78.1. SUMMARY OF THE PROBLEM 2657

where m(ω) is some nonnegative constant, δ (ω) > 0. In fact, it is often enough toassume the left side is given by

inf(∫ t

0⟨u∗,u⟩+λ (ω)⟨Bu,u⟩ds : u∗ ∈ A(u,ω)

)for some λ (ω) by using a suitable exponential shift argument and changing thedependent variable. We will sometimes denote weak convergence by ⇀.

• limit condition

If ui ⇀ u in V and (Bui)′⇀ (Bu)′ in V ′, u∗i ∈ A(ui) , ⇀ denoting weak convergence,

then iflim sup

i→∞

⟨u∗i ,ui−u⟩V ′,V ≤ 0

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

lim infi→∞⟨u∗i ,ui− v⟩V ′,V ≥ ⟨u

∗ (v) ,u− v⟩V ′,V (78.1.7)

• measurability condition

For ω → u(·,ω) measurable into V ,

ω → A(u(·,ω) ,ω) has a measurable selection into V ′. (78.1.8)

This last condition means there is a function ω → u∗ (ω) which is measurable into V ′

such that u∗ (ω) ∈ A(u(·,ω) ,ω) . This is assured to take place if the following standardmeasurability condition is satisfied for all O open in V ′:

{ω : A(u(·,ω) ,ω)∩O ̸= /0} ∈F (78.1.9)

See for example [70], [10] or the chapter on measurable multifunctions Chapter 48. Ourassumption is implied by this one but they are not equivalent. Thus what is considered hereis more general than an assumption that ω → A(u(·,ω) ,ω) is set valued measurable.

Note that this condition would hold if u→ A(t,u,ω) is bounded and pseudomonotoneas a single valued map from V to V ′ and (t,ω)→ A(t,u,ω) is product measurable intoV ′ for each u. One would use the demicontinuity of u→ A(t,u,ω) which comes froma pseudo monotone and bounded assumption and consider a sequence of simple functionsun (t,ω)→ u(t,ω) in V for u measurable, each un (·,ω) being in V , Then the measurabilityof A(t,un,ω) would attach to A(t,u,ω) in the limit. In the situation where A(·,ω) satisfiesa suitable upper semicontinuity condition, it is enough to assume only that ω→A(u,ω) hasa measurable selection for each u ∈V . This is a straightforward exercise in approximatingwith simple functions and then using upper semicontinuity instead of continuity.

We assume always that the norm on the various reflexive Banach spaces is strictlyconvex.