In this section gives surjectivity of the sum of a pseudomonotone set valued map with a
linear maximal monotone map and also with another maximal monotone operator added
in. It generalizes the surjectivity results given earlier because one could have 0 for the
maximal monotone linear operator. The theorems developed here lead to nice results on
evolution equations because the linear maximal monotone operator can be something like
a time derivative and X can be some sort of an L^{p} space for functions having values in
a suitable Banach space. This is presented later in the material on Bochner
integrals.
The notation
∗
〈z,u〉
_{V ′,V } will mean z^{∗}
(u)
in this section. We will not worry about the
order either. Thus
〈u,z∗〉 ≡ z∗(u) ≡ 〈z∗,u〉
This is just convenient in writing things down. Also, it is assumed that all Banach spaces
are real to simplify the presentation. It is also usually assumed that the Banach spaces
are reflexive. Thus we can regard
′
(V × V′)= V ′ × V
and
〈(y∗,x ),(u,v∗)〉
≡
〈y∗,u〉
+
〈x,v∗〉
. It is known [?] that for a reflexive Banach space,
there is always an equivalent strictly convex norm. It is therefore, assumed that the norm
for the reflexive Banach space is strictly convex.
Definition 23.8.1Let L : D
(L )
⊆ V → V^{′}be a linear map where we always assumeD
(L)
is dense in V . Then
D (L ∗) ≡ {u ∈ V : |〈Lv,u〉| ≤ C∥v∥ for all v ∈ D (L)}
For such u, it follows that on a dense subset of V, namely D
(L)
,v →
〈Lz,u〉
is acontinuous linear map. Hence there exists a unique element of V^{′}, denoted as L^{∗}u suchthat for all v ∈ D
(L)
,
〈Lv,u〉 ′ = 〈L∗u,v〉 ′
V ,V V,V
Thus
L : D (L) ⊆ V → V′
L∗ : D (L ∗) ⊆ V → V ′
There is an interesting description of L^{∗} in terms of L which will be quite
useful.
Proposition 23.8.2Let τ : V × V^{′}→ V^{′}× V be given by τ
Theorem 23.5.4 is a very nice surjectivity result for set valued pseudomonotone
operators. We recall what it said here. Recall the meaning of coercive.
{ }
lim inf 〈z∗,v〉: z∗ ∈ Tv = ∞
∥v∥→∞ ||v||
In this section, we use the convenient notation
〈z∗,x〉
_{V ′,V }≡ z^{∗}
(x)
.
Theorem 23.8.3Let V be a reflexive Banach space and let T : V → P
(V ′)
be pseudomonotone, bounded and coercive. Then T is onto. More generally, thiscontinues to hold if T is modified bounded pseudomonotone.
Recall the definition of pseudomonotone.
Definition 23.8.4For X a reflexive Banach space, we say A : X → P
(X ′)
ispseudomonotone if the following hold.
The set Au is nonempty, closed and convex for all u ∈ X.
If F is a finite dimensional subspace of X, u ∈ F, and if U is a weakly openset in V^{′}such that Au ⊆ U, then there exists a δ > 0 such that if v ∈ B_{δ}(u)∩Fthen Av ⊆ U. (Weakly upper semicontinuous on finite dimensional subspaces.)
If u_{i}→ u weakly in X and u_{i}^{∗}∈ Au_{i}is such that
Also recall the definition of modified bounded pseudomonotone. It is just the above
except that the limit condition is replaced with the following condition: If u_{i}→ u weakly
in X and
Also recall that this more general limit condition along with the assumption 1 and the
assumption that A is bounded is sufficient to obtain condition 2. This was Lemma 23.4.9
proved earlier and stated here for convenience.
Lemma 23.8.5Let A : X →P
(X′)
satisfy conditions 1and 3above andsuppose A is bounded. Also suppose the condition that if x_{n}→ x weakly and
limsup_{n→∞}
〈zn,xn − x〉
≤ 0 implies there exists a subsequence
{xnk}
such that for anyy,
lim nin→f∞〈znk,xnk − y〉 ≥ 〈z(y),x− y〉
for z
(y)
some element of Ax. Then if this weaker condition holds, you havethat if U is a weakly open set containing Ax, then Ax_{n}⊆ U for all n largeenough.
