2616 CHAPTER 77. STOCHASTIC INCLUSIONS

It follows that J−1Luε ∈ D(Λ∗) and so for all v ∈ D(Λ) ,

ε⟨Λ∗J−1Luε ,v⟩+ ⟨Fuε ,v⟩+ ⟨Luε ,v⟩= ⟨ f ,v⟩. (77.4.12)

Since D(Λ) is dense in V , this equation holds for all v ∈ V and so in particular, it holdsfor v = J−1Luε . Therefore,

−||Fuε ||V ′ ||Luε ||V ′ + ||Luε ||2V ′ ≤ || f ||V ′ ||Luε ||V ′ . (77.4.13)

It follows from 77.4.13, 77.4.11 that ||Luε ||V ′ is bounded independent of ε . Therefore,there exists a sequence ε → 0 such that

uε ⇀ u in V , (77.4.14)

Kuε ⇀ Ku in X ′, (77.4.15)

Fuε ⇀ u∗ in V ′, (77.4.16)

Buε (0)⇀ Bu(0) in W ′. (77.4.17)

In 77.4.10 replace v with uε −u. Using J−1 is monotone,

ε⟨Luε −Lu,J−1Lu⟩+ ⟨Fuε +Kuε ,uε −u⟩

≤ ⟨ f ,uε −u⟩+ ⟨B(uε −u)(0) ,u0⟩ (77.4.18)

Formula 77.4.17 applied to the last term of 77.4.18 implies

lim supε→0⟨Fuε +Kuε ,uε −u⟩ ≤ 0. (77.4.19)

By pseudomonotonicity,

lim infε→0⟨Fuε +Kuε ,uε −u⟩ ≥ ⟨Fu+Ku,u−u⟩= 0

so limε→0⟨Fuε +Kuε ,uε −u⟩= 0 and so

⟨u∗+Ku,u− v⟩

lim infε→0

(⟨Fuε +Kuε ,uε −u⟩+ ⟨Fuε +Kuε ,u− v⟩) =

lim infε→0⟨Fuε +Kuε ,uε − v⟩ ≥ ⟨Fu+Ku,u− v⟩

and so u∗ = Fu and from 77.4.10,

⟨Ku,v⟩+ ⟨Fu,v⟩= ⟨ f ,v⟩+ ⟨Bv(0) ,u0⟩ (77.4.20)

Thus for every v ∈ X ,∫ T

0

⟨(Bu)′ ,v

⟩ds+ ⟨Bu,v⟩(0)+

∫ T

0⟨Fu,v⟩ds =

∫ T

0⟨ f ,v⟩ds+ ⟨Bv(0) ,u0⟩