2650 CHAPTER 77. STOCHASTIC INCLUSIONS

Then returning to 77.8.82, all terms are bounded except∫ t

0⟨Gµ u,u

⟩ds, so this term must

also be bounded for t = T also. Thus∣∣∣∣∫ T

0

⟨Gµ u,u

⟩dt∣∣∣∣≤C

where C is independent of µ . We denote by uµ the solution to the above equation.Here is the definition of quasi-bounded.

Definition 77.8.8 A set valued operator G is quasi-bounded if whenever x ∈ D(G) andx∗ ∈ Gx are such that

|⟨x∗,x⟩| , ∥x∥ ≤M,

it follows that ∥x∗∥ ≤ KM . Bounded would mean that if ∥x∥ ≤M, then ∥x∗∥ ≤ KM . Hereyou only know this if there is another condition.

Assumption 77.8.9 G : D(G)→P (V ′) is quasi-bounded and maximal monotone.

By Proposition 25.7.23 an example of a quasi-bounded operator is a maximal monotoneoperator G for which 0 ∈ int(D(G)).

Now Gµ uµ ∈ GJµ uµ as noted above. Therefore, there exists gµ ∈ G(Jµ uµ

)such that

C≥⟨Gµ uµ ,uµ

⟩V ′,V =

⟨gµ ,uµ

⟩V ′,V =

⟨gµ ,Jµ uµ

⟩V ′,V +

⟨gµ ,uµ − Jµ uµ

⟩V ′,V (77.8.85)

≥ −|G(0)|∥∥Jµ uµ

∥∥V+

⟨− 1

µ p−1 F(Jµ uµ −uµ

),uµ − Jµ uµ

⟩V ′,V

= −|G(0)|∥∥Jµ uµ

∥∥V+

1µ p−1

∥∥Jµ uµ −uµ

∥∥pV

(77.8.86)

Thus the fact that∥∥uµ

∥∥ is bounded independent of µ implies that∥∥Jµ uµ

∥∥ is also boundedand that in fact

∥∥uµ − Jµ uµ

∥∥V→ 0 as µ → 0. This follows from consideration of the last

line of the above formula. Note also that⟨gµ ,uµ − Jµ uµ

⟩V ′,V =

1µ p−1

∥∥Jµ uµ −uµ

∥∥pV

is bounded. (77.8.87)

Then from 77.8.85, it follows that⟨gµ ,Jµ uµ

⟩V ′,V is bounded. By the assumption that G is

quasi-bounded, gµ must also be bounded.Then we have shown

Buµ (t,ω)+∫ t

0zµ (s,ω)ds+

∫ t

0gµ (s,ω)ds =

∫ t

0f (s,ω)ds+Bu0 (ω) (77.8.88)

where∥∥gµ

∥∥V ′ +

∥∥zµ

∥∥V ′ + sup

t∈[0,T ]

⟨Buµ ,uµ

⟩(t)+

∥∥Jµ uµ

∥∥V+∥∥uµ

∥∥V+∥∥∥(Buµ

)′∥∥∥V ′≤C

(77.8.89)