1400 CHAPTER 42. MAXIMAL MONOTONE OPERATORS, HILBERT SPACE

Theorem 42.6.1 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. See [24] and [116].

Theorem 42.6.2 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) . (42.6.40)

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

Φ(Jλ x)≤Φ(x)+Cλ (42.6.41)

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 ) (42.6.42)

and so(y− xλ −Aλ xλ ,Jλ xλ − xλ )≤Φ(Jλ xλ )−Φ(xλ )≤Cλ (42.6.43)

which implies

−(y− xλ −Aλ xλ ,Aλ xλ ) = |Aλ xλ |2−|y− xλ | |Aλ xλ | ≤C. (42.6.44)

By 42.6.42 and monotonicity of Aλ ,

Φ(ξ )−Φ(xλ )≥

 ∈∂Φ(xλ )︷ ︸︸ ︷y− xλ −Aλ xλ ,ξ − xλ

= (y− xλ ,ξ − xλ )− (Aλ xλ ,ξ − xλ )

≥ (y− xλ ,ξ − xλ )− (Aλ ξ ,ξ − xλ )

≥ (y−ξ ,ξ − xλ )+ |ξ − xλ |2− (Aλ ξ ,ξ − xλ )

= |ξ − xλ |2 +(y−ξ −Aλ ξ ,ξ − xλ )