932 CHAPTER 25. NONLINEAR OPERATORS

Thus, taking limsup of both sides,

lim supn→∞

⟨Bnxn,xn− x⟩= lim supn→∞

⟨Bnxn,Jnxn− x⟩ ≤ 0

Hencelim sup

n→∞

⟨Bnxn,Jnxn⟩ ≤ ⟨g∗,x⟩

Letting [a,b∗] ∈ G (B) ,

⟨Bnxn−b∗,Jnxn−a⟩= ⟨Bnxn,Jnxn⟩−⟨Bnxn,a⟩−⟨b∗,Jnxn⟩+ ⟨b∗,a⟩

Then taking limsup,

0 ≤ lim supn→∞

⟨Bnxn−b∗,Jnxn−a⟩

≤ ⟨g∗,x⟩−⟨g∗,a⟩−⟨b∗,x⟩+ ⟨b∗,a⟩= ⟨g∗−b∗,x−a⟩

It follows that g∗ ∈ B(x) and x ∈ D(B).Thus, passing to the limit in the equation 25.8.95 where, as explained λ = 1/n, one

obtainsy∗ = Lu+ z∗+g∗

where z∗ ∈ Ax and g∗ ∈ Bx. This proves the following nice generalization of the aboveperturbation theorem.

Theorem 25.8.15 Let B be maximal monotone from X to P (X ′), 0 ∈ D(B) , and B isquasi-bounded as explained above. Let A : X →P (X ′) be pseudomonotone, bounded,and coercive. Also let L be a densely defined linear operator such that both L and L∗ aremonotone. (That is, L is linear and maximal monotone.) Then L+A+B is onto X ′.