2612 CHAPTER 77. STOCHASTIC INCLUSIONS

Theorem 77.3.2 If A and B are pseudo monotone and bounded then A+B is also pseudomonotone and bounded.

Also the following result, found in [91] is well known.

Theorem 77.3.3 If a single valued map, A : X → X ′ is monotone, hemicontinuous, andbounded, then A is pseudo monotone. Furthermore, the duality map, J−1 : X → X ′ whichsatisfies ⟨J−1 f , f ⟩ = || f ||2 , ||J−1 f ||X = || f ||X is strictly monotone hemicontinuous andbounded. So is the duality map F : X → X ′ which satisfies ∥F f∥X ′ = ∥ f∥p−1

X , ⟨F f , f ⟩ =∥ f∥p

X for p > 1.

The following fundamental result will be of use in what follows. There is somewhatmore in this than will be needed. In this paper, B is a possibly degenerate operator satisfyingonly the following:

B ∈L(W,W ′

), ⟨Bu,u⟩ ≥ 0,⟨Bu,v⟩= ⟨Bv,u⟩ (77.3.6)

where here V ⊆W and V is dense in W . In the case where B = B(ω) , we will assume forthe sake of simplicity that

B(ω) = k (ω)B, k (ω)≥ 0,k being F measurable

Allowing B to depend on ω introduces some technical considerations so if there is nointerest in this, simply assume B is independent of ω . This includes all cases of mostinterest.

Lemma 77.3.4 Suppose V,W are separable Banach spaces such that V is dense in W andB ∈L (W,W ′) satisfies

⟨Bx,x⟩ ≥ 0, ⟨Bx,y⟩= ⟨By,x⟩ ,B ̸= 0.

Then there exists a countable set {ei} of vectors in V such that⟨Bei,e j

⟩= δ i j

and for each x ∈W,

⟨Bx,x⟩=∞

∑i=1|⟨Bx,ei⟩|2 ,

and also

Bx =∞

∑i=1⟨Bx,ei⟩Bei,

the series converging in W ′. If B = B(ω) and B is F measurable into L (W,W ′) and ifthe ei = ei (ω) are as described above, then these ei are measurable into V . If t→ B(t,ω)is C1 ([0,T ] ,L (W,W ′)) and if for each w ∈W,⟨

B′ (t,ω)w,w⟩≤ kw,ω (t)⟨B(t,ω)w,w⟩

Where kw,ω ∈ L1 ([0,T ]) , then the vectors ei (t) can be chosen to also be right continuousfunctions of t.