Existence and Multiplicities of Solutions for Asymptotically Linear Ordinary Differential Equations Satisfying Sturm-Liouville BVPs with Resonance ()
1. Introduction
In this paper, we investigate the nontrivial solutions of asymptotically linear ordinary differential equations satisfying Sturm-Liouville BVPs with resonance. Various boundary value problems of asymptotically linear ordinary differential equations have been studied before. Most of them are gotten by the topological degree theory. There are also some papers about resonant problem. But the asymptotically linear ordinary differential equations with resonance aren’t concerned ago. Here, we concern asymptotically linear ordinary differential equations satisfying both Sturm-Liouville boundary value and resonance. We solve the problem to get Theorem 1.1 in the following section.
Now, we consider solutions of the following Sturm-Liouville boundary value problem:
(1.1)
(1.2)
(1.3)
where
. In this paper,
denotes the derivative with respect to x. Our main result is the following theorem.
Theorem 1.1 Assume that
in [1], i.e.
and (1.2)-(1.3) has one nontrivial solution
,
,
. f satisfies the following two conditions:
(H1)
as
, r is a constant,
,
,
,
;
(H2)
,
, where
;
For the sake of convenience, we denote
,
(H3)
,
where
,
, and
is a nontrivial solution of
and (1.2)-(1.3),
. Then (1.1-1.3) has at least one nontrivial solution. Moreover, if we assume
(H4)
.
Then (1.1-1.3) has at least two nontrivial solutions.
In this paper, for any
,
and
denote its index and nullity of the associated linear ordinary differential equation (see [2] [3] for reference). In Section 2, we will briefly recall the index and its properties. For the readers’ convenience, we give an example: Assume
is a constant,
and
. Then
and
In [2], an index for second order linear Hamiltonian systems was defined. And in [3], an index for more general linear self-adjoint operator equations was developed. In [4] [5] [6] [7], by Conley, Zehnder and Long, an index theory for sympletic paths was defined. More applications about these index theories can be found in [8] - [13]. As in [11], throughout this paper, for
, we write
, if
, for
; we write
, if
, and
holds on a subset of
with nonzero measure.
It is well known [14] that under non-resonant conditions
the existence of solutions of a second order nonlinear ordinary differential equation. Such conditions are called nonresonant. Resonant conditions in [15] [16] :
are not enough for existence solutions of (1.1-1.2). An additional condition called the (LL) condition like (H3) is usually needed. For resonant conditions, we refer to [15] [16] [17]. These three papers [14] [15] [16] are about existence of solutions.
In [18], under resonance conditions, periodic solutions of nonlinear second order ordinary differential equations are considered. Second order Hamiltonian systems satisfying Sturm-Liouville boundary vale with the nonresanonce are considered in [3]. First order asymptotical linear Hamiltonian systems satisfying Sturm-Liouville boundary vale with the nonresanonce are studied in [19]. In [20] [21],
, the existence of solutions of (1.1-1.3) is investigated.
In this paper, we study the existence of equations with resonance conditions. In order to prove our theorem, we construct the corresponding functional:
(1.4)
where
,
as
or
π, and E will be described in Section 2. This functional
is continuous differentiable on E, and any critical point of
corresponds to a solution of (1.1)-(1.3).
In Section 3, we will give proofs by the Morse theory following [11] [17].
2. Index Theory for Linear Duffing Equations
For any
, consider the following equation:
(2.1)
where
. Define a Hilbert space
. Here
as
;
as
;
as
and
. With norm
and a bilinear form as
as follows
(2.2)
From proposition 2.1.1 and 2.3.3 in [3], we have the following properties.
Proposition 2.1 For any
,
1) The E can be divided into three parts:
such that
is positive definite, null and negative definite on
and
respectively. Furthermore,
and
are finitely dimensional. We call
and
the nullity and index respectively.
2)
.
3)
is the dimension of the solution subspace of (2.1-2.3), and
.
4) If
, then
and
; if
, then
.
5) There exists
such that
Remarks: 1) The notation
means that the space E is the direct sum of some subspaces.
2) By 4), we can see the index has monotonicity.
