882 CHAPTER 25. NONLINEAR OPERATORS

25.7.3 Surjectivity TheoremsAs an interesting example of this theorem, here is another result in Barbu [13]. It is inter-esting because it is not assumed B is bounded.

Theorem 25.7.15 Let B : X→ X ′ be monotone hemicontinuous. Then B is maximal mono-tone. If B is coercive, then B is also onto. Here X is a strictly convex reflexive Banachspace.

Proof: Suppose B is not maximal monotone. Then there exists (x0,x∗0) ∈ X ×X ′ suchthat for all x,

⟨Bx− x∗0,x− x0⟩ ≥ 0

and yet x∗0 ̸= Bx0. This is going to be a contradiction. Let u ∈ X and consider xt ≡ tx0 +(1− t)u, t ∈ (0,1). Then consider

⟨Bxt − x∗0,xt − x0⟩

However, xt − x0 = tx0 +(1− t)u− x0 = (1− t)(u− x0) and so, for each t ∈ (0,1) ,

0≤ ⟨Bxt − x∗0,xt − x0⟩= (1− t)⟨Bxt − x∗0,u− x0⟩

Divide by (1− t) and then let t ↑ 1. This yields the following by hemicontinuity.

⟨Bx0− x∗0,u− x0⟩ ≥ 0

which holds for all u. Hence Bx0 = x∗0 after all. Thus B is indeed maximal monotone.Next suppose B is coercive. Let F be the duality map (or the duality map for arbitrary

p > 1). Then from Theorem 25.7.13 there exists a solution xλ to

λFxλ +Bxλ = x∗0 ∈ X ′ (25.7.50)

Then the xλ are bounded because, doing both sides to xλ ,

λ ∥xλ∥2 + ⟨Bxλ ,xλ ⟩= ⟨x∗0,xλ ⟩

and so⟨Bxλ ,xλ ⟩∥xλ∥

≤ ∥x∗0∥

Thus the coercivity of B implies that the xλ are bounded. There exists a subsequence suchthat

xλ → x weakly.

Then from the equation 25.7.50 ∥λFxλ∥= λ ∥xλ∥ and so,

Bxλ → x∗0 strongly.

Since B is monotone and hemicontinuous, it satisfies the pseudomonotone condition, The-orem 25.1.4. The above strong convergence implies

limλ→0⟨Bxλ ,xλ − x⟩= 0