In the following, V will be a reflexive separable Banach space. Following [?], here is a
definition of a pseudomonotone operator. Actually, we will consider a slight generalization
of the usual definition in 23.4.18 which involves an assumption that there exists a
subsequence such that the liminf condition holds rather than use the original
sequence.
Definition 47.5.1Let V be a reflexive Banach space. Then A : V →P
(V ′)
ispseudomonotoneif the following conditions hold.
Au is closed, nonempty, convex. (47.5.19)
(47.5.19)
If F is a finite dimensional subspace of V , then if u ∈ F and W ⊇ Au for W a weaklyopen set in V^{′}, then there exists δ > 0 such that
v ∈ B(u,δ)∩ F implies Av ⊆ W . (47.5.20)
(47.5.20)
If u_{k}⇀ u and if u_{k}^{∗}∈ Au_{k}is such that
∗
lim sk→up∞ 〈u k,uk − u〉 ≤ 0,
Then there exists a subsequence still denoted with k such that for all v ∈ V , there existsu^{∗}
If one assumes A is bounded, then the weak upper semicontinuity condition 47.5.20
can be proved from the other conditions. It has been known for a long time that these
operators are useful in the study of variational inequalities. In this section, we give a
short example to show how one can obtain measurable solutions to variational
inequalities from the measurable Browder lemma given above. This is the following
theorem which gives a measurable version of old results of Brezis dating from the late
1960s. This will involve the following assumptions.
Measurability condition
For each u ∈ V, there is a measurable selection z
(ω)
such that
z(ω) ∈ A (u,ω).
Values of A
A
(⋅,ω)
: V →P
′
(V )
has bounded, closed, nonempty, convex values. A
(⋅,ω)
maps
bounded sets to bounded sets.
Limit conditions
If u ⇀ u and lim sup 〈z ,u − u〉 ≤ 0, z ∈ A (u ,ω )
n n→∞ n n n n
then for given v, there exists z
(v)
∈ A
(u,ω)
such that
lim kin→f∞ 〈zn,un − v〉 ≥ 〈z(v),u − v〉
Thus, for fixed ω,A
(⋅,ω)
is a set valued bounded pseudomonotone operator.
Recall that the sum of two of these is also a set valued bounded pseudomonotone
operator.
has a measurable
selection. Also, the limit condition implies that A
(⋅,ω)
is upper semicontinuous from the
strong to the weak topology. The overall approach to the following theorem is well
known. The new ingredients are Lemma 47.2.2 and Theorem 47.3.3 which are
what allows us to obtain measurable solutions. First is a standard result on the
sum of two pseudomonotone bounded operators. See Theorem 23.5.1 on Page
2690.
Theorem 47.5.2Say A,B are set valued bounded pseudomonotone operators.Thentheir sum is also a set valued bounded pseudomonotone operator. Also, if u_{n}→ uweakly, z_{n}→ z weakly, z_{n}∈ A
(un)
, and w_{n}→ w weakly with w_{n}∈ B
(un)
, thenif
lim nsu→p∞〈zn +wn, un − u〉 ≤ 0,
it follows that
lim inf 〈z + w ,u − v〉 ≥ 〈z(v)+ w(v),u − v〉, z(v) ∈ A (u),w(v) ∈ B (u),
n→∞ n n n
and z ∈ A
(u)
,w ∈ B
(u)
.
Theorem 47.5.3Let V be a reflexive separable Banach space. Let ω → K
(ω )
be ameasurablemultifunction, K
(ω)
convex, closed, and bounded. Also for A = B,Clet A