1) Let
,
, and
. Then (2.1) has a nontrivial solution
. So
,
. If
,
, and
, then (2.1) has a nontrivial solution
,
. So
,
. If
,
, and
, then (2.1) has a nontrivial solution
,
. So by Proposition 2.1 (3),
,
.
The following lemmas are useful for us to prove the results.
Lemma 2.2 The norm
, for any
, where
is a positive constant.
Lemma 2.3 If (H1) holds, then we have that
, where
is given in Theorem 1.1.
Proof By Proposition 2.1 (1) and conditions
, we have that
(2.3)
By Proposition 2.1 (1) and (3), we know that with respect to
, the following decomposition holds,
(2.4)
Since
is one dimensional space, we can assume that
is a base of
, i.e.
. So for any
, we have
where
,
, and
is a constant. For
, we have the decomposition
, where
and
is a constant. It is obvious that
. Indeed, if
, then
. By definition of
, we will have a contradiction that
and
. So we obtain
We have proved that
. It is also obvious that
. In fact that if
, we have that on the one hand for
. Then there exists a
such that
and
on the other hand for
,
By Proposition 2.1 (1) and (2.2), we have
. This is a contradiction. So the proof is completed.
Remark: For
, we can define
.
In order to prove Theorem 1.1, we need some lemmas. Let X be a Hilbert space and
. As in [17], let
,
. For an isolated critical point
, the critical group is defined by
for
, where U is a neighborhood of
such that
and
.
When
and
, we have
is a self-adjoint operator. We call the dimension of negative space corresponding to the spectral decomposing the Morse index of p and denote it by
, and denote by
. If
has a bounded inverse we say that p is nondegenerate.
From Theorem 3.1 in Chapter 3, Theorem 5.1, 5.2, Corollary 5.2 in Chapter 5 in [17], one can prove the following lemma.
Lemma 2.4. Assume
satisfies the (PS) condition,
, where
is the zero vector in X and
is the zero vector in
which is the dual space of X, and there is a positive integer
such that
and
for
some regular
, here
. Then
has a critical point
with
. Moreover, if
is a nondegenerate critical point, and
, then
has another critical point
.
The following lemma is also useful for us to prove the main result.
Lemma 2.5 (Fatou’s lemma). Given a measure space
and a set
, let
be a sequence of
-measurable non-negative functions
, where
denotes the σ-algebra of Borel sets on
. Define the function
by setting
for every
. Then f is
-measurable, and
Remark The integrals may be finite or infinite.
3. Proof of the Main Result
The proof of Theorem 1.1 will depend on the following lemma.
Lemma 3.1 Under (H1),(H2), and (H3), the functional
satisfies the (PS) condition.
Proof For
, and
is bounded, we shall find a convergent subsequence in E. By (1.3), for
, we have
(3.1)
Next, we will prove
is bounded. Indeed, it suffices to prove that
is bounded. By a contradiction, we assume that
, as
.
Defining
(3.2)
from (H1), (H2) and
is continuous, we have
(3.3)
where
is a constant. Then we get
(3.4)
By (3.1), it follows that
(3.5)
Assuming
, by (3.4), and multiplying
on both sides of (3.5), we can get that
(3.6)
Furthermore, we add
on two sides of (3.6) to obtain that
So, by
and (3.3) we have
where
,
and
are constants. So
is bounded. Then
has a convergent subsequence. Without loss of generality, we also denoted by
. Then
in E and
in
. By inequality
, we have
in
. Then taking the limits on both sides of (3.6), we have, for any
,
(3.7)
From (3.7) and [3], we have that
is a solution of the following problem:
(3.8)
What’s more, since
, we have
. In fact, by the meaning of the notation “<” and “
”, on the one hand, if
, then
. Therefore, by the definition
, this means that (3.8) only has a trivial solution. In fact, by
, we obtain
. So (3.8) has a nontrivial solution. This is a contradiction. On the other hand, if
, then
holds. While
is a nontrivial solution of (3.8), this leads
. So by Proposition 2.1 (4), we get
. This is also a contradiction. From discussion above, we obtain the conclusion that
. So we immediately get
.
Since
, there are two cases about
. One is that
, the other is that
. Without loss of generality