920 CHAPTER 25. NONLINEAR OPERATORS

I claim {∥xλ∥}are bounded independent of λ .By 25.7.84 and monotonicity of Aλ ,

Φ(ξ )−Φ(xλ )≥ ⟨y∗−Fxλ −Aλ xλ ,ξ − xλ ⟩

≥ ⟨y∗−Fxλ ,ξ − xλ ⟩−⟨Aλ xλ ,ξ − xλ ⟩

≥ ⟨y∗−Fxλ ,ξ − xλ ⟩−⟨Aλ ξ ,ξ − xλ ⟩

= ⟨y∗,ξ ⟩−⟨y∗,xλ ⟩−⟨Fxλ ,ξ ⟩+∥xλ∥2−∥ξ − xλ∥∥Aλ ξ∥

≥ −∥y∗∥∥ξ∥−∥y∗∥∥xλ∥−∥xλ∥∥ξ∥−∥ξ∥|Aξ |−∥xλ∥|Aξ |+∥xλ∥2

Therefore, there exist constants, C1 and C2, depending on ξ and y∗ but not on λ such that

Φ(ξ )≥Φ(xλ )+∥xλ∥2−C1 ∥xλ∥−C2.

Since Φ ≥ 0, the above shows that ∥xλ∥ is indeed bounded. Now from 25.7.86 it followsthat {Aλ xλ} is bounded for small positive λ . By Theorem 25.7.43, there exists a solutionx to

y∗ ∈ Fx+Ax+∂Φ(x)

and since y∗ is arbitrary, this shows that A+∂Φ is maximal monotone.

25.8 Perturbation TheoremsIn this section gives surjectivity of the sum of a pseudomonotone set valued map with alinear maximal monotone map and also with another maximal monotone operator addedin. It generalizes the surjectivity results given earlier because one could have 0 for themaximal monotone linear operator. The theorems developed here lead to nice results onevolution equations because the linear maximal monotone operator can be something likea time derivative and X can be some sort of an Lp space for functions having values in asuitable Banach space. This is presented later in the material on Bochner integrals.

The notation ⟨z∗,u⟩V ′,V will mean z∗ (u) in this section. We will not worry about theorder either. Thus

⟨u,z∗⟩ ≡ z∗ (u)≡ ⟨z∗,u⟩

This is just convenient in writing things down. Also, it is assumed that all Banach spacesare real to simplify the presentation. It is also usually assumed that the Banach spaces arereflexive. Thus we can regard (

V ×V ′)′=V ′×V

and ⟨(y∗,x) ,(u,v∗)⟩ ≡ ⟨y∗,u⟩+ ⟨x,v∗⟩. It is known [8] that for a reflexive Banach space,there is always an equivalent strictly convex norm. It is therefore, assumed that the normfor the reflexive Banach space is strictly convex.

Definition 25.8.1 Let L : D(L)⊆V →V ′ be a linear map where we always assume D(L)is dense in V . Then

D(L∗)≡ {u ∈V : |⟨Lv,u⟩| ≤C∥v∥ for all v ∈ D(L)}