48.5. SOME VARIATIONAL INEQUALITIES 1567

48.5 Some Variational InequalitiesIn the following, V will be a reflexive separable Banach space. Following [99], here isa definition of a pseudomonotone operator. Actually, we will consider a slight general-ization of the usual definition in 25.4.17 which involves an assumption that there exists asubsequence such that the liminf condition holds rather than use the original sequence.

Definition 48.5.1 Let V be a reflexive Banach space. Then A : V →P (V ′) is pseudomono-tone if the following conditions hold.

Au is closed, nonempty, convex. (48.5.15)

If F is a finite dimensional subspace of V , then if u ∈ F and W ⊇ Au for W a weakly openset in V ′, then there exists δ > 0 such that

v ∈ B(u,δ )∩F implies Av⊆W. (48.5.16)

If uk ⇀ u and if u∗k ∈ Auk is such that

lim supk→∞

⟨u∗k ,uk−u⟩ ≤ 0,

Then there exists a subsequence still denoted with k such that for all v ∈ V , there existsu∗ (v) ∈ Au such that

lim infk→∞

⟨u∗k ,uk− v⟩ ≥ ⟨u∗ (v) ,(u− v)⟩. (48.5.17)

We say A is coercive if

lim∥v∥→∞

inf{⟨z∗,v⟩∥v∥

: z∗ ∈ Av}= ∞. (48.5.18)

If one assumes A is bounded, then the weak upper semicontinuity condition 48.5.16can be proved from the other conditions. It has been known for a long time that theseoperators are useful in the study of variational inequalities. In this section, we give a shortexample to show how one can obtain measurable solutions to variational inequalities fromthe measurable Browder lemma given above. This is the following theorem which gives ameasurable version of old results of Brezis dating from the late 1960s. This will involvethe following assumptions.

1. Measurability condition

For each u ∈V, there is a measurable selection z(ω) such that

z(ω) ∈ A(u,ω) .

2. Values of A

A(·,ω) : V →P (V ′) has bounded, closed, nonempty, convex values. A(·,ω) mapsbounded sets to bounded sets.