33.1 Basic Theory
Here is provided a short introduction to some of the most important properties of
maximal monotone operators in Hilbert space. The following definition describes them. It
is more specialized than the earlier material on maximal monotone operators from a
Banach space to its dual and therefore, better results can be obtained. More on this can
be read in [?] and [?].
Definition 33.1.1 Let H be a real Hilbert space and let A : D
the following properties.
- For each y ∈ H there exists x ∈ D
such that y ∈ x + Ax.
- A is monotone. That is, if z ∈ Ax and w ∈ Ay then
Such an operator is called a maximal monotone operator.
It turns out that whenever A is maximal monotone, so is λA for all λ > 0.
Lemma 33.1.2 Suppose A is maximal monotone. Then so is λA. Also Jλ ≡
−1 makes sense for each λ >
0 and is Lipschitz continuous.
Proof: To begin with consider
Then y ∈
and so y − xi ∈ Axi
. By monotonicity
which shows J1 ≡
makes sense. In fact this is Lipschitz with Lipschitz
constant 1. Here is why. x ∈
and y ∈
and so by monotonicity
which yields the result.
Next consider the claim that λA is maximal monotone. The monotone part is
immediate. The only thing in question is whether I + λA is onto. Let r ∈
f ∈ H
. Consider solving the equation for u
This is equivalent to finding u such that
or in other words finding u such that
is a contraction mapping and so there exists a unique solution to
It follows for any
is maximal monotone. This takes care
of all λ ∈
). Now do the same thing for
to get the result for all
Now apply the same argument to
to get the result
for all λ ∈
Next consider the same argument to
get the desired result for all λ ∈
Continuing this way shows
is maximal monotone for all λ >
0. Also from the first part of the proof
is Lipschitz continuous with Lipschitz constant 1. This proves the
A maximal monotone operator can be approximated with a Lipschitz continuous
operator which is also monotone and has certain salubrious properties. This operator is
called the Yosida approximation and as in the case of linear operators it is obtained by
If you do the division formally you get the definition for Aλ,
where Jλ =
as above. It is obvious that Aλ
is Lipschitz continuous with
Lipschitz constant no more than 2∕λ.
Actually you can show 1∕λ
also works but this is
not important here.
Lemma 33.1.3 Aλx ∈ AJλx and
for all y ∈ Ax whenever x ∈ D
Also Aλ is monotone.
Proof: Consider the first claim. From the definition,
Certainly so. This is how Jλ is defined.
Now consider the second claim. Let y ∈ Ax for some x ∈ D
monotonicity and what was just shown
Finally, to show Aλ is monotone,
and this proves the lemma.
Proposition 33.1.4 Suppose D
is dense in H. Then for all x ∈ H,
Proof: From the above, if u ∈ D
y ∈ Au,
Hence Jλu → u. Now for x arbitrary,
where the last term converges to 0 as λ →
0. Since ε
is arbitrary, this shows the
Thus in the case where D
is dense, if you have
so that xε = Jεx, then
The next lemma gives a way to determine whether a pair
is in the graph of
Here I am writing
to avoid confusion with the inner product. It is
the conclusion of this lemma which accounts for the use of the term “maximal”. It
essentially says there is no larger monotone graph which includes the one for
Lemma 33.1.5 Suppose
0 for all
where A is
maximal monotone. Then x1 ∈ D
and y1 ∈ Ax1. Also if
xk → x,yk ⇀ y where the half arrow denotes weak convergence, then
Proof: I want to show y1 ∈ Ax1 or in other words I want to show
or in other words
This is the motivation for the following argument.
From Lemma 33.1.3 Aλ
and so by the above
and this says x1 ∈ D
maps into D
. Also it says
and so y1 ∈ Ax1.
This makes the last claim pretty easy. Suppose xk → x where xk ∈ D
yk ∈ Axk
and yk ⇀ y.
I need to verify y
and x ∈ D
and so, by the first part, x ∈ D
y ∈ Ax
. Why does that limit hold? It is
The second term is no larger than
which converges to 0 since yk is weakly convergent, hence bounded. The first term
converges to 0 because of the assumption that yk converges weakly to y. This proves the
What about the sum of maximal monotone operators? This might not be maximal
monotone but what you can say is the following.
Proposition 33.1.6 Let A be maximal monotone and let B be Lipschitz and
monotone. Then A + B is maximal monotone.
Proof: First suppose B has a Lipschitz constant less than 1. The monotonicity is
obvious. I need to show that for any y there exists x ∈ D
This hapens if and only if
if and only if x =
Then T is clearly a contraction mapping because
is Lipschits with Lipschitz
constant 1. Therefore, there exists a unique fixed point and this shows A
monotone. Now the same argument applied to A
shows that A
monotone. Continuing this way A
is maximal monotone. Now for arbitrary B
be large enough that n−1B
has Lipschitz constant less than 1. Then as
just explained, A
is maximal monotone. This proves the
The following is a useful result for determining conditions under which A + B is
maximal monotone or more particularly whether a given y is in
are both maximal monotone.
Theorem 33.1.7 Let A and B be maximal monotone, let
and suppose Bλxλ is bounded independent of λ. Then there exists x ∈ D
Proof: First of all, it follows from Proposition 33.1.6 that there exists a unique xλ.
and so by monotonicity of A,
I want to write as many things as possible in terms of the Bλ
. Denote as Jλ
Thus 33.1.3 becomes
Now recall Bμx ∈ BJμ
Then by monotonicity the first and last terms to the right
of the equal sign in the above are negative. Therefore,
where C is some constant which comes from the assumption the Bλxλ are bounded.
Therefore, letting λ denote a sequence converging to 0 it follows
for some x, the convergence being strong convergence. Also taking a further subsequence
and using weak compactness it can be assumed
where this time the convergence is weak. Taking another subsequence, it can also be
the convergence being weak convergence. Recall Bλxλ ∈ BJλ
and also note that
by assumption there is a constant C
independent of λ
also. Now it follows from Lemma 33.1.5 that x1 ∈ D
z1 ∈ Bx1
and so by the same lemma again,
By 33.1.4 it follows
and this proves the theorem.