890 CHAPTER 25. NONLINEAR OPERATORS

Theorem 25.7.21 to conclude that Fτ is indeed maximal monotone because it has convexclosed values, the hemicontinuity condition, and is monotone.

Suppose T is a bounded pseudomonotone operator and S is a maximal monotone op-erator, both defined on a strictly convex reflexive Banach space. What of their sum? Is(T +S)(x) convex and closed? Say ti ∈ T x and si ∈ Sx is it the case that θ (s1 + t1) +(1−θ)(s2 + t2) ∈ (T +S)(x) whenever θ ∈ [0,1]? Of course this is so. Thus T + S hasconvex values. Does it have closed values? Suppose {sn + tn} converges to z ∈ X ′,sn ∈Sx, tn ∈ T x. Is z ∈ (T +S)(x)? Taking a subsequence, and using the assumption that T isbounded, it can be assumed that tn → t ∈ T x weakly. Therefore, sn must also convergeweakly and so it converges to some s = z− t ∈ Sx. Convex and closed implies weaklyclosed. Thus T +S has closed convex values. Is it upper semicontinuous on finite dimen-sional subspaces? Suppose xn → x in a finite dimensional subspace F . Does it followthat

(S+T )xn ⊆ (S+T )x+B(0,r)

for all n sufficiently large? It is known that Sxn ⊆ Sx+B(0,r/2) and T xn ⊆ T x+B(0,r/2)whenever n is sufficiently large and so it follows that

(S+T )xn ⊆ (S+T )x+B(0,r/2)+B(0,r/2)⊆ (S+T )x+B(0,r)

whenever n is large enough.What of the pseudomonotone condition? Suppose

lim supn→∞

⟨u∗n + v∗n,xn− x⟩ ≤ 0

where u∗n ∈ Sxn and v∗n ∈ T xn where xn → x weakly. Is it the case that for every y, thereexists u∗ ∈ Sx and v∗ ∈ T x such that

lim infn→∞⟨u∗n + v∗n,xn− y⟩ ≥ ⟨u∗+ v∗,x− y⟩?

By monotonicity,

0 ≥ lim supn→∞

⟨u∗n + v∗n,xn− x⟩ ≥ lim supn→∞

⟨u∗+ v∗n,xn− x⟩

= lim supn→∞

⟨v∗n,xn− x⟩

Hencelim sup

n→∞

⟨v∗n,xn− x⟩ ≤ 0

which implies

lim infn→∞⟨v∗n,xn− x⟩ ≥ ⟨v̂∗,x− x⟩= 0≥ lim sup

n→∞

⟨v∗n,xn− x⟩

showing thatlimn→∞⟨v∗n,xn− x⟩= 0 (25.7.54)