2642 CHAPTER 77. STOCHASTIC INCLUSIONS

77.8 Adding A Quasi-bounded OperatorRecall the following conditions for the various operators.

Bounded and coercive conditions

A(·,ω) . A(·,ω) : VI → V ′I for each I a subinterval of [0,T ] I =[0, T̂], T̂ ≤ T

A(·,ω) : VI →P(V ′I) is bounded, (77.8.73)

If, for u ∈ V ,u∗X[0,T̂ ] ∈ A

(uX[0,T̂ ],ω

)for each T̂ in an increasing sequence converging to T, then

u∗ ∈ A(u,ω) (77.8.74)

Assume the specific estimate

sup{∥u∗∥V ′I : u∗ ∈ A(u,ω)

}≤ a(ω)+b(ω)∥u∥p−1

VI(77.8.75)

where a(ω) ,b(ω) are nonnegative. Also assume the following coercivity estimate validfor each t ≤ T and for some λ (ω)≥ 0,

inf(∫ t

0⟨u∗,u⟩+λ (ω)⟨Bu,u⟩dt : u∗ ∈ A(u,ω)

)≥ δ (ω)

∫ t

0∥u∥p

V ds−m(ω) (77.8.76)

where m(ω) is some nonnegative constant, δ (ω)> 0.

Limit condition

Let U be a Banach space dense in V and that if ui ⇀ u in VI and u∗i ∈A(ui) with u∗i ⇀ u∗

in V ′I and (Bui)′⇀ (Bu)′ in U ′

rI , ⇀ denoting weak convergence, then if

lim supi→∞

⟨u∗i ,ui−u⟩V ′I ,VI≤ 0

it follows that for all v ∈ VI , there exists u∗(v) ∈ Au such that

lim infi→∞⟨u∗i ,ui− v⟩V ′I ,VI

≥ ⟨u∗ (v) ,u− v⟩V ′I ,VI(77.8.77)

where r > max(p,2) , and we replace p with r and I an arbitrary subinterval of the form[0, T̂], T̂ < T, for [0,T ], and U for V where indicated. Here

UrI ≡ Lr (I;U)

Typically, U is compactly embedded in V .

Measurability condition