880 CHAPTER 25. NONLINEAR OPERATORS

Then from the above argument, KV ̸= /0. You just choose your subspaces Xn to all include V .Also, from coercivity of B and the above argument, these KV are all bounded and weaklyclosed. Hence they are weakly compact. Then if you have finitely many of them, youcan let your subspaces include each V and conclude that these KV have finite intersectionproperty and so there exists x ∈ ∩V KV which gives the desired x.

Note that there is only one place where 0∈D(A) was used and it was to get the estimate.In the argument,

⟨v,u− xn⟩ ≥ ⟨Bxn,xn−u⟩

and it was convenient to be able to take u= 0. However, you could also assume other thingson B such as that it satisfies an estimate of the form

∥Bx∥ ≤C∥x∥+C

and if you did this, you could also obtain the necessary estimate as follows.

⟨v,u− xn⟩ ≥ ⟨Bxn,xn−u⟩⟨v,u− xn⟩+ ⟨Bxn,u⟩ ≥ ⟨Bxn,xn⟩

∥v∥(∥u∥+∥xn∥)+([C∥xn∥+C]∥u∥) ≥ ⟨Bxn,xn⟩

and then pick some [u,v]. Thus the following corollary comes right away. This would haveworked just as well if you had an estimate of the form

∥Bx∥ ≤C∥x∥p−1 +C, p > 1

Corollary 25.7.11 Let X be a reflexive Banach space and let K be a closed convex subsetof X. Let A,B be monotone such that

1. D(A)⊆ K

2. B is single valued, hemicontinuous, bounded and coercive mapping X to X ′ whichsatisfies the estimate

∥Bx∥ ≤C∥x∥+C or more generally ∥Bx∥ ≤C∥x∥p−1 +C, p > 1

Then there exists x ∈ K such that

⟨Bx+ v,u− x⟩X ′,X ≥ 0 for all [u,v] ∈ G (A)

Now here is the equivalence between maximal monotone graph and having F +A beonto. It was already shown that if λF +A is onto, then the graph of A is maximal monotonein the sense that there is no monotone operator whose graph properly contains the graph ofA. This was Theorem 25.7.2 above which is stated here as a reminder of what it said.

Theorem 25.7.12 Let X, X ′ be reflexive and have strictly convex norms. Let A be a mono-tone set valued map as just described. Then if

λF +A onto,