1404 CHAPTER 42. MAXIMAL MONOTONE OPERATORS, HILBERT SPACE

42.8 A More Complicated Perturbation TheoremIn this section is a simple perturbation theorem which is a small generalization of one foundin [24] and [116].

Recall that for B a maximal monotone operator, Bλ ,the Yosida approximation, is de-fined by

Bλ x≡ 1λ(x− Jλ x) , Jλ x≡ (I +λB)−1 x.

This follows from Theorem 42.1.7 on Page 1372

Theorem 42.8.1 Let A and B be maximal monotone operators and let xλ be the solution to

y ∈ xλ +Bλ xλ +Axλ .

Then y ∈ x+Bx+Ax for some x ∈ D(A)∩D(B) if Bλ xλ is bounded independent of λ .

The following is the perturbation theorem of this section. It generalizes a well knownresult in [24] and [116].

Theorem 42.8.2 Let H be a real Hilbert space and let Φ be non negative, convex, proper,and lower semicontinuous. Suppose also that A is a maximal monotone operator and thereexists

ξ ∈ D(A)∩D(Φ) . (42.8.51)

Suppose also that for Jλ x≡ (I +λA)−1 x,

Φ(Jλ x)≤Φ(x)+C (x)λ (42.8.52)

where for some constants, K1,K2,

K2 +K1

(Φ(x)+ |x|2

)≥C (x) . (42.8.53)

Then A+∂Φ is maximal monotone.

Proof: Letting Aλ be the Yosida approximation of A,

Aλ x =1λ(x− Jλ x) ,

and letting y ∈ H, it follows from the Hilbert space version of Proposition 42.1.6 thereexists xλ ∈ H such that

y ∈ xλ +Aλ xλ +∂Φ(xλ ) .

Consequently,y− xλ −Aλ xλ ∈ ∂Φ(xλ ) (42.8.54)

and so

(y− xλ −Aλ xλ ,Jλ xλ − xλ )≤Φ(Jλ xλ )−Φ(xλ )≤C (xλ )λ (42.8.55)