1 Introduction
The present paper is devoted to the study of the following eigenvalue problem with a set-valued perturbation:
(1.1)
{
L
x
-
λ
C
x
+
ε
ϕ
(
x
)
∋
0
,
x
∈
∂
Ω
.
Here L:E→F is a Fredholm linear operator of index 0 between two real Banach spaces E and F such that kerL≠0, C is another bounded linear operator, Ω is an open subset of E not necessarily bounded and containing 0, ϕ:Ω¯→2F is a locally compact, upper semicontinuous (u.s.c. for short) set-valued map of CJ-type (see Section 4 for a precise definition), and λ,ε∈ℝ are parameters.
Problem (1.1) can be seen as a set-valued perturbation of a linear eigenvalue problem (which is retrieved for ε=0):
(1.2)
{
L
x
-
λ
C
x
=
0
,
x
∈
∂
Ω
.
So, it is reasonable to expect that, under suitable assumptions, solutions of (1.1) appear in a neighborhood of the eigenpairs (x,λ) of (1.2). In fact, we show that this is the case for the trivial eigenpairs (x,0), provided dim(kerL) is odd, the set Ω¯∩kerL is compact, and the following transversality condition holds:
(1.3)
im
L
+
C
(
ker
L
)
=
F
.
More precisely, we denote 𝒮0=∂Ω∩kerL the set of trivial solutions of (1.2). We prove that there exist a rectangle ℛ=[-a,a]×[-b,b] (a,b>0) and c>0 such that for all ε∈[-a,a] the set of real parameters λ∈[-b,b] for which (1.1) admits a nontrivial solution x∈E with dist(x,𝒮0)<c is nonempty and depends on ε by means of an u.s.c. set-valued map. Similarly, for all ε∈[-a,a] the set of vectors x∈E with dist(x,𝒮0)<c that solve (1.1) for some λ∈[-b,b] is nonempty and depends on ε by means of an u.s.c. set-valued map. This is usually referred to as a persistence result for eigenpairs. Using such persistence, we prove that 𝒮0 contains at least one bifurcation point, i.e., a trivial solution x0 such that any neighborhood of x0 in E contains a nontrivial solution.
The origin of this type of investigation of nonlinear eigenvalue problems goes back to a work of Chiappinelli [11], in which the author investigates a persistence property of the eigenvalues and eigenvectors of the system
(1.4)
{
L
x
+
ε
N
(
x
)
=
λ
x
,
∥
x
∥
=
1
,
where L is a self-adjoint operator defined on a real Hilbert space H, N:H→H is a nonlinear continuous (single-valued) map, ε, λ still are real parameters. Under the assumptions that λ0∈ℝ is an isolated simple eigenvalue of L and that N is Lipschitz continuous, Chiappinelli proves that there exist two H-valued Lipschitz curves, ε↦xε1 and ε↦xε2, defined in a neighborhood V of 0 in ℝ, as well as two real Lipschitz functions, ε↦λε1 and ε↦λε2, such that for i=1,2 and ε∈V one has
L
x
ε
i
+
ε
N
(
x
ε
i
)
=
λ
ε
i
x
ε
i
,
∥
x
ε
i
∥
=
1
,
i.e., the triples (xεi,ε,λεi) solve (1.4) for all ε∈V. In particular, when ε=0, these four functions satisfy x0i=xi, λ0i=λ0, where x1 and x2 are the two unit eigenvectors of L corresponding to the simple eigenvalue λ0. After the result of Chiappinelli, in a series of papers [12, 13, 14, 15] the above property of local persistence of the eigenvalues and eigenvectors was extended to the case in which the multiplicity of the eigenvalue λ0 is bigger than one.
In particular, in [4] the first author, with Calamai, Furi, and Pera, proved a persistence result for (1.2) under a single-valued nonlinear map in general Banach spaces. The approach in [4] is topological, based on a concept of degree, developed in [2, 3], for a class of noncompact (single-valued) perturbations of Fredholm maps of index zero between Banach spaces.
We proceed here in the general spirit of [4], extending the result to the case of a set-valued perturbation. Such an extension requires a more general degree theory for set-valued maps, which extends Brouwer’s degree for nonlinear maps on C1-manifolds. Such a degree theory has been introduced in [30] and redefined in [9] by a precise notion of orientation for set-valued perturbations of nonlinear Fredholm maps between Banach spaces. The concept of orientation used in [9] (and reproduced here) is a natural extension of a notion of orientation for nonlinear Fredholm maps in Banach spaces presented in [5, 6] and on which is also based the approach in [4]. This orientation actually simplifies the method followed to define the degree in [30], based on the so called concept of oriented Fredholm structure, introduced by Elworty and Tromba in [20, 19] (where an orientation is constructed on the source and targets Banach spaces and manifolds).
Our abstract results find a natural application to differential inclusions. This type of problems, arising from control theory and differential equations with discontinuous nonlinearities (see [1, 10, 21]), extends classical differential equations by means of set-valued terms usually representing some degree of uncertainty of the problem. Problem (1.1) can model several differential inclusions, with L being a Fredholm differential operator of index 0 between two function spaces, C being some linear operator, ∂Ω representing some constraint, and ϕ being a set-valued mapping satisfying convenient conditions.
To fix ideas, we will consider the following ordinary differential inclusion with Neumann boundary conditions and an integral constraint:
{
u
′′
+
u
′
-
λ
u
+
ε
Φ
(
u
)
∋
0
in
[
0
,
1
]
,
u
′
(
0
)
=
u
′
(
1
)
=
0
,
∥
u
∥
1
=
1
.
Here Φ(u):[0,1]→2ℝ is a set-valued map depending on u, to be chosen according to several requirements (three different examples will be presented). We shall prove that the transversality condition (1.3) holds, and hence the above problem admits at least one bifurcation point. This is by no means the only possible application of our method, for instance one could define the operator
L
u
=
-
Δ
u
-
λ
1
u
,
where λ1>0 denotes the first eigenvalue of the negative Laplacian on a domain with homogeneous Dirichlet conditions. Then kerL has dimension 1, hence, by appropriately choosing the function spaces E and F, one can rephrase L as a Fredholm operator of index 0, and accordingly define C and ϕ in (1.1). Other examples for the dynamic part of the problem are shown in [4]. Nevertheless, since the novelty of our work lies in the set-valued term, we will restrict our attention on the above inclusion problem, focusing on the possible choices of Φ.
For the convenience of the reader, most of our paper (Sections 2–5) is devoted to the construction of the orientation and degree for the set-valued perturbations of Fredholm maps. Then, in Section 6, we prove our persistence and bifurcation results. Finally, in the large Section 7 we will prove bifurcation results for differential inclusions.
Notation.
Whenever E, F are Banach spaces, we denote by ℒ(E,F) the space of bounded linear operators from E into F (in particular, ℒ(E)=ℒ(E,E)). We shall use the term operator for linear functions, and map for nonlinear ones.
2 A Remark on Orientation and Transversality
In this preliminary section we recall some facts regarding the classical notions of orientation and transversality in finite dimension. We assume that the reader is familiar with the notion of orientation for finite-dimensional Banach manifolds and spaces. Let M be a real C1-manifold, and let F be a real vector space such that
dim
(
M
)
=
dim
(
F
)
<
∞
.
Definition 2.1.
A subspace F1⊆F and a map g∈C1(M,F) are transverse if for all x∈M,
im
D
g
(
x
)
+
F
1
=
F
.
In the situation described above, the map g is backward orientation-preserving on F1:
Lemma 2.2.
Let M, F be oriented, and let F1⊆F, g∈C1(M,F) be transverse. Then any orientation of F1 induces an orientation of M1=g-1(F1).
Proof.
Since g is C1, M1 is a C1-submanifold of M with
dim
(
M
1
)
=
dim
(
F
1
)
.
Fix x∈M1, and let Tx(M), Tx(M1) be the tangent spaces to M, M1, respectively, at x. Then we have
T
x
(
M
1
)
=
(
D
g
(
x
)
)
-
1
(
F
1
)
.
Let E0 be a direct complement to Tx(M1) in Tx(M), and F0=Dg(x)(E0). The restriction Dg(x)|E0∈ℒ(E0,F0) is an isomorphism and F0⊕F1=F. Now let F1 be oriented so that any two positively oriented bases of F0, F1 (in this order) form a positively oriented basis of F. Thus, we can orient E0 so that Dg(x)|E0 is orientation-preserving.
Similarly, we can orient Tx(M1) so that any two positively oriented bases of E0, Tx(M1) (in this order) form a positively oriented basis of Tx(M). Then this pointwise choice induces a global orientation on M1 (see [24, p. 100] for details). ∎
By Lemma 2.2 we have a natural way to orient M1:
Definition 2.3.
Let M, F, F1⊆F be oriented and g∈C1(M,F) transverse to F1. The manifold M1=g-1(F1), with the orientation induced by that of F1 is an oriented g-preimage of F1.
Now let f∈C(M,F), and choose y∈F such that f-1(y)⊂M is compact. Brouwer’s degree for the triple (f,M,y) is defined and denoted by
deg
B
(
f
,
M
,
y
)
∈
ℤ
.
For the definition and properties of Brouwer’s degree (both on open sets and manifolds) we refer to [28, 29]. We only need to add the following reduction property:
Proposition 2.4.
Let M, F, F1⊂F be oriented, let g∈C1(M,F) be transverse to F1, and let M1 be the oriented g-preimage of F1. In addition, let f∈C(M,F), y∈F1 be such that f-1(y) is compact and
(
f
-
g
)
(
M
)
⊆
F
1
.
Finally, let f1=f|M1. Then
deg
B
(
f
,
M
,
y
)
=
deg
B
(
f
1
,
M
1
,
y
)
.
Proof.
First we note that for all x∈M1,
f
(
x
)
=
g
(
x
)
+
(
f
-
g
)
(
x
)
∈
F
1
,
so f1∈C(M1,F1). In particular, f1-1(y)=f-1(y) is a compact subset of M1. We orient M1 and F1 as in Lemma 2.2, so we can define Brouwer’s degree for the triple (f1,M1,y). Now, the conclusion follows from [28, Lemma 4.2.3]. ∎
3 Orientation for Fredholm Maps
In order to develop a degree theory, we need a precise notion of orientability for Fredholm operators and maps. The one we are going to recall here was introduced in [5, 6].