Definition 23.8.6Now let L : D
(L)
⊆ V → V^{′}such that L is linear,monotone, D
(L)
is dense in V , L is closed, and L^{∗}is monotone. Let A : V →P
′
(V )
be a boundedoperator. Then A is called L pseudomonotone if Av is closed and convex in V^{′}and forany sequence
{un}
⊆ D
(L)
such that u_{n}→ u weakly in V and Lu_{n}→ Lu weakly in V^{′},and for z_{n}^{∗}∈ Au_{n},
lim sup 〈z∗n,un − u〉 ≤ 0
n→∞
then for every v ∈ V, there exists z^{∗}
(v)
∈ Au such that
lim inf 〈z∗n,un − v〉 ≥ 〈z∗(v),u− v〉
n→ ∞
It is called L modified bounded pseudomonotone if the above liminf conditionholds for some subsequence whenever u_{n}→ u weakly and Lu_{n}→ Lu weakly and
limsup_{n→∞}
〈z∗n,un − u〉
≤ 0.
Lemma 23.8.7Suppose X is the Banach space
X = D (L), ∥u∥X ≡ ∥u∥V + ∥Lu ∥V′
where L is as described in the above definition. Also assume that A is bounded. Then if Ais L pseudomonotone, it follows that A is pseudomonotone as a map from X to P
(X′)
.If A is L modified bounded pseudomonotone, then A is modified bounded pseudomonotoneas a map from X to P
(X ′)
.
Proof: Is A bounded? Of course, because the norm of X is stronger than the norm on
V . Is Au convex and closed? This also follows because X ⊆ V . It is clear that Au is
convex. If
{zn}
⊆ Au and z_{n}→ z in X^{′}, then does it follow that z ∈ Au? Since A is
bounded, there is a further subsequence which converges weakly to w in V^{′}. However, Au
is convex and closed so it is weakly closed. Hence w ∈ Au and also w = z. It only remains
to verify the pseudomonotone limit condition. Suppose then that u_{n}→ u weakly in X
and for z_{n}^{∗}∈ Au_{n},
lim sup 〈z∗n,un − u〉 ≤ 0
n→∞
Then it follows that Lu_{n}→ Lu weakly in V^{′} and u_{n}→ u weakly in V so u ∈ X. Hence
the assumption that A is L pseudomonotone implies that for every v ∈ V, and for every
v ∈ X, there exists z^{∗}
(v)
∈ Au ⊆ V^{′}⊆ X^{′} such that
∗ ∗
lim nin→f∞ 〈zn,un − v〉 ≥ 〈z (v),u− v〉
The last claim goes the same way. You just have to take a subsequence. ■
Then we have the following major surjectivity result. In this theorem, we will assume
for simplicity that all spaces are real spaces. Versions of this appear to be due to Brezis
[?] and Lions [?]. Of course the theorem holds for complex spaces as well. You just need
to use Re
〈 〉
instead of
〈 〉
.
Theorem 23.8.8Let L : D
(L)
⊆ V → V^{′}where D
(L)
is dense, L is monotone,L is closed, and L^{∗}is monotone, L a linear map. Let A : V → P
′
(V )
be Lpseudomonotone, bounded, coercive. Then L + A is onto. Here V is a reflexiveBanach space such that the norms for V and V^{′}are strictly convex.In case that A isstrictly monotone (
〈Au − Av,u − v〉
> 0 implies u≠v) the solution u to f ∈ Lu+Auis unique. If, in addition to this,
〈Au − Av,u− v〉
≥ r
(∥u − v∥U)
where U is someBanach space containing V, and r is a positive strictly increasing function for which
lim_{t→0+}r
(t)
= 0, then the map f → u where f ∈ Lu + Au is continuous as a mapfrom V^{′}to U. The conclusion holds if A is only L modified bounded pseudomonotone.
Proof: Let F be the duality map for p = 2. Consider the Banach space X given
by
X = D (L), ∥u∥X ≡ ∥u ∥V + ∥Lu ∥V ′
This is isometric with the graph of L with the graph norm and so X is reflexive. Now
define a set valued map G_{ε} on X as follows. z^{∗}∈ G_{ε}
(u)
means there exists w^{∗}∈ Au such
that.
〈z∗,v〉 ′ = ε〈Lv,F −1(Lu)〉 ′ + 〈Lu, v〉 ′ + 〈w∗,v〉 ′
X,X V ,V V ,V V ,V
It follows from Lemma 23.8.7 that G_{ε} is the sum of a set valued L modified bounded
pseudomonotone operator with an operator which is demicontinuous, bounded, and
monotone, hence pseudomonotone. Thus by Lemma 23.5.2 it is L modified bounded
pseudomonotone. Is it coercive?
{ 〈 〉 }
〈z∗,u〉 +ε Lu,F−1 (Lu )V ′,V + 〈Lu,u〉V′,V ∗
∥ul∥im→ ∞inf ----------------||u||-----------------: z ∈ Au = ∞?
X X