23.1 Some Nonlinear Single Valued Operators
Here is an assortment of nonlinear operators which are useful in applications
to nonlinear partial differential equations. Generalizations of the notion of a
pseudomonotone map will be presented later to include the case of set valued
pseudomonotone maps. This is on the single valued version of some of these and these
ideas originate with Brezis in the 1960’s. A good description is given in Lions
Definition 23.1.1 For V a real Banach space, A : V → V ′ is a pseudomonotone map if
it follows that for all v ∈ V,
The half arrows denote weak convergence.
If V is finite dimensional, then pseudomonotone maps are continuous. Also the
property of being pseudomonotone is preserved when restriction is made to finite
dimensional spaces. The notation is explained in the following diagram.
The map i is just the inclusion map. iw ≡ w and i∗ is the usual adjoint map.
. Thus i∗Ai
and it is defined
in other words, you restrict A to W and only consider what the resulting functional does
to things in W.
Proposition 23.1.2 Let V be finite dimensional and let A : V → V ′ be
pseudomonotone and bounded (meaning A maps bounded sets to bounded sets). Then
A is continuous. Also, if A : V → V ′ is pseudomonotone and bounded, and if
W ⊆ V is a finite dimensional subspace, then i∗Ai is pseudomonotone as a map
from W to W′.
Proof: Say un → u. Does it follow that Aun → Au? If not, then there is a
subsequence such that Aun → ξ≠Au thanks to
being bounded. Then the limsup
condition holds obviously. In fact the limit of
exists and equals 0. Hence for
for all v and so in fact ξ = Au after all. Thus A must be continuous.
As to the second part of this proposition, if you have wn ⇀ w in W, then in fact
convergence takes place strongly because weak and strong convergence are the same in
finite dimensions. Hence the same argument given above holds to show that i∗Ai is
Definition 23.1.3 A : V → V ′ is monotone if for all v,u ∈ V,
and A is Hemicontinuous if for all v,u ∈ V,
Theorem 23.1.4 Let V be a Banach space and let A : V → V ′ be monotone and
hemicontinuous. Then A is pseudomonotone.
Proof: Let A be monotone and Hemicontinuous. First here is a claim.
Claim: If 23.1.1 and 23.1.2 hold, then limn→∞〈Aun,un − u〉 = 0.
Proof of the claim: Since A is monotone,
Now using that A is monotone again, then letting t > 0,
Taking the liminf on both sides and using the claim and t > 0,
Next divide by t and use the Hemicontinuity of A to conclude that
From the claim,
|liminf n→∞〈Aun,u − v〉 || = liminf n→∞
| || = liminf n→∞〈Aun,un − v〉≥〈Au,u − v〉.■ |
Monotonicity is very important in the above proof. The next example shows that even
if the operator is linear and bounded, it is not necessarily pseudomonotone.
Example 23.1.5 Let H be any Hilbert space (complete inner product space, more on
these later) and let A : H → H′ be given by
Then A fails to be pseudomonotone.
be an orthonormal set of vectors in H.
and so for any x ∈ H,limn→∞
Thus xn ⇀
0 ≡ x.
If A were pseudomonotone, we would need to be able to conclude that for all
y ∈ H,
The following proposition is useful.
Proposition 23.1.6 Suppose A : V → V ′ is pseudomonotone and bounded where
V is separable. Then it must be demicontinuous. This means that if un → u, then
Aun ⇀ Au. In case that V is reflexive, you don’t need the assumption that V is
Proof: Since un → u is strong convergence and since Aun is bounded, it
Suppose this is not so that Aun converges weakly to Au. Since A is bounded, there exists
a subsequence, still denoted by n such that Aun ⇀ ξ weak ∗. I need to verify ξ = Au.
From the above, it follows that for all v ∈ V
There is another type of operator which is more general than pseudomonotone.
Definition 23.1.7 Let A : V → V ′ be an operator. Then A is called type M if whenever
un ⇀ u and Aun ⇀ ξ, and
it follows that Au = ξ.
Proposition 23.1.8 If A is pseudomonotone, then A is type M.