Let E, F be two (possibly, infinite-dimensional) real Banach spaces. We first recall a basic definition:
Definition 3.1.
A bounded linear operator L∈ℒ(E,F) is a Fredholm operator of index k∈ℤ if
dim
(
ker
L
)
,
dim
(
coker
L
)
<
∞
,
dim
(
ker
L
)
-
dim
(
coker
L
)
=
k
.
The set of such operators is denoted Φk(E,F).
It is known that Φk(E,F)⊂ℒ(E,F) is open for all k∈ℤ. We are mainly interested in Φ0(E,F), the set of Fredholm operators of index 0, also denoted Φ0-operators. The following construction leads to a notion of orientation for such operators:
Definition 3.2.
Let L∈Φ0(E,F), A∈ℒ(E,F). Then A is a corrector of L if
dim
(
im
A
)
<
∞
(finite rank),
L
+
A
∈
ℒ
(
E
,
F
)
is an isomorphism.
The set of correctors of L is denoted 𝒞(L).
Clearly, 𝒞(L)≠∅ for all L∈Φ0(E,F). Following [5], we define an equivalence relation in 𝒞(L). Let A,B∈𝒞(L), and set
T
=
(
L
+
B
)
-
1
(
L
+
A
)
,
K
=
I
-
T
=
(
L
+
B
)
-
1
(
B
-
A
)
.
By Definition 3.2, T∈ℒ(E) is an automorphism, and K∈ℒ(E) has finite rank. Let E0⊆E be a non-trivial finite-dimensional subspace such that imK⊆E0, and set T0=T|E0. We note that T0∈ℒ(E0) and is an automorphism as well. Indeed, T0 is injective by injectivity of T, and for all x∈E0 we have
T
0
(
x
)
=
x
-
K
(
x
)
∈
E
0
,
so T0 is surjective as well (recall that dim(E0)<∞). Thus, as soon as we fix a basis for E0, the determinant of T0 is well defined and denoted detT0∈ℝ∖{0}. A remarkable fact is that detT0 does not depend on the choice of E0 (by choosing the same basis in E0 both as the domain and as the codomain of T0), so we can provide T with a uniquely defined determinant by setting
det
T
=
det
T
0
.
The above notion of determinant for linear operators between (possibly) infinite dimensional spaces can be found in [27].
Definition 3.3.
Let L∈Φ0(E,F). Two correctors A,B∈𝒞(L) are L-equivalent if
det
(
(
L
+
B
)
-
1
(
L
+
A
)
)
>
0
.
It is easily seen that L-equivalence is actually an equivalence relation, splitting 𝒞(L) into two equivalence classes. Now we can define a notion of orientation for Φ0-operators:
Definition 3.4.
Let L∈Φ0(E,F).
An orientation of L is any L-equivalence class α⊂𝒞(L), then the pair (L,α) is an orientedΦ0-operator, and a corrector A∈𝒞(L) is positive for (L,α) if A∈α, negative if A∈𝒞(L)∖α.
If L is an isomorphism, then α⊂𝒞(L) is the natural orientation of L if 0∈α, and in such case (L,α) is naturally oriented.
If (L,α) is an oriented Φ0-operator, its sign is defined as follows:
sign
(
L
,
α
)
=
{
+
1
if
(
L
,
α
)
is a naturally oriented isomorphism
,
-
1
if
(
L
,
α
)
is a non-naturally oriented isomorphism
,
0
if
(
L
,
α
)
is not an isomorphism.
Let (L,α) be an oriented Φ0-operator, and let A∈α be a positive corrector. Since the set of isomorphisms is open in ℒ(E,F), we can find a neighborhood 𝒰⊂Φ0(E,F) of L such that A∈𝒞(T) for all T∈𝒰. Then any operator T∈𝒰 can be oriented so that A∈𝒞(T) is a positive corrector. In such a way, any orientation of L induces orientations of nearby Φ0-operators, which allows us to define orientability of Φ0(E,F)-valued maps:
Definition 3.5.
Let X be a topological space, h∈C(X,Φ0(E,F)). An orientation of h is a map α that associates to every x∈X an orientation, say α(x), of h(x)∈Φ0(E,F) satisfying the following continuity condition: there exist A∈α(x) and a neighborhood V⊂X of x such that A∈α(y) for all y∈V. The map h is orientable if it admits an orientation, and in such case (h,α) is an orientedΦ0(E,F)-valued map.
Now we can consider (nonlinear) Fredholm maps:
Definition 3.6.
Let Ω⊆E be open. A map g∈C1(Ω,F) is a Φ0-map if Dg(x)∈Φ0(E,F) for all x∈Ω.
For instance, any Fredholm operator L∈Φ0(E,F) is a Φ0-map, since DL(x)=L for all x∈E.
Definition 3.7.
Let Ω⊆E be open, and let g∈C1(Ω,F) be a Φ0-map.
An orientation of g is any orientation of Dg∈C(Ω,Φ0(E,F)) (Definition 3.5).
The map g is orientable if it admits an orientation α, and in such case (g,α) is an orientedΦ0-map.
The existence (and number) of orientations of a Φ0-map depend mainly on the topology of its domain (see [5] for the proof):
Proposition 3.8.
Let Ω⊆E be open, and let g∈C1(Ω,F) be a Φ0-map.
If
g
is orientable, then it admits at least two orientations.
If
g
is orientable and Ω is connected, then g admits exactly two orientations.
If Ω is simply connected, then g is orientable.
Another important use of Definition 3.5 is towards orientation of Fredholm homotopies:
Definition 3.9.
Let Ω⊆E be open. A map h∈C(Ω×[0,1],F) is a Φ0-homotopy if
h
(
⋅
,
t
)
is a Φ0-map for all t∈[0,1],
the map (x,t)↦Dxh(x,t) is continuous from Ω×[0,1] into Φ0(E,F), where we denote by Dxh(x,t) the derivative of h(⋅,t) at x.
Note that no differentiability in t is required. Condition (ii) here is crucial, as it allows us to apply Definition 3.5 to the map (x,t)↦Dxh(x,t), and thus define a notion of orientation for Φ0-homotopies:
Definition 3.10.
Let Ω⊆E be open, and let h∈C(Ω×[0,1],F) be a Φ0-homotopy.
An orientation of h is any orientation of Dxh∈C(Ω×[0,1],Φ0(E,F)) (Definition 3.5).
The homotopy h is orientable if it admits an orientation α, and in such case (h,α) is an orientedΦ0-homotopy.
Let (h,α) be an oriented Φ0-homotopy. Clearly, α induces an orientation αt of the Φ0-map h(⋅,t), for all t∈[0,1]. Remarkably, the converse is also true, as shown by the following result on continuous transportation of orientations (see [5, Theorem 3.14]):
Proposition 3.11.
Let Ω⊆E be open, let h∈C(Ω×[0,1],F) be a Φ0-homotopy, and let t∈[0,1] be such that h(⋅,t)∈C1(Ω,F) admits an orientation αt. Then there exists a unique orientation α of h which induces αt.
We conclude this section by establishing a link between the orientation of Fredholm maps and that of manifolds:
Proposition 3.12.
Let Ω⊆E be open, let g∈C1(Ω,F) be an orientable Φ0-map, let F1⊆F be a finite-dimensional subspace, transverse to g, and let M1=g-1(F1). Then:
M
1
⊆
E
is a
C
1
-
manifold with
dim
(
M
1
)
=
dim
(
F
1
)
.
Any orientation of
g
and any orientation of
F
1
induce an orientation of
M
1
.
Proof.
Assertion (i) is obvious (see Section 2). Assertion (ii) follows from [5, Remark 2.5, Lemma 3.1].
We prove (iii). Let α be an orientation of g, and x∈M1. By Definition 3.7, α(x) is an orientation of Dg(x)∈Φ0(E,F). By transversality (Definition 2.1), we can find A∈α(x) such that imA⊆F1. Indeed, since Dg(x)∈Φ0(E,F), we can split both Banach spaces as follows:
E
=
D
g
(
x
)
-
1
(
F
1
)
⊕
E
2
,
F
=
F
1
⊕
F
2
,
where E2 is any direct complement of Dg(x)-1(F1) and F2:=Dg(x)(E2). Observe that kerDg(x)⊆Dg(x)-1(F1) and the latter has the same dimension as F1. So we rephrase Dg(x) as
D
g
(
x
)
=
[
L
1
,
1
0
0
L
2
,
2
]
,
where L2,2∈ℒ(E2,F2) is an isomorphism. We may choose A∈ℒ(E,F) with the structure
A
=
[
A
1
,
1
0
0
0
]
,
where A1,1+L1,1∈ℒ(L-1(F1),F1) is an isomorphism. So A∈𝒞(Dg(x)) and imA⊆F1. Choosing A1,1 in such a way that A∈α(x) and assigning an orientation to F1, we orient the tangent space Tx(M1)⊂E so that the isomorphism
(
D
g
(
x
)
+
A
)
|
T
x
(
M
1
)
∈
ℒ
(
T
x
(
M
1
)
,
F
1
)
is orientation-preserving. As proved in [7], such orientation of Tx(M1) does not depend on A. This pointwise choice induces a global orientation on M1. ∎
We can now give a Fredholm analogue of Definition 2.3:
Definition 3.13.
Let Ω⊆E be open, let (g,α) be an oriented Φ0-map, let F1⊆F be a finite-dimensional subspace, transverse to g, and let M1=g-1(F1). With the orientation induced by α and the orientation of F1, M1 is an oriented (Φ0,g)-preimage of F1.
Remark 3.14.
In what follows, we will denote an oriented Φ0-operator (L,α) simply by L, as long as no confusion arises. We will do the same for oriented Φ0-maps, Φ0-homotopies, and so on.
4 Topological Properties of Set-Valued Maps
In this section, for the reader’s convenience, we recall some definitions and properties of set-valued maps between metric spaces, referring to [22] for details. Let X, Y be metric spaces with distance functions dX, dY, respectively. Then X×Y is a metric space under the distance
d
(
(
x
,
y
)
,
(
x
′
,
y
′
)
)
=
max
{
d
X
(
x
,
x
′
)
,
d
Y
(
y
,
y
′
)
}
.
