924 CHAPTER 25. NONLINEAR OPERATORS
Theorem 25.8.8 Let L : D(L)⊆V →V ′ where D(L) is dense, L is monotone, L is closed,and L∗ is monotone, L a linear map. Let A : V →P (V ′) be L pseudomonotone, bounded,coercive. Then L+A is onto. Here V is a reflexive Banach space such that the norms for Vand V ′ are strictly convex. In case that A is strictly monotone (⟨Au−Av,u− v⟩> 0 impliesu ̸= v) the solution u to f ∈ Lu+Au is unique. If, in addition to this, ⟨Au−Av,u− v⟩ ≥r (∥u− v∥U ) where U is some Banach space containing V, and r is a positive strictly in-creasing function for which limt→0+ r (t) = 0, then the map f → u where f ∈ Lu+Au iscontinuous as a map from V ′ to U. The conclusion holds if A is only L modified boundedpseudomonotone.
Proof: Let F be the duality map for p = 2. Consider the Banach space X given by
X = D(L) , ∥u∥X ≡ ∥u∥V +∥Lu∥V ′
This is isometric with the graph of L with the graph norm and so X is reflexive. Now definea set valued map Gε on X as follows. z∗ ∈ Gε (u) means there exists w∗ ∈ Au such that.
⟨z∗,v⟩X ′,X = ε⟨Lv,F−1 (Lu)
⟩V ′,V + ⟨Lu,v⟩V ′,V + ⟨w∗,v⟩V ′,V
It follows from Lemma 25.8.7 that Gε is the sum of a set valued L modified bounded pseu-domonotone operator with an operator which is demicontinuous, bounded, and monotone,hence pseudomonotone. Thus by Lemma 25.5.2 it is L modified bounded pseudomonotone.Is it coercive?
lim∥u∥X→∞
inf
{⟨z∗,u⟩+ ε
⟨Lu,F−1 (Lu)
⟩V ′,V + ⟨Lu,u⟩V ′,V
||u||X: z∗ ∈ Au
}= ∞?
It equals
lim∥u∥X→∞
inf
{⟨z∗,u⟩+ ε
⟨FF−1 (Lu) ,F−1 (Lu)
⟩V ′,V + ⟨Lu,u⟩V ′,V
||u||X: z∗ ∈ Au
}and this is
≥ lim∥u∥X→∞
inf
{⟨z∗,u⟩+ ε
∥∥F−1 (Lu)∥∥2
V||u||X
: z∗ ∈ Au
}
= lim∥u∥X→∞
inf
{⟨z∗,u⟩+ ε ∥Lu∥2
V ′
∥u∥V +∥Lu∥V ′: z∗ ∈ Au
}because L is monotone. Now let M be an arbitrary positive number. By assumption, thereexists R such that if ∥u∥V > R, then
inf{⟨z∗,u⟩∥u∥V
: z∗ ∈ Au}> M
and so for every z∗ ∈ Au,
⟨z∗,u⟩∥u∥V
> M, ⟨z∗,u⟩> M ∥u∥V