Proof: Suppose A is pseudomonotone and un ⇀ u and Aun ⇀ ξ, and
for all v ∈ V . Consequently, for all v ∈ V,
and so Au
An interesting result is the following which states that a monotone linear function
added to a type M is also type M.
Proposition 23.1.9 Suppose A : V → V ′ is type M and suppose L : V → V ′
is monotone, bounded and linear. Then L + A is type M. Let V be separable or
reflexive so that the weak convergences in the following argument are valid.
Proof: Suppose un ⇀ u and Aun + Lun ⇀ ξ and also that
Does it follow that ξ = Au + Lu? Suppose not. There exists a further subsequence, still
called n such that Lun ⇀ Lu. This follows because L is linear and bounded. Then from
Hence with this further subsequence, the limsup is no larger and so
It follows since A is type M that Au = ξ − Lu, which contradicts the assumption that
ξ≠Au + Lu. ■
There is also the following useful generalization of the above proposition.
Corollary 23.1.10 Suppose A : V → V ′ is type M and suppose L : W → W′ is
monotone, bounded and linear where V ⊆ W and V is dense in W so that W′⊆ V ′.
Then for u0 ∈ W define M
. Then M
+ A is type M. Let V be
separable or reflexive so that the weak convergences in the following argument are
Proof: Suppose un ⇀ u and Aun + Mun ⇀ ξ and also that
Does it follow that ξ = Au + Mu? Suppose not. By assumption, un ⇀ u and
since L is bounded, there is a further subsequence, still called n such that
Since M is monotone,
Hence with this further subsequence, the limsup is no larger and so
It follows since A is type M that Au = ξ − Mu, which contradicts the assumption that
ξ≠Au + Mu. ■
The following is Browder’s lemma. It is a very interesting application of the Brouwer
fixed point theorem.
Lemma 23.1.11 (Browder) Let K be a convex closed and bounded set in ℝn and let
A : K → ℝn be continuous and f ∈ ℝn. Then there exists x ∈ K such that for all
y ∈ K,
If K is convex, closed, bounded subset of V a finite dimensional vector space,
then the same conclusion holds. If f ∈ V ′, there exists x ∈ K such that for all
y ∈ K,
Proof: Let PK denote the projection onto K. Thus PK is Lipschitz continuous.
is a continuous map from K to K. By the Brouwer fixed point theorem, it has a fixed
point x ∈ K. Therefore, for all y ∈ K,
As to the second claim. Consider the following diagram.
Thus θ and θ∗ are both continuous linear and one to one and onto. Hence there is
x ∈ θ−1K a closed convex and bounded subset of ℝn such that x = θ−1u,u ∈ K,
for all y ∈ K. ■
From this lemma, there is an interesting theorem on surjectivity.
Proposition 23.1.12 Let A : V → V ′ be continuous and coercive,
for some v0. Then for all f ∈ V ′, there exists v ∈ V such that Av = f.
Proof: Define the closed convex sets Bn ≡B
. By Browder’s lemma, there
exists xn such that
for all y ∈ Bn. Then taking y = v0,
letting wn = vn − v0,
which implies that the
and hence the
are bounded. It follows that for large
is an interior point of Bn
for all z in some open ball centered at v0. Hence f − Avn = 0. ■
Lemma 23.1.13 Let A : V → V ′ be type M and bounded and suppose V is
reflexive or V is separable. Then A is demicontinuous.
Proof: Suppose un → u and Aun fails to converge weakly to Au. Then there is a
further subsequence, still denoted as un such that Aun ⇀ ζ≠Au. Then thanks to the
strong convergence, you have
which implies ζ = Au after all. ■
With these lemmas and the above proposition, there is a very interesting surjectivity
Theorem 23.1.14 Let A : V → V ′ be type M, bounded, and coercive
for some u0, where V is a separable reflexive Banach space. Then A is surjective.
Proof: Since V is separable, there exists an increasing sequence of finite
and each V n
. Then consider the following diagram.
The map i is the inclusion map. Consider the map i∗Ai. By Lemma 23.1.13 this map is
Hence i∗Ai is coercive. Let f ∈ V ′. Then from Proposition 23.1.12, there exists xn such
In other words,
for all y ∈ V n. Letting y ≡ vn − u0 ≡ wn,