For all A⊂X, x∈X we set
dist
(
x
,
A
)
=
inf
z
∈
A
d
X
(
x
,
z
)
,
and for all ε>0 we set
B
ε
(
A
)
=
{
x
∈
X
:
dist
(
x
,
A
)
<
ε
}
(if A={x}, then we set Bε(A)=Bε(x)). A set-valued map ϕ:X→2Y is a map from X to the set of all parts of Y. We will always assume that ϕ is compact-valued, i.e., that ϕ(x)⊆Y is either ∅ or compact, for all x∈X. The graph of ϕ is defined by
graph
ϕ
=
{
(
x
,
y
)
∈
X
×
Y
:
y
∈
ϕ
(
x
)
}
.
We also recall a classical definition:
Definition 4.1.
A set-valued map ϕ:X→2Y is upper semicontinuous (u.s.c.) if for all open V⊆Y the set
ϕ
+
(
V
)
=
{
x
∈
X
:
ϕ
(
x
)
⊆
V
}
is open.
Any (single-valued) map f:X→Y coincides with the set-valued map ϕ(x)={f(x)}, in such case ϕ is u.s.c. iff f is continuous. A remarkable property of u.s.c. set-valued maps is that they preserve compactness, in the following sense: if ϕ:X→2Y is a u.s.c. set-valued map and C is a compact subset of X, then ϕ(C) is compact.
Remark 4.2.
Any compact subset of a product space can be seen as the graph of a u.s.c. set-valued mapping. Precisely, if 𝒦⊂X×Y is compact, define for all x∈X,
ϕ
(
x
)
=
{
y
∈
Y
:
(
x
,
y
)
∈
𝒦
}
.
Then ϕ:X→2Y is u.s.c. (see [22, Proposition 14.5] or [26, Theorem 1.1.5]).
We introduce the notion of approximability:
Definition 4.3.
Let ϕ:X→2Y.
For all ε>0, f∈C(X,Y) is an ε-approximation of ϕ if for all x∈X there exists x′∈Bε(x) such that f(x)∈Bε(ϕ(x′)) (the set of ε-approximations of ϕ is denoted Bε(ϕ)).
ϕ is approximable if Bε(ϕ)≠∅ for all ε>0.
Note that all approximations of a set-valued map are required to be continuous. A characterization (whose proof is an obvious consequence of Definition 4.3):
Lemma 4.4.
Let ϕ:X→2Y, ε>0, f∈C(X,Y). Then the following are equivalent:
f
(
x
)
∈
B
ε
(
ϕ
(
B
ε
(
x
)
)
)
for all
x
∈
X
.
graph
f
⊆
B
ε
(
graph
ϕ
)
.
Approximation of an u.s.c. set-valued map is a special case, enjoying several properties (see [22, Proposition 22.3]):
Proposition 4.5.
Let ϕ:X→2Y be u.s.c. Then:
For all compact
X
1
⊆
X
, ε>0 there exists δ>0 such that for all f∈Bδ(ϕ) we have f|X1∈Bε(ϕ|X1).
If
X
is compact, then for any metric space
Z, g∈C(Y,Z), and ε>0 there exists δ>0 such that for all f∈Bδ(ϕ) we have g∘f∈Bε(g∘ϕ).
If
X
is compact, then for any u.s.c. set-valued map
ψ
:
X
×
[
0
,
1
]
→
2
Y
, ε>0, and t∈[0,1] there exists δ>0 such that for all f∈Bδ(ψ) we have f(⋅,t)∈Bε(ψ(⋅,t)).
For any metric space
Z
, any u.s.c. set-valued map
ψ
:
X
→
2
Z
, and
ε
>
0
there exists
δ
>
0
such that for all
f
∈
B
δ
(
ϕ
)
, g∈Bδ(ψ) we have (f,g)∈Bε(ϕ×ψ).
Approximability of a set-valued map is strongly influenced by the topology of its values, the easiest case being in general that of convex-valued maps between Banach spaces. In the general case of a metric space, convexity makes no sense and it must be replaced by a more general notion, of topological nature. We recall from [22] some definitions and properties (here 𝕊n-1, 𝔹n denote the unit sphere and closed ball, respectively, in ℝn):
Definition 4.6.
A set A⊂Y is aspheric if for any ε>0 there exists δ∈(0,ε) such that for all n∈ℕ and all g∈C(𝕊n-1,Bδ(A)) there is g~∈C(𝔹n,Bε(A)) such that g~|𝕊n-1=g.
The following characterization of aspheric sets holds in ANR-spaces (absolute neighborhood retracts, see [22, Definition 1.7]):
Proposition 4.7.
Let Y be an ANR-space, A⊆Y. Then the following are equivalent:
There exists a decreasing sequence
(
A
n
)
of compact, contractible subsets of
Y
such that
⋂
n
=
1
∞
A
n
=
A
(
R
δ
-
set).
We go back to set-valued maps:
Definition 4.8.
A set-valued map ϕ:X→2Y is a J-map if ϕ is u.s.c. and ϕ(x) is aspheric for all x∈X. The set of J-maps from X to Y is denoted by J(X,Y).
Some sufficient conditions (see [22, Definition 1.7] the definition of AR-set):
Lemma 4.9.
Let Y be an ANR-space, let ϕ:X→2Y be u.s.c., and let one of the following hold:
ϕ
(
x
)
an
R
δ
-
set for all
x
∈
X
;
ϕ
(
x
)
is an
AR
-
set (absolute retract) for all x∈X.
Then ϕ∈J(X,Y).
By Lemma 4.9, in particular, if either ϕ has contractible values, or Y is a Banach space and ϕ has convex values, then ϕ∈J(X,Y).
In our results, we shall need a slightly more general class of set-valued maps:
Definition 4.10.
A set-valued map ϕ:X→2Y is a CJ-map if there exist a metric space Z, ψ∈J(X,Z), and k∈C(Z,Y) such that ϕ=k∘ψ. The set of CJ-maps from X to Y is denoted by CJ(X,Y).
The following result ensures that CJ-maps are approximable (the proof is easily deduced from [22, Theorems 23.8, 23.9, and Section 26]):
Proposition 4.11.
Let X be a compact ANR-space, ϕ∈CJ(X,Y). Then:
For all
ε
>
0
there exists
δ
ε
>
0
such that for all
δ
∈
(
0
,
δ
ε
)
and all
f
,
g
∈
B
δ
(
ϕ
)
we can find a homotopy
h
∈
C
(
X
×
[
0
,
1
]
,
Y
)
such that
h
(
⋅
,
0
)
=
f
, h(⋅,1)=g, and h(⋅,t)∈Bε(ϕ) for all t∈[0,1].
5 Degree for Multitriples
In this section we develop a degree theory for set-valued maps, extending Brouwer’s degree. This degree has been presented in [9] and its construction basically follows [30], except for the notion of orientation. In fact, our approach is based on the notion of orientation for Fredholm maps, introduced in [6, 7] and recalled here in Section 3, while the construction in [30] makes use of the concept of oriented Fredholm structures, introduced in [20, 19]. For a comprehensive presentation of degree theory for set-valued maps the reader can see the very rich textbook of Väth [32]. Throughout this section E, F are real Banach spaces and Ω⊆E is an open set.
Definition 5.1.
Let g∈C1(Ω,F) be an oriented Φ0-map, let U⊆Ω be open, and let ϕ∈CJ(Ω,F) be locally compact. Then (g,U,ϕ) is an admissible (multi)triple if the coincidence set
C
(
g
,
U
,
ϕ
)
=
{
x
∈
U
:
g
(
x
)
∈
ϕ
(
x
)
}
is compact.
We construct our degree as an integer-valued function defined on the set of admissible triples. First we assume
(5.1)
dim
(
ϕ
(
U
)
)
<
∞
.
Since C(g,U,ϕ) is compact, we can find an open neighborhood W⊂U of C(g,U,ϕ) and a subspace F1⊆F such that dim(F1)=m<∞, ϕ(U)⊂F1 (by virtue of (5.1)), and F1 is transverse to g in W (Definition 2.1), as it can be seen as follows: given any x∈C(g,U,ϕ), take a finite-dimensional subspace Fx of F containing ϕ(U) and transverse to g at x. This is possible since Dg(x) is Fredholm. By the continuity of z↦Dg(z), there exists a neighborhood Wx of x in U such that g is transverse to Fx at any z∈Wx. Then F1 and W as above are obtained by the compactness of C(g,U,ϕ).
We orient F1 and set M=g-1(F1), hence M is an orientable C1-manifold in E with dim(M)=m. We then orient M so that it is an oriented (Φ0,g)-preimage of F1 (Definition 3.13). Then C(g,U,ϕ)⊂M is compact even as a subset of M, and the following open covering of C(g,U,ϕ) exists:
Lemma 5.2.
Let (g,U,ϕ) be an admissible triple satisfying (5.1), and let W, F1, M be defined as above. Then there exist k∈N, and bounded open sets V1,…,Vk⊂M such that
V
¯
j
⊂
M
, j=1,…,k (by V¯j we denote the closure of Vj in E),
C
(
g
,
U
,
ϕ
)
⊂
V
:=
⋃
j
=
1
k
V
j
,
V
¯
j
is diffeomorphic to a closed convex subset of
ℝ
m
, j=1,…,k.
By (iii), V¯1,…,V¯k,V¯ are compact ANR-spaces. So, Lemma 4.9 implies that ϕ|V¯∈CJ(V¯,F1). Thus, by Proposition 4.11, ϕ|V¯ is approximable. In addition, observe that, by the construction of V, one has
g
(
∂
V
)
∩
ϕ
(
∂
V
)
=
∅
.
Recalling that g(∂V) and ϕ(∂V) are compact sets (since ∂V is compact and ϕ is u.s.c.), they have positive distance, say d>0. Hence, taking (for example) ε∈(0,d2), every f∈Bε(ϕ|V¯) satisfies
dist
(
0
,
(
g
-
f
)
(
∂
V
)
)
>
0
.
So, Brouwer’s degree for the triple (g|V¯-f,V,0) is well defined and it enjoys the reduction property displayed in Proposition 2.4. Now we prove that such degree is invariant:
Lemma 5.3.
