closed and convex for all u ∈ X . A takes bounded sets tobounded sets.

2. If ui→ u weakly in X and u∗i ∈ Aui is such that

lim supi→∞

⟨u∗i ,ui−u⟩ ≤ 0, (77.5.23)

then, for each v ∈ X , there exists u∗ (v) ∈ Au such that

lim infi→∞⟨u∗i ,ui− v⟩ ≥ ⟨u∗(v),u− v⟩. (77.5.24)