and let A be maximal monotone and let B : X →X^{′}be monotone hemicontinuous, bounded, and coercive. Then B+A is also maximalmonotone. Also B + A is onto.
and so y^{∗}∈ Bx + Ax + Fx showing that B + A is maximal monotone because it added to
F is onto. As to the last claim, just don’t add in F in the argument. Thus for all
[u,u∗]
,
∗ ∗
〈Bx − y + u ,u − x〉 ≥ 0
Then the rest is as before. You find that y^{∗}− Bx ∈ Ax. ■
Corollary 23.7.40Suppose instead of 0 ∈ D
(A )
, it is known that x_{0}∈ D
(A)
and
lim 〈B-(x0 +-x),x〉-= ∞
∥x∥→ ∞ ∥x∥
Then if B is monotone and hemicontinuous and A is maximal monotone, then B + A isonto.
Proof:Let Â
(x)
≡ A
(x0 + x)
so in fact 0 ∈ D
( )
Aˆ
. Then letting
ˆB
be defined
similarly, it follows from the above lemma that if y^{∗}∈ X^{′}, there exists x such
that
y∗ ∈ Aˆx + Bˆx ≡ A (x0 + x)+ B (x0 + x) ■
Lemma 23.7.41Let 0 be on the interior of D
(A)
and also in D
(B)
. Also let
0 ∈ B
(0)
and 0 ∈ A
(0)
. Then if A,B are maximal monotone, so is A + B.
Proof:Note that, since 0 ∈ A
(0)
, if x^{∗}∈ Ax, then
∗
〈x ,x〉
≥ 0. Also note that
∥Bλ(0)∥
≤
|B (0)|
= 0 and so also
〈Bλx,x〉
≥ 0. It is necessary to show that F + A + B
is onto. However, B_{λ} is monotone hemicontinuous, bounded and coercive. Hence, by
Lemma 23.7.39, B_{λ} + A is maximal monotone. If x^{∗}∈ X^{′} is given, there exists a solution
to
∗
x ∈ F xλ + B λxλ + Axλ
Do both sides to x_{λ} and let x_{λ}^{∗}∈ Ax_{λ} be such that equality holds in the above.
Thus from 23.7.62, x_{λ},x_{λ}^{∗},Fx_{λ} are all bounded. Hence it follows from 23.7.61 that B_{λ}x_{λ}
is also bounded. Therefore, there is a sequence, λ_{n}→ 0 such that
lim sup 〈F xλn + x∗λ − (F xλm + x∗λ ),xλn − xλm〉 ≤ 0
m,n→∞ n m
Now from Lemma 23.7.39, F + A is maximal monotone. Hence Proposition 23.7.38
applies again and it follows that u^{∗} + w^{∗}∈ Fz + Az. Then passing to the limit as n →∞
in
x∗ = F x + B x + x∗
λn λn λn λn
it follows that
x∗ = u∗ + b∗ + w ∗ = Fz + Az + Bz
and this shows that A + B is maximal monotone because x^{∗} was arbitrary.
■
You don’t need to assume all that stuff about 0 ∈ A
(0)
,0 ∈ B
(0)
,0 on interior of
D
(A)
and so forth.
Theorem 23.7.42Suppose A,B are maximal monotone and the interior of D
(A)
has nonempty intersection with D
(B)
. Then A + B is maximal monotone.
Proof:Let x_{0} be on the interior of D
(A)
and also in D
(B)
. Let Â
(x)
= A
(x0 + x)
−x_{0}^{∗}
where x_{0}^{∗}∈ A
(x0)
. Thus 0 ∈ D
( )
Aˆ
and 0 ∈Â
(0)
. Do the same thing for B
to get
ˆB
defined similarly. Are these still maximal monotone? Suppose for all
[u,u∗]
∈G
( )
ˆA
〈y∗ − u∗,y− u〉 ≥ 0
Does it follow that y^{∗}∈Ây? It is given that u^{∗}∈ A
(x0 + u)
. The above implies for all
[u,u∗]
∈G
( )
ˆA
〈y∗ + x∗0 − (u∗ + x∗0),(y+ x0)− (u +x0)〉 ≥ 0
and since u + x_{0} is a generic element of D
(A)
for u ∈ D
( )
ˆA
, the above implies
y^{∗} + x_{0}^{∗}∈ A
(y+ x0)
and so y ∈ A
(y + x0)
− x_{0}^{∗}≡Â
(y)
. Hence the graph is maximal.
Similar for
ˆ
B
. Thus the lemma can be applied to Â,
ˆ
B
to conclude that the sum of these
is maximal monotone. Now a repeat of the above reasoning which shows that Â is
maximal monotone shows that the fact that Â +
ˆ
B
is maximal monotone implies that
A + B is also. You just shift with −x_{0} instead of x_{0}. It amounts to nothing more than the
observation that maximal graphs don’t lose their maximality by shifting their ranges and
domains. ■
Suppose B,A are maximal monotone. Does there always exist a solution x
to
∗
x ∈ F x+ B λx+ Ax? (23.7.64)
(23.7.64)
Consider the monotone hemicontinuous and bounded operator F + B_{λ}.Is
ˆ
F
+
ˆ
B
_{λ} defined
by
( ) ( )
Fˆ+ Bˆλ (x) ≡ Fˆ+ Bˆλ (x+ x0)
also coercive for some x_{0}∈ D
(A)
? If so, the existence of the desired solution to the above
inclusion follows from Corollary 23.7.40. Then for all
and so by Corollary 23.7.40, there exists a solution to 23.7.64. This shows half of
the following interesting theorem which is another version of the above major
result.
Theorem 23.7.43Suppose A,B are maximal monotone operators. Then for eachx^{∗}∈ X^{′}, there exists a solution x_{λ}to
x∗ ∈ F xλ + B λxλ + Axλ, λ > 0 (23.7.65)
(23.7.65)
If for λ ∈
(0,δ)
,
{Bλxλ}
is bounded, then there exists a solution x to
x∗ ∈ F x+ Bx + Ax
Proof: The existence of a solution to the inclusion 23.7.65 comes from the above
discussion. The last claim follows from almost a repeat of the last part of the proof of the
above theorem. Since
Now from Corollary 23.7.40, F + A is maximal monotone (In fact, F + A is onto). Hence
Proposition 23.7.38 applies again and it follows that u^{∗} + w^{∗}∈ Fz + Az. Then passing to
the limit as n →∞ in