Let (g,U,ϕ) be an admissible triple satisfying (5.1), and let F1, V, f be defined as above. Then degB(g|V¯-f,V,0) does not depend on F1, V, and f.
Proof.
We prove our assertion in three steps (backward):
(a) Let F1, V be fixed, and let f′,f′′∈Bε(ϕ|V¯) be two approximations of ϕ. By homotopy invariance of Brouwer’s degree and Proposition 4.11, by reducing ε>0 if necessary we can apply [30, Lemma 3.4] and get
deg
B
(
g
|
V
¯
-
f
′
,
V
,
0
)
=
deg
B
(
g
|
V
¯
-
f
′′
,
V
,
0
)
.
(b) Let F1 be fixed, and let V′,V′′⊂M be open such that C(g,U,ϕ)⊂V′∩V′′ and V′¯, V′′¯ are compact ANR-spaces. Without loss of generality we may assume V′⊂V′′. By Proposition 4.5 (i), by reducing ε>0 if necessary we can find f∈Bε(ϕ|V′′¯) such that f|V′¯∈Bε(ϕ|V′¯). So, by the excision property of Brouwer’s degree, we have
deg
B
(
g
|
V
′
¯
-
f
|
V
′
¯
,
V
′
,
0
)
=
deg
B
(
g
|
V
′′
¯
-
f
,
V
′′
,
0
)
.
(c) Finally, let F1′, F1′′ be finite-dimensional subspaces of F, transverse to g in W, such that ϕ(U)⊂F1′∩F1′′. Then, by Proposition 2.4, we have for any choice of V, f the same degB(g|V¯-f,V,0).
So, degB(g|V¯-f,V,0) is independent of F1, V, and f. ∎
By virtue of Lemma 5.3, we can define a degree for the triple (g,U,ϕ):
Definition 5.4.
Let (g,U,ϕ) be an admissible triple satisfying (5.1), and let F1, V, f be defined as above. The degree of (g,U,ϕ) is defined by
deg
(
g
,
U
,
ϕ
)
=
deg
B
(
g
|
V
¯
-
f
,
V
,
0
)
.
The following is a special homotopy invariance result, which will be useful in the forthcoming construction:
Lemma 5.5.
Let U⊆E be open, let h:U×[0,1]→F be an oriented Φ0-homotopy, and let ϕ∈CJ(U×[0,1],F) be locally compact such that
the coincidence set
C
(
h
,
U
×
[
0
,
1
]
,
ϕ
)
=
{
(
x
,
t
)
∈
U
×
[
0
,
1
]
:
h
(
x
,
t
)
∈
ϕ
(
x
,
t
)
}
is compact,
dim
(
ϕ
(
U
×
[
0
,
1
]
)
)
<
∞
.
Then the map t↦deg(h(⋅,t),U,ϕ(⋅,t)) is constant in [0,1].
Proof.
By (i) and (ii) we can find an open neighborhood W⊂U×[0,1] of C(h,U×[0,1],ϕ) and a subspace F1⊆F such that dim(F1)=m<∞, ϕ(U×[0,1])⊂F1, and for all t∈[0,1], F1 is transverse to h(⋅,t) in the set
W
t
:=
{
x
∈
U
:
(
x
,
t
)
∈
W
}
.
Set M1=h-1(F1)∩W; then M1 is an (m+1)-dimensional C1-manifold in E×ℝ with boundary
∂
M
1
=
{
(
x
,
t
)
∈
M
1
:
t
=
0
,
1
}
.
We orient F1, so that the orientations of h, F1 induce an orientation of M1 in a unique way (Proposition 3.11). Now let V⊂M1 be an open (in M1) neighborhood of C(h,U×[0,1],ϕ) such that V¯⊂M1 is a compact ANR-space (the construction is analogous to that of Lemma 5.2). By Propositions 4.5 and 4.11, the restriction ϕ|V¯∈CJ(V¯,F1) is approximable, and for all ε>0 small enough we can find f∈Bε(ϕ|V¯) such that for all t∈[0,1],
deg
(
h
(
⋅
,
t
)
,
U
,
ϕ
(
⋅
,
t
)
)
=
deg
B
(
h
(
⋅
,
t
)
|
V
¯
-
f
(
⋅
,
t
)
,
V
,
0
)
(Definition 5.4). By homotopy invariance of Brouwer’s degree, the latter does not depend on t∈[0,1], which concludes the proof. ∎
Now we remove assumption (5.1). Let (g,U,ϕ) be an admissible triple, not necessarily satisfying (5.1). Since g is locally proper, ϕ is locally compact, and C(g,U,ϕ) is compact (Definition 5.1), we can find a bounded open neighborhood U1⊂U of C(g,U,ϕ) such that g|U¯1 is proper and ϕ|U¯1 is compact. Recalling that ϕ has closed graph, being u.s.c., one can see that g-ϕ:U¯1→2F is a closed set-valued map, and 0∉(g-ϕ)(∂U1). Since (g-ϕ)(∂U1) is closed, there exists δ>0 such that
B
δ
(
0
)
∩
(
g
-
ϕ
)
(
∂
U
1
)
=
∅
.
The set K=ϕ(U¯1)¯ is compact. So we can find a finite-dimensional subspace F1⊂F and a (single-valued) map jδ∈C(K,F1) such that for all x∈K,
∥
j
δ
(
x
)
-
x
∥
F
<
δ
2
(this is a classical result in nonlinear functional analysis, see e.g. [17, Proposition 8.1]). Set
ϕ
1
=
j
δ
∘
ϕ
∈
C
J
(
U
¯
1
,
F
)
(Definition 4.10); then it satisfies
B
δ
/
2
(
0
)
∩
ϕ
1
(
∂
U
1
)
=
∅
,
and C(g,U1,ϕ1) is compact. So, (g,U1,ϕ1) is an admissible triple satisfying (5.1). Definition 5.4 then applies, and produces a degree deg(g,U1,ϕ1). Moreover, such a degree is invariant:
Lemma 5.6.
Let (g,U,ϕ) be an admissible triple, and let U1, jδ be defined as above. Then deg(g,U1,ϕ1) does not depend on U1, jδ.
Proof.
Just as in Lemma 5.3, we divide the proof in two steps backward:
(a) Let U1 be fixed, let F1, K, δ be defined as above, and let jδ′,jδ′′∈C(K,F1) be such that for all x∈K,
∥
j
δ
′
(
x
)
-
x
∥
F
,
∥
j
δ
′′
(
x
)
-
x
∥
F
<
δ
2
.
Set for all (x,t)∈U1×[0,1],
h
(
x
,
t
)
=
g
(
x
)
,
ϕ
~
(
x
,
t
)
=
(
1
-
t
)
j
δ
′
(
ϕ
(
x
)
)
+
t
j
δ
′′
(
ϕ
(
x
)
)
.
Then the map h:U1×[0,1]→F is a Φ0-homotopy (Definition 3.9). A more delicate question is proving that ϕ~∈CJ(U1×[0,1],F), since this map is not explicitly defined as a composition of a J-map and a continuous single-valued function (Definition 4.10). Since ϕ∈CJ(U,F), there exist a metric space Z, ψ∈J(U1,Z), and k∈C(Z,F) such that ϕ=k∘ψ. Set for all (x,t)∈U1×[0,1],
ψ
~
(
x
,
t
)
=
ψ
(
x
)
×
{
t
}
,
so clearly ψ~∈J(U1×[0,1],Z×[0,1]); and set for all (z,t)∈Z×[0,1],
k
~
(
z
,
t
)
=
(
1
-
t
)
j
δ
′
(
k
(
z
)
)
+
t
j
δ
′′
(
k
(
z
)
)
,
so k~∈C(Z×[0,1],F). Then
ϕ
~
=
k
~
∘
ψ
~
∈
C
J
(
U
1
×
[
0
,
1
]
,
F
)
.
Now we prove that the coincidence set
C
(
h
,
U
1
×
[
0
,
1
]
,
ϕ
~
)
=
{
(
x
,
t
)
∈
U
1
×
[
0
,
1
]
:
h
(
x
,
t
)
∈
ϕ
~
(
x
,
t
)
}
is compact. Let (xn,tn) be a sequence in C(h,U1×[0,1],ϕ~). Passing to a subsequence, we have tn→t. For all n∈ℕ there exist yn′,yn′′∈ϕ(xn) such that
g
(
x
n
)
=
(
1
-
t
n
)
j
δ
′
(
y
n
′
)
+
t
n
j
δ
′′
(
y
n
′′
)
.
By compactness of ϕ|U¯1, passing again to a subsequence we have yn′→y′, yn′′→y′′, which implies
g
(
x
n
)
→
(
1
-
t
)
j
δ
′
(
y
′
)
+
t
j
δ
′′
(
y
′′
)
.
By properness of g|U¯1, we can find x∈U¯1 such that up to a further subsequence xn→x. We need to prove that x∈U1. Arguing by contradiction, let x∈∂U1. Then, by the choice of δ>0, we have
dist
(
g
(
x
)
,
ϕ
(
x
)
)
⩾
δ
.
Besides, since ϕ is u.s.c., we have y′,y′′∈ϕ(x), hence by the metric properties of the maps jδ′, jδ′′ we have
dist
(
g
(
x
)
,
ϕ
(
x
)
)
⩽
(
1
-
t
)
dist
(
j
δ
′
(
y
′
)
,
ϕ
(
x
)
)
+
t
dist
(
j
δ
′′
(
y
′′
)
,
ϕ
(
x
)
)
⩽
(
1
-
t
)
∥
j
δ
′
(
y
′
)
-
y
′
∥
F
+
t
∥
j
δ
′′
(
y
′′
)
-
y
′′
∥
F
⩽
δ
2
,
a contradiction. So, x∈U1 and we deduce that C(h,U1×[0,1],ϕ~) is compact. In addition, ϕ~ has a finite-dimensional rank. Then, by Lemma 5.5, deg(h(⋅,t),U1,ϕ~(⋅,t)) is independent of t∈[0,1]. In particular, taking t=0,1 we get
deg
(
g
,
U
1
,
j
δ
′
∘
ϕ
)
=
deg
(
g
,
U
1
,
j
δ
′′
∘
ϕ
)
.
(b) Let U1′,U1′′⊂U be open neighborhoods of C(g,U,ϕ) such that the restrictions of g to both U1′¯, U1′′¯ are proper and the restrictions of ϕ to U1′¯, U1′′¯ are compact, respectively. Without loss of generality we may assume U1′⊆U1′′, hence we continue the construction in U1′′. Then independence of the degree follows from the excision property of Brouwer’s degree.
So, deg(g,U1,ϕ1) does not depend on the choice of U1, jδ. ∎
By virtue of Lemma 5.6, we can define a degree for the triple (g,U,ϕ) extending Definition 5.4:
Definition 5.7.
Let (g,U,ϕ) be an admissible triple, and let U1, ϕ1 be defined as above. The degree of (g,U,ϕ) is defined by
deg
(
g
,
U
,
ϕ
)
=
deg
(
g
,
U
1
,
ϕ
1
)
.
The degree theory we just introduced enjoys some classical properties:
Proposition 5.8.
The following properties hold:
(Normalization) If U⊂E is open such that 0∈U, and I is the naturally oriented identity of E, then
deg
(
I
,
U
,
0
)
=
1
.
(Domain additivity) If (g,U,ϕ) is an admissible triple, U1,U2⊂U are open such that U1∩U2=∅, C(g,U,ϕ)⊂U1∪U2, then (g,U1,ϕ), (g,U2,ϕ) are admissible triples and
deg
(
g
,
U
,
ϕ
)
=
deg
(
g
,
U
1
,
ϕ
)
+
deg
(
g
,
U
2
,
ϕ
)
.
(Homotopy invariance) If U⊂E is open, h:U×[0,1]→F is an oriented Φ0-homotopy, ϕ∈CJ(U×[0,1],F) is locally compact such that C(h,U×[0,1],ϕ) is compact, then for all t∈[0,1], (h(⋅,t),U,ϕ(⋅,t)) is an admissible triple and the function
t
↦
deg
(
h
(
⋅
,
t
)
,
U
,
ϕ
(
⋅
,
t
)
)
is constant in
[
0
,
1
]
.
Proof.
Properties (i) and (ii) follow from Definition 5.7 and the corresponding properties of Brouwer’s degree (the proof is straightforward, so we omit it).
To prove (iii), we first fix t∈[0,1]. By compactness, we can find σ>0 and a bounded open neighborhood W⊂U of the section
C
t
=
{
x
∈
U
:
(
x
,
t
)
∈
C
(
h
,
U
×
[
0
,
1
]
,
ϕ
)
}
such that h|W¯×Iσ is proper and ϕ|W¯×Iσ is compact, where we have set
I
σ
=
[
t
-
σ
,
t
+
σ
]
∩
[
0
,
1
]
.
We also introduce a finite rank map j∈C(K,F), close enough to the identity of K=ϕ(W¯×Iσ)¯. By the excision property of Brouwer’s degree and the construction above, for all s∈Iσ we have
deg
(
h
(
⋅
,
s
)
,
U
,
ϕ
(
⋅
,
s
)
)
=
deg
(
h
(
⋅
,
s
)
,
W
,
j
∘
ϕ
(
⋅
,
s
)
)
.
Besides, by Lemma 5.5 the function
s
↦
deg
(
h
(
⋅
,
s
)
,
W
,
j
∘
ϕ
(
⋅
,
s
)
)
is constant in Iσ, hence deg(h(⋅,s),U,ϕ(⋅,s)) turns out to be locally constant in [0,1]. Since [0,1] is connected, we get the conclusion. ∎
Remark 5.9.
Proposition 5.8 (iii) holds in a stronger form, i.e., for subsets of E×ℝ which are not necessarily products, as it can be seen from the proof. In fact, consider the case in which the domain of h and ϕ is an open subset U of the product E×[0,1]. Fix t∈[0,1] and call
U
t
:=
{
x
∈
E
:
(
x
,
t
)
∈
U
}
.
By the compactness of C(h,U,ϕ) we can find an open subset Wt of E and a positive σ such that
C
t
(
h
,
U
,
ϕ
)
⊂
W
t
⊂
W
¯
t
⊂
U
t
,
C
s
(
h
,
U
,
ϕ
)
⊂
W
t
for every s∈Iσ,
where Cs(h,U,ϕ):={x∈E:(x,t)∈C(h,U,ϕ)} and Iσ is as in the proof of (iii). Now, item (iii) of Proposition 5.8 implies that
s
↦
deg
(
h
(
⋅
,
s
)
,
W
t
,
ϕ
(
⋅
,
s
)
)
is constant in Iσ. The constance of deg(h(⋅,s),Us,ϕ(⋅,s)) follows from the excision property of the degree, which is straightforward (we omit the proof). This property is usually called generalized homotopy invariance.
6 Persistence Results and Bifurcation Points
We can now prove the main results of the present paper, as announced in the Introduction. Throughout this section, E and F are two real Banach spaces, Ω⊂E is an open (not necessarily bounded) set such that 0∈Ω, L∈Φ0(E,F) satisfy kerL≠0, C∈ℒ(E,F) is another bounded linear operator, and ϕ∈CJ(Ω¯,F) is locally compact. The linear operators L, C satisfy the transversality condition (1.3). For all ε,λ∈ℝ we consider the perturbed problem (1.1), whose solutions are meant in the following sense:
Definition 6.1.
A solution of (1.1) is a triple (x,ε,λ)∈∂Ω×ℝ×ℝ such that
L
x
-
λ
C
x
+
ε
ϕ
(
x
)
∋
0
.
The set of solutions is denoted by 𝒮. A solution (x,ε,λ)∈𝒮 is a trivial solution if ε=λ=0. Finally, we say that x0∈∂Ω is a bifurcation point if (x0,0,0)∈𝒮 and any neighborhood of (x0,0,0) in E×ℝ×ℝ contains at least one non-trivial solution.
Clearly, any trivial solution (x,0,0) of (1.1) identifies with its first component x. The set of such vectors is
𝒮
0
=
∂
Ω
∩
ker
L
.
Regarding our definition of a bifurcation point, we note that it is analogous to that of [4], and fits in the very general definition given in [16, p. 2]. Finally, we note that, whenever (x,0,λ)∈𝒮, (x,λ) is an eigenpair of the eigenvalue problem (1.2): thus, we keep the names eigenvector for x and eigenvalue for λ, respectively, for any triple (x,ε,λ)∈𝒮.
As observed in [4, Remark 5.1], transversality condition (1.3) is in fact equivalent to
im
L
⊕
C
(
ker
L
)
=
F
.
Thus, we can find b>0 such that L-λC∈Φ0(E,F) is invertible for all 0<|λ|⩽b. Besides, since 0∉∂Ω, for any bifurcation point x0∈𝒮0 we can find a neighborhood W⊂E×ℝ×ℝ of (x0,0,0) such that any triple (x,ε,λ)∈𝒮∩W actually must have ε≠0.
The map λ↦L-λC (which is orientable according to Definition 3.5 since its domain is simply connected, see Proposition 3.8iii) exhibits a sign jump property (a special case of [8, Corollary 5.1]):
Lemma 6.2.
Let b>0 be defined as above, and let h∈C([-b,b],Φ0(E,F)) be defined by
h
(
λ
)
=
L
-
λ
C
,
and oriented. Then:
The map
λ
↦
sign
h
(
λ
)
is constant in both
[
-
b
,
0
)
and
(
0
,
b
]
.
sign
h
(
b
)
≠
sign
h
(
-
b
)
iff
dim
(
ker
L
)
is odd.
Lemma 6.2 above is the reason why the assumption that dim(kerL) is odd is so important in our theory. Now we prove an existence result on bounded subdomains, which is the core of our argument:
Proposition 6.3.
Let dim(kerL) be odd, let (1.3) hold, and let U⊆Ω be an open, bounded set such that 0∈U and ϕ|U¯∈CJ(U¯,F) is compact. Then there exist a,b>0 such that for all ε∈[-a,a] there exist λ∈[-b,b], x∈∂U such that
L
x
-
λ
C
x
+
ε
ϕ
(
x
)
∋
0
.
Proof.
Let b>0 be as in Lemma 6.2, and fix a>0 (to be better determined later). Set
ℛ
=
[
-
a
,
a
]
×
[
-
b
,
b
]
,
and define the set
(6.1)
𝒦
=
{
(
x
,
ε
,
λ
)
∈
∂
U
×
ℛ
:
L
x
-
λ
C
x
+
ε
ϕ
(
x
)
∋
0
}
.
The set 𝒦⊂E×ℝ×ℝ is compact. Indeed, let (xn,εn,λn) be a sequence in 𝒦. Then (εn,λn) is a bounded sequence in ℛ, hence passing to a subsequence we have (εn,λn)→(ε,λ) for some (ε,λ)∈ℛ. As seen above, we have eventually εn≠0. Now set for all n∈ℕ
y
n
=
-
1
ε
n
(
L
x
n
-
λ
n
C
x
n
)
∈
ϕ
(
x
n
)
.
Since ϕ(U¯)¯ is compact, passing if necessary to a further subsequence, we have yn→y for some y∈F, which implies
lim
n
(
L
x
n
-
λ
C
x
n
)
=
lim
n
(
L
x
n
-
λ
n
C
x
n
)
+
lim
n
(
λ
n
-
λ
)
C
x
n
=
-
ε
y
(recall that (xn) is a bounded sequence and C is a bounded operator). The operator L-λC∈Φ0(E,F) is proper on closed and bounded subsets of E, hence passing again to a subsequence if necessary we have xn→x for some x∈∂U. Thus, (xn,εn,λn)→(x,ε,λ) for some (x,ε,λ)∈𝒦.
Clearly, the projection of 𝒦 onto ℛ, namely the set
Γ
=
{
(
ε
,
λ
)
∈
ℛ
:
(
x
,
ε
,
λ
)
∈
𝒦
for some
x
∈
∂
U
}
,
is compact as well. Now we choose an orientation of L∈Φ0(E,F) (Definition 3.4), and fix (ε,λ)∈ℛ∖Γ. Then (L-λC,U,-εϕ) is an admissible triple (Definition 5.1), since the coincidence set
C
(
L
-
λ
C
,
U
,
-
ε
ϕ
)
=
{
x
∈
U
:
L
x
-
λ
C
x
+
ε
ϕ
(
x
)
∋
0
}
is compact. Indeed, arguing as above, for any sequence (xn) in C(L-λC,U,-εϕ) we can find a relabeled subsequence such that xn→x for some x∈U¯. It remains to prove that x∈U. Otherwise, we would have x∈∂U, hence (ε,λ)∈Γ, a contradiction.
So, the integer-valued map
(
ε
,
λ
)
↦
deg
(
L
-
λ
C
,
U
,
-
ε
ϕ
)
is well defined in the relatively open set ℛ∖Γ (Definition 5.7), and constant on any connected component of ℛ∖Γ by homotopy invariance (Proposition 5.8 iii).
By the choice of b>0, both operators L±bC are invertible. Hence, (0,±b)∈ℛ∖Γ (recall that 0∉∂U). We claim that
(6.2)
deg
(
L
+
b
C
,
U
,
0
)
≠
deg
(
L
-
b
C
,
U
,
0
)
.
Indeed, let L+bC∈Φ0(E,F) be naturally oriented; then
sign
(
L
+
b
C
)
=
1
(Definition 3.4 (ii), iii). We fix a non-trivial, finite-dimensional subspace F1⊂F and set E1=(L+bC)-1(F1), then we orient F1 and E1 so that E1 is the oriented (L+bC)-preimage of F1. With such an orientation of the involved spaces and maps, recalling Definitions 5.7 and 5.4, we have
deg
(
L
+
b
C
,
U
,
0
)
=
deg
B
(
(
L
+
b
C
)
|
E
1
,
U
∩
E
1
,
0
)
=
1
.
The second of the above two equalities is consequence of classical properties in Brouwer degree (see e.g. [25, Chapter 5, Section 1]). Since dim(kerL) is odd, by Lemma 6.2(ii) we have
sign
(
L
-
b
C
)
=
-
1
,
which, repeating the construction above with the same orientations, leads to
deg
(
L
-
b
C
,
U
,
0
)
=
-
1
.
Similar arguments can be developed if different orientations are chosen, so (6.2) holds in any case.
By (6.2), we deduce that (0,±b) lie in different connected components of ℛ∖Γ. By reducing further a>0 if necessary, we may assume that (ε,±b) lie in different connected components of ℛ∖Γ, for all ε∈[-a,a] (the situation is depicted in Figure 1). So, for all ε∈[-a,a] we can find λ∈[-b,b] such that (ε,λ)∈Γ, which concludes the proof. ∎
Proposition 6.3 is the main tool for proving persistence of the eigenpairs under a set-valued perturbation, with the additional assumption that the set Ω0:=Ω¯∩kerL is compact (note that Ω0≠∅ as 0∈Ω). We begin with eigenvalues:
Theorem 6.4.
Let dim(kerL) be odd, let (1.3) hold, and let Ω0:=Ω¯∩kerL be non-empty and compact. Then, for all c>0 small enough, there exist a,b>0 such that the set-valued map Γ:[-a,a]→2[-b,b] defined by
Γ
(
ε
)
=
{
λ
∈
[
-
b
,
b
]
:
(
x
,
ε
,
λ
)
∈
𝒮
for some
x
∈
∂
Ω
∩
B
c
(
𝒮
0
)
}
has the following properties:
Γ
(
ε
)
≠
∅
for all
ε
∈
[
-
a
,
a
]
,
Proof.
Since the set Ω0 is compact, we can find a bounded open neighborhood W⊂E of Ω0 such that ϕ|U¯ is compact, where we have set U=W∩Ω. Clearly, U is a bounded open set such that 0∈U. Then we can apply Proposition 6.3 and thus find a rectangle
ℛ
=
[
-
a
,
a
]
×
[
-
b
,
b
]
(
a
,
b
>
0
)
such that for all ε∈[-a,a] there exist λ∈[-b,b], x∈∂U such that
L
x
-
λ
C
x
+
ε
ϕ
(
x
)
∋
0
.
Besides, let c>0 be small enough that Bc(𝒮0)⊂W (recall that 𝒮0=∂Ω∩kerL). We define 𝒦⊂E×ℛ as in (6.1). As in the proof of Proposition 6.3, we see that 𝒦 is compact. We define a set-valued map ψ:ℛ→2F by setting
ψ
(
ε
,
λ
)
=
{
x
∈
∂
U
:
(
x
,
ε
,
λ
)
∈
𝒦
}
.
We claim that
ψ
(
0
,
0
)
=
𝒮
0
⊂
B
c
(
𝒮
0
)
.
Indeed, for all x∈𝒮0 we have x∈Ω0⊂W, which along with x∈∂Ω implies x∈∂U, while (x,0,0)∈𝒮, so (x,0,0)∈𝒦. Conversely, if x∈ψ(0,0), then x∈∂U⊆∂Ω∪∂W, while x∈Ω0⊂W, so we deduce x∈∂Ω and since (x,0,0)∈𝒦 we have x∈𝒮0.
Plus, the set graphψ⊂ℛ×E is obtained as a continuous image of 𝒦 (by a swap of coordinates) and hence is compact. So, recalling Remark 4.2, ψ is u.s.c.
Therefore by reducing a,b>0 if necessary we have for all (ε,λ)∈ℛ,
(6.3)
ψ
(
ε
,
λ
)
⊂
B
c
(
𝒮
0
)
.
Now we can prove both assertions. Fix ε∈[-a,a]. By Proposition 6.3 there exist λ∈[-b,b] and x∈∂U such that x∈ψ(ε,λ), so by (6.3) we have x∈Bc(𝒮0). Then x∈∂U∩W⊂∂Ω, so (x,ε,λ)∈𝒮. Thus λ∈Γ(ε), which proves (i).
To prove (ii), we just need to note that
graph
Γ
=
{
(
ε
,
λ
)
∈
ℛ
:
(
x
,
ε
,
λ
)
∈
𝒦
for some
x
∈
∂
Ω
∩
B
c
(
𝒮
0
)
}
is but the projection of 𝒦 onto ℛ, hence compact. As above, Γ:[-a,a]→2[-b,b] is u.s.c. ∎
A similar persistence result holds for the eigenvectors:
Theorem 6.5.
Let dim(kerL) be odd, let (1.3) hold, and let Ω0 be compact. Then, for all c>0 small enough, there exist a,b>0 such that the set-valued map Σ:[-a,a]→2E defined by
Σ
(
ε
)
=
{
x
∈
∂
Ω
∩
B
c
(
𝒮
0
)
:
(
x
,
ε
,
λ
)
∈
𝒮
for some
λ
∈
[
-
b
,
b
]
}
has the following properties:
Σ
(
ε
)
≠
∅
for all
ε
∈
[
-
a
,
a
]
,
Proof.
As in the proof of Theorem 6.4, for all c>0 small enough we find a rectangle ℛ=[-a,a]×[-b,b] (a,b>0) and an open neighborhood W⊂E of Ω0 such that, setting U=W∩Ω, the set 𝒦 defined by (6.1) is compact, and in addition x∈Bc(𝒮0) whenever (x,ε,λ)∈𝒦 (see (6.3)).
In particular, for all (x,ε,λ)∈𝒦 we have x∈Σ(ε). Then Proposition 6.3 implies (i). Besides, since
graph
Σ
=
{
(
ε
,
x
)
∈
[
-
a
,
a
]
×
(
∂
Ω
∩
B
c
(
𝒮
0
)
)
:
(
x
,
ε
,
λ
)
∈
𝒦
for some
λ
∈
[
-
b
,
b
]
}
is the projection of 𝒦 onto [-a,a]×E, hence compact, we also deduce (ii) (recall Remark 4.2). ∎
As a consequence, we prove that the set 𝒮0 contains at least one bifurcation point (Definition 6.1):
Theorem 6.6.
Let dim(kerL) be odd, let (1.3) hold, and let Ω0 be compact. Then problem (1.1) has at least one bifurcation point.
Proof.
We argue by contradiction: assume that 𝒮0 contains no bifurcation points, i.e., for all x∈𝒮0 there exists an open neighborhood 𝒰x⊂E×ℝ×ℝ of (x,0,0) such that for all (x,ε,λ)∈𝒮∩𝒰x we have (ε,λ)=(0,0). The family (𝒰x)x∈𝒮0 is an open covering of the compact set 𝒮0×{(0,0)} in E×ℝ×ℝ, so we can find a finite sub-covering, which we relabel as (𝒰i)i=1m.
Let a,b,c>0 be such that
B
c
(
𝒮
0
)
×
ℛ
⊂
⋃
i
=
1
m
𝒰
i
,
where as usual ℛ=[-a,a]×[-b,b]. Thus, we have
𝒮
∩
(
B
c
(
𝒮
0
)
×
ℛ
)
=
𝒮
0
×
{
(
0
,
0
)
}
(i.e., there are no solutions in Bc(𝒮0)×ℛ except the trivial ones). By reducing a,b,c>0 if necessary, Theorem 6.5 applies. So, for all ε∈[-a,a]∖{0} there exist x∈∂Ω∩Bc(𝒮0), λ∈[-b,b] such that (x,ε,λ)∈𝒮, a contradiction. ∎
Remark 6.7.
Since kerL has finite dimension, compactness of Ω0 (which is assumed in the statements of the last theorems) is clearly verified as long as Ω is bounded. On the other hand, trivial examples in Euclidean spaces show that, if Ω is unbounded, then Ω0 may fail to be compact. We want to present a special type of (possibly unbounded) domains which satisfy our assumption: let γ:E→ℝ be a continuous norm (not necessarily coinciding with the norm ∥⋅∥ of E) and set
Ω
=
{
x
∈
E
:
γ
(
x
)
<
1
}
.
Let (xn) be a sequence in Ω0=Ω¯∩kerL. Without loss of generality we may assume xn≠0 for all n∈ℕ. Setting yn=xn/∥xn∥, we define a bounded sequence (yn) in the finite-dimensional space kerL, so passing to a subsequence if necessary we have yn→y, ∥y∥=1. By continuity, γ(yn)→γ(y)>0, so
∥
x
n
∥
=
γ
(
x
n
)
γ
(
y
n
)
⩽
1
γ
(
y
n
)
is bounded. Passing to a further subsequence, we have ∥xn∥→μ⩾0, and hence xn→μy. So, Ω0 is compact.
We conclude this section by presenting a special case of Theorem 6.6:
Corollary 6.8.
Let dim(kerL) be odd, let (1.3) hold, let γ∈C(E,R) be a norm, and let
Ω
=
{
x
∈
E
:
γ
(
x
)
<
1
}
.
Then problem (1.1) has at least one bifurcation point.
Proof.
Clearly, Ω⊂E is an open set such that 0∈Ω and ∂Ω=γ-1(1). In addition, by Remark 6.7, the set Ω0=Ω¯∩kerL is compact. Thus, we can apply Theorem 6.6 and conclude. ∎
7 Applications to Differential Inclusions
We devote this final section to an application of our abstract results in the field of differential inclusions. We consider the problem stated in the Introduction:
(7.1)
{
u
′′
+
u
′
-
λ
u
+
ε
Φ
(
u
)
∋
0
in
[
0
,
1
]
u
′
(
0
)
=
u
′
(
1
)
=
0
∥
u
∥
1
=
1
.
We recall that Φ(u):[0,1]→2ℝ is a set-valued mapping depending on u, to be defined later, while ε,λ∈ℝ are parameters and ∥⋅∥1 is the usual L1-norm on [0,1]. Problem (7.1) falls into the general pattern (1.1), with the following definitions. Set
E
=
{
u
∈
C
2
(
[
0
,
1
]
,
ℝ
)
:
u
′
(
0
)
=
u
′
(
1
)
=
0
}
,
F
=
C
0
(
[
0
,
1
]
,
ℝ
)
,
endowed with the usual norms. Then E, F are real Banach spaces, in particular E is a 2-codimensional subspace of C2([0,1],ℝ). Set for all u∈E,
L
u
=
u
′′
+
u
′
,
C
u
=
u
.
Then L,C∈ℒ(E,F). Besides, L∈Φ0(E,F) as the composition of the embedding E↪C2([0,1],ℝ) (which is Fredholm of index -2) and the linear differential operator u↦u′′+u′ (which is Fredholm of index 2 between C2([0,1],ℝ) and F). In order to check the transversality condition (1.3), we need more detailed information about L. It is easily seen that kerL is the space of constant functions, i.e.,
ker
L
=
ℝ
,
in particular dim(kerL)=1 (odd). In addition, we have
im
L
=
{
f
∈
F
:
∫
0
1
f
(
t
)
e
t
𝑑
t
=
0
}
.
Indeed, for all f∈imL there exists u∈E such that u′′+u′=f, so integrating by parts we deduce
∫
0
1
f
(
t
)
e
t
𝑑
t
=
∫
0
1
u
′′
(
t
)
e
t
𝑑
t
+
∫
0
1
u
′
(
t
)
e
t
𝑑
t
=
0
.
Besides, since L∈Φ0(E,F), we have dim(cokerL)=1 (Definition 3.1), so the condition above is also sufficient. Now we prove (1.3), or equivalently
im
L
⊕
C
(
ker
L
)
=
F
.
Indeed, C(kerL)=ℝ is not contained in the 1-codimensional subspace imL, hence it is a (direct) complement for it in F.
The integral constraint rephrases as u∈∂Ω, where we have set
Ω
=
{
u
∈
E
:
∥
u
∥
1
<
1
}
.
Since ∥⋅∥1 is a continuous norm on E, Ω is an (unbounded) open set such that Ω0=Ω¯∩kerL is compact (Remark 6.7). Besides, from the characterization of kerL we have 𝒮0={±1}.
The construction of Φ requires some care. We are going to consider a set-valued map ϕ∈CJ(Ω¯,F), and then set for all u∈Ω¯, t∈[0,1],
Φ
(
u
)
(
t
)
=
{
w
(
t
)
:
w
∈
ϕ
(
u
)
}
(this can be seen as a set-valued superposition operator). Details will be given in Examples 7.2, 7.3, and 7.7 below. We can now apply our abstract results to prove existence of a bifurcation point:
Theorem 7.1.
Let E, F, L, C, Ω, and Φ be as above, being ϕ∈CJ(Ω¯,F) locally compact. Then there exist sequences (un) in ∂Ω, (εn) in R∖{0}, (λn) in R such that (un,εn,λn) is a solution of (7.1) for all n∈N, and
u
n
→
±
1
,
ε
n
→
0
,
λ
n
→
0
.
Proof.
By Remark 6.7 and Corollary 6.6, problem (1.1) has at least one bifurcation point in 𝒮0, that is, either the constant 1 or -1. So, we can find a sequence (un,εn,λn) if non-trivial solutions of (1.1) (more precisely, with εn≠0) converging to either (1,0,0) or (-1,0,0) in E×ℝ×ℝ. By the definition of Φ, for all n∈ℕ and all t∈[0,1] we have
u
n
′′
(
t
)
+
u
n
′
(
t
)
-
λ
n
u
n
(
t
)
+
ε
n
Φ
(
u
n
)
(
t
)
∋
0
,
so (un,εn,λn) solves (7.1). ∎
We present three examples of locally compact CJ-maps ϕ:Ω¯→2F. The first and second examples are quite easy, ϕ being defined by means of finite-dimensional reduction.
Example 7.2.
We define a set-valued map ϕ:Ω¯→2F whose values consist of piecewise affine functions along a decomposition of [0,1], satisfying some bounds at the nodal points. Fix m∈ℕ, points 0=t0<t1<⋯<tm=1, and ρ∈(0,1). For all u∈Ω¯ we define ϕ(u) as the set of all functions w∈F such that
w is affine in [tj-1,tj], j=1,…,m,
u
(
t
j
)
-
ρ
⩽
w
(
t
j
)
⩽
u
(
t
j
)
+
ρ
, j=0,…,m.
We first prove that ϕ has convex values. For any u∈Ω¯, w0,w1∈ϕ(u), and μ∈[0,1], by (a) the function w=(1-μ)w0+μw1 is affine in any interval [tj-1,tj] (j=1,…,m), while (b) implies
u
(
t
j
)
-
ρ
⩽
w
(
t
j
)
⩽
u
(
t
j
)
+
ρ
(
j
=
0
,
…
,
m
)
.
Then we prove that ϕ has compact values. Let u∈Ω¯, and let (wn) be a sequence in ϕ(u). By (a) we can find α1,…,αm>0 such that |wn′(t)|⩽αj for all t∈(tj-1,tj), j=1,…,m, and n∈ℕ. So the sequence (wn) is uniformly bounded and equi-continuous (by (b)), hence by Ascoli’s theorem we can pass to a subsequence such that wn→w in F. Due to uniform convergence, w is piecewise affine and satisfies the bounds at tj (j=0,…,m), so w∈ϕ(u). Thus, ϕ(u) is compact.
In addition, the set-valued map ϕ is locally compact. Indeed, let (un) be a bounded sequence in Ω¯ and let (wn) be a sequence in F such that wn∈ϕ(un) for all n∈ℕ. Recalling that (un′) is uniformly bounded, we can argue as above to find a relabeled subsequence wn such that wn→w in F. Thus, ϕ(un)¯ is compact.
We prove finally that graphϕ is closed in Ω¯×F. Indeed, let (un,wn) be a sequence in Ω¯×F such that wn∈ϕ(un) for all n∈ℕ, and (un,wn)→(u,w). Then w∈ϕ(u). By [22, Proposition 4.15], ϕ is u.s.c. We conclude that ϕ∈CJ(Ω¯,F) and is locally compact.
Example 7.3.
In this second example, ϕ(u) depends on u in a single-valued sense, but is multiplied by an interval depending on the mean value of u (non-local dependence). Fix f∈C0(ℝ,ℝ), α,β:ℝ→ℝ such that α is lower semicontinuous, β is upper semicontinuous, and α(s)⩽β(s) for all s∈ℝ. For all u∈Ω¯ set
u
¯
=
∫
0
1
u
(
τ
)
𝑑
τ
.
We define ϕ(u) as the set of all functions w∈F of the form
w
(
t
)
=
c
f
(
u
(
t
)
)
,
α
(
u
¯
)
⩽
c
⩽
β
(
u
¯
)
.
Obviously, ϕ:Ω¯→2F has convex values.
We prove that ϕ has compact values. Let u∈Ω¯, (wn) be a sequence in ϕ(u). Then, for all n∈ℕ, there exists cn∈[α(u¯),β(u¯)] such that wn=cnf(u). The sequence (cn) is bounded, so passing to a subsequence we have cn→c for some c∈ℝ. Set w=cf(u), then clearly wn→w in F and w∈ϕ(u). Thus, ϕ(u) is compact.
The map ϕ is locally compact. Indeed, let (un) be a bounded sequence in Ω¯ and let (wn) be a sequence in F such that wn∈ϕ(un) for all n∈ℕ. Then for all n∈ℕ we can find cn∈[α(u¯n),β(u¯n)] such that wn=cnf(un). Since (un) is uniformly bounded and equi-continuous, passing to a subsequence we have un→u uniformly in [0,1] (note that u∉E in general). Hence, u¯n→u¯. So (cn) turns out to be bounded, and up to a subsequence cn→c. Passing to the limit, due to the properties of α and β, we have
α
(
u
¯
)
⩽
c
⩽
β
(
u
¯
)
.
So, setting w=cf(u), we deduce wn→w in F. Thus, ϕ(un)¯ is compact.
Alternatively, we can prove that graphϕ is closed in Ω¯×F. Indeed, let (un,wn) be a sequence in Ω¯×F such that wn∈ϕ(un) for all n∈ℕ, and (un,wn)→(u,w). For all n∈ℕ we find cn∈[α(u¯n),β(u¯n)] such that wn=cnf(un). Then u¯n→u¯, and f(un)→f(u) uniformly in [0,1]. We prove now that (cn) converges, indeed avoiding trivial cases we may assume that f(u(t))≠0 at some t∈[0,1], then
lim
n
c
n
=
w
n
(
t
)
f
(
u
n
(
t
)
)
=
w
(
t
)
f
(
u
(
t
)
)
=
c
,
with c∈[α(u¯),β(u¯)]. So w∈ϕ(u). By [22, Proposition 4.15] again, ϕ is u.s.c. We conclude that ϕ∈CJ(Ω¯,F) and is locally compact.
The last example is more sophisticated, since in the construction of ϕ we preserve the infinite dimension, and we apply some classical results from functional analysis to prove all required compactness properties. We recall such results, starting from a weak notion of compactness in L1 (see [23, Definition 2.94] and [26, Definition 4.2.1]):
Definition 7.4.
A sequence (vn) in L1([0,1],ℝ) is said to be semicompact if
it is integrably bounded, i.e., if there exists g∈L1([0,1],ℝ) such that |vn(t)|⩽g(t) for a.e. t∈[0,1] and all n∈ℕ,
the image sequence (vn(t)) is relatively compact in ℝ for a.e. t∈[0,1].
The following result follows from the Dunford–Pettis Theorem (see also [26, Proposition 4.21]):
Proposition 7.5.
Every semicompact sequence is weakly compact in L1(0,1).
We also recall Mazur’s well-known theorem (see e.g. [18]):
Theorem 7.6.
Let E be a normed space, and let (xn) be a sequence in E weakly converging to x. Then there exists a sequence of convex linear combinations
y
n
=
∑
k
=
1
n
a
n
,
k
x
k
,
a
n
,
k
∈
(
0
,
1
]
,
such that yn→x (strongly) in E.
We can now present our last example:
Example 7.7.
Let α,β:[0,1]×ℝ→ℝ be continuous functions such that
α
(
t
,
s
)
⩽
β
(
t
,
s
)
for all
(
t
,
s
)
∈
[
0
,
1
]
×
ℝ
,
and for all u∈Ω¯ define ϕ(u) as the set of all functions w∈F for which there exists v∈L1(0,1) such that for all t∈[0,1],
w
(
t
)
=
∫
0
t
v
(
τ
)
𝑑
τ
,
and for a.e. t∈[0,1],
v
(
t
)
∈
[
α
(
t
,
u
(
t
)
)
,
β
(
t
,
u
(
t
)
)
]
.
In short, we might define ϕ(u) as a set-valued integral in the sense of Aumann
ϕ
(
u
)
(
t
)
=
∫
0
t
[
α
(
τ
,
u
(
τ
)
)
,
β
(
τ
,
u
(
τ
)
)
]
𝑑
τ
(see [9, 26]). Clearly, for all u∈Ω¯, any w∈ϕ(u) is absolutely continuous and hence a.e. differentiable in [0,1] with derivative v. We first prove that ϕ has convex values. Let u∈Ω¯, w0,w1∈ϕ(u), and μ∈[0,1]. There exist v0,v1∈L1(0,1) such that
w
i
(
t
)
=
∫
0
t
v
i
(
τ
)
𝑑
τ
,
v
i
(
t
)
∈
[
α
(
t
,
u
(
t
)
)
,
β
(
t
,
u
(
t
)
)
]
(a.e.)
.
Set w=(1-μ)w0+μw1 and v=(1-μ)v0+μv1; then we have in [0,1]
w
(
t
)
=
∫
0
t
v
(
τ
)
𝑑
τ
,
v
(
t
)
∈
[
α
(
t
,
u
(
t
)
)
,
β
(
t
,
u
(
t
)
)
]
(a.e.)
,
which implies w∈ϕ(u).
We prove now that ϕ has compact values (this is not immediate and will require several steps). Let u∈Ω¯, and let (wn) be a sequence in ϕ(u); then there exists a sequence (vn) in L1(0,1) such that for all n∈ℕ,
w
n
(
t
)
=
∫
0
t
v
n
(
τ
)
𝑑
τ
,
v
n
(
t
)
∈
[
α
(
t
,
u
(
t
)
)
,
β
(
t
,
u
(
t
)
)
]
(a.e.)
.
Clearly, (wn) is bounded in F. Also, since (vn) is essentially bounded, (wn) turns out to be equi-absolutely continuous. By Ascoli’s theorem, passing if necessary to a subsequence, we have wn→w in F and w is the primitive of some v∈L1(0,1), i.e., for all t∈[0,1] we have
(7.2)
w
(
t
)
=
∫
0
t
v
(
τ
)
𝑑
τ
.
On the other side, (vn) is a semicompact sequence in L1(0,1) (Definition 7.4), so by Proposition 7.5 we can pass to a further subsequence and have (vn) weakly converging in L1(0,1) to some v^∈L1(0,1). For all t∈[0,1], the linear functional
f
↦
∫
0
t
f
(
τ
)
𝑑
τ
is bounded in L1(0,1), so weak convergence is enough to deduce that for all t∈[0,1],
w
n
(
t
)
=
∫
0
t
v
n
(
τ
)
𝑑
τ
→
∫
0
t
v
^
(
τ
)
𝑑
τ
.
Comparing to (7.2), we see that v=v^ in L1(0,1). So (vn) converges weakly to v in L1(0,1). By Theorem 7.6, we can find a sequence (v~n) of convex linear combinations of (vn) such that v~n→v in L1(0,1) (strongly). Clearly, for all n∈ℕ and a.e. t∈[0,1] we have
v
~
n
(
t
)
∈
[
α
(
t
,
u
(
t
)
)
,
β
(
t
,
u
(
t
)
)
]
,
so we can pass to the limit and deduce the same property for v. So, by (7.2), we have w∈ϕ(u). Thus, ϕ(u) is compact. By similar arguments, we prove that ϕ is locally compact.
Upper semicontinuity of ϕ can be proved as in the previous cases, applying [22, Proposition 4.1.5]. Nevertheless, in order to give the reader a more complete picture, we present a direct proof based on Definition 4.1.
Let V⊂F be open, and let u¯∈ϕ+(V). We claim that there exists a neighborhood of u¯ contained in ϕ+(V). Indeed, by compactness of ϕ(u¯), there exist w1,…,wn∈ϕ(u¯) and ε1,…,εn>0 such that
B
ε
i
(
w
i
)
⊆
V
(i=1,…,n),
ϕ
(
u
¯
)
⊂
⋃
i
=
1
n
B
ε
i
/
2
(
w
i
)
.
Consider now the following compact subset of ℝ2:
C
=
{
(
t
,
y
)
∈
ℝ
2
:
t
∈
[
0
,
1
]
,
α
(
t
,
u
¯
(
t
)
)
⩽
y
⩽
β
(
t
,
u
¯
(
t
)
)
}
,
and let A be the open ball of [0,1]×ℝ centered at C with radius 1. Then set
ε
¯
:=
min
{
ε
1
,
…
,
ε
n
}
2
.
Since α, β are uniformly continuous in A¯, we can find δ∈(0,1) such that
(7.3)
max
{
|
α
(
t
,
y
)
-
α
(
t
,
z
)
|
,
|
β
(
t
,
y
)
-
β
(
t
,
z
)
|
}
<
ε
¯
for all
(
t
,
y
)
,
(
t
,
z
)
∈
A
¯
,
dist
(
(
t
,
y
)
,
(
t
,
z
)
)
<
δ
.
We claim that ϕ(Bδ(u¯))⊆V. Indeed, fix u∈Bδ(u¯). Clearly, we have |u(t)-u¯(t)|<δ for all t∈[0,1]. Take now w∈ϕ(u), which can be written as
w
(
t
)
=
∫
0
t
v
(
τ
)
𝑑
τ
with v∈L1(0,1) satisfying for a.e. t∈[0,1],
α
(
t
,
u
(
t
)
)
⩽
v
(
t
)
⩽
β
(
t
,
u
(
t
)
)
.
If for a.e. t∈[0,1],
α
(
t
,
u
¯
(
t
)
)
⩽
v
(
t
)
⩽
β
(
t
,
u
¯
(
t
)
)
,
then w∈ϕ(u¯)⊆V and we are done. Otherwise, consider the truncated map v¯:[0,1]→ℝ defined by
v
¯
(
t
)
=
{
α
(
t
,
u
¯
(
t
)
)
if
v
(
t
)
<
α
(
t
,
u
¯
(
t
)
)
,
v
(
t
)
if
α
(
t
,
u
¯
(
t
)
)
⩽
v
(
t
)
⩽
β
(
t
,
u
¯
(
t
)
)
,
β
(
t
,
u
¯
(
t
)
)
if
v
(
t
)
>
β
(
t
,
u
¯
(
t
)
)
,
which is an L1-function (see e.g. [31]), and denote
w
¯
(
t
)
=
∫
0
t
v
¯
(
τ
)
𝑑
τ
,
so w¯∈ϕ(u¯). By the bounds above we have for all t∈[0,1] that (t,u(t)),(t,u¯(t))∈A¯ with
dist
(
(
t
,
u
(
t
)
)
,
(
t
,
u
¯
(
t
)
)
)
<
δ
,
so by (7.3) we have |v(t)-v¯(t)|<ε¯ for a.e. t∈[0,1]. This in turn implies ∥w-w¯∥∞<ε¯. By (ii), we can find i∈{1,…,n} such that w¯∈Bεi/2(wi). So, recalling the definition of ε¯, we have
∥
w
-
w
i
∥
∞
⩽
∥
w
-
w
¯
∥
∞
+
∥
w
¯
-
w
i
∥
∞
<
ε
i
,
hence by (i)w∈V. Thus, ϕ(Bδ(u¯))⊆V and ϕ turns out to be u.s.c. In conclusion, ϕ∈CJ(Ω¯,F) and it is locally compact.
Remark 7.8.
Comparing Definition 6.1 and problem (7.1), one may be left in doubt that the non-trivial solutions ensured by Theorem 7.1 might be triples (±1,ε,0) with ε≠0 (quite trivial in fact). But this case may only occur if 0∈ϕ(±1). Easy computations show that in Example 7.2 we have 0∉ϕ(±1), due to the choice ρ∈(0,1). Similarly, in Example 7.3 it is enough to choose functions f, α, and β to be positive in order to have 0∉ϕ(±1), thus avoiding such difficulty. Also in Example 7.7, we can easily find α, β such that 0∉ϕ(±1).
Acknowledgements
This paper has gone through a long process before being concluded, and several people deserve to be acknowledged for their contributions: we thank L. Gasiński for help on Example 7.2, I. Benedetti for Example 7.3, P. Zecca and V. Obukhovskii for Example 7.7, and S. Mosconi for the figure. We also thank O. Rio Branco de Oliveira for fruitful discussions.
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und
Antonio Iannizzotto
