25.7. MAXIMAL MONOTONE OPERATORS 919

Then one can show that Jλ is Lipschitz continuous and many other nice things happen.Next is an interesting result about when the sum of a maximal monotone operator and

a subgradient is also maximal monotone. A version of this is well known in the case of asingle Hilbert space. In the case of a single Hilbert space, this result can be used to producevery regular solutions to evolution equations for functions which have values in the Hilbertspace. You would get this by letting X = X ′ equal to a Hilbert space and your maximalmonotone operator A would be defined on L2 (0,T ;H) = X a space of Hilbert space valuedfunctions which are square integrable. Then you could take Lu = u′ with domain equal tothose functions in X which are equal to 0 at the left end of the interval for example. This isdone more generally later. In this case the duality map is just the identity. The next theoremincludes the case of two different spaces. I am not sure whether this is a useful result at thistime, in terms of evolution equations. However, it is good to have conditions which showthat the sum of two maximal monotone operators is maximal monotone.

Theorem 25.7.55 Let X be a reflexive Banach space with strictly convex norm and let Φ

be non negative, convex, proper, and lower semicontinuous. Suppose also that A : D(A)→P (X ′) is a maximal monotone operator and there exists

ξ ∈ D(A)∩D(Φ) . (25.7.82)

Suppose also thatΦ(Jλ x)≤Φ(x)+Cλ (25.7.83)

Then A+∂Φ is maximal monotone.

Proof: Recall that

Aλ x =−λ−1F (Jλ x− x) , where 0 ∈ F (Jλ x− x)+λ∂A(Jλ x)

Let y∗ ∈ X ′ . From Theorem 25.7.43 there exists xλ ∈ H such that

y∗ ∈ Fxλ +Aλ xλ +∂Φ(xλ ) .

It is desired to show that Aλ xλ is bounded. From the above,

y∗−Fxλ −Aλ xλ ∈ ∂Φ(xλ ) (25.7.84)

and so⟨y∗−Fxλ −Aλ xλ ,Jλ xλ − xλ ⟩ ≤Φ(Jλ xλ )−Φ(xλ )≤Cλ (25.7.85)

which implies⟨y∗−Fxλ −Aλ xλ ,(−λ )F−1 (Aλ x)

⟩≤Φ(Jλ xλ )−Φ(xλ )≤Cλ

and so ⟨y∗−Fxλ −Aλ xλ ,−F−1 (Aλ x)

⟩≤C

Hence ⟨y∗−Fxλ ,−F−1 (Aλ xλ )

⟩+∥Aλ xλ∥2 ≤C (25.7.86)