25.5. SUM OF PSEUDOMONOTONE OPERATORS 863

for all u∗ ∈ Tu. This follows from the separation theorems. (These theorems imply thereexists z ∈V such that

Re⟨w∗,z⟩< Re⟨u∗,z⟩

for all u∗ ∈ Tu. Define u− v≡ z.)Now let F ⊇ {u,v}. Since u ∈WF , a weakly sequentially compact set, there exists a

sequence, {uk}, such thatuk ⇀ u, uk ∈WF .

Then since F ⊇ {u,v}, there exists u∗k ∈ Tuk such that

⟨u∗k ,uk−u⟩= ⟨w∗,uk−u⟩ .

Therefore,lim sup

k→∞

Re⟨u∗k ,uk−u⟩= lim supk→∞

Re⟨w∗,uk−u⟩= 0.

It follows by the assumption that T is modified bounded pseudomonotone or generalizedbounded pseudomonotone and the pseudomonotone limit condition, a further subsequencecorresponding to v such that the following holds for the v defined above in 25.5.30.

lim infk→∞

Re⟨u∗k ,uk− v⟩ ≥ Re⟨u∗ (v) ,u− v⟩ , u∗ (v) ∈ Tu.

But since v ∈ F,Re⟨u∗k ,uk− v

⟩= Re⟨w∗,uk− v⟩ and so

lim infk→∞

Re⟨u∗k ,uk− v⟩= lim infk→∞

Re⟨w∗,uk− v⟩= Re⟨w∗,u− v⟩,

so from 25.5.30, Re⟨w∗,u− v⟩< Re⟨u∗,u− v⟩ for all u∗ ∈ Tu,

Re⟨w∗,u− v⟩= lim infk→∞

Re⟨u∗k ,uk− v⟩

≥ Re⟨u∗ (v) ,u− v⟩> Re⟨w∗,u− v⟩,

a contradiction. Thus, w∗ ∈ Tu.This is likely a good place to put an extremely interesting convergence theorem. It is a

version of one in Aubin and Cellina [9]. It is a perfectly marvelous use of the fact that theweak and strong closures of a convex set are the same.

Proposition 25.5.5 Let X ,Y be Banach spaces, and let F : (0,T )×X →P (Y ) be a mul-tifunction such that

1. The values of F are nonempty, closed and convex subsets of Y

2. For a.e. t ∈ (0,T ) ,F (t, ·) is upper semicontinuous from X into Y with the weaktopology

Then let xn : (0,T )→ X ,yn : (0,T )→Y be measurable functions such that the sequence{xn} converges a.e. on (0,T ) to a function x : (0,T )→ X and yn converges weakly inL1 (0,T ;Y ) to y ∈ L1 (0,T,Y ). If yn (t) ∈ F (t,xn (t)) for all n ∈ N and a.e.t, then y(t) ∈F (t,x(t)) for a.e.t ∈ (0,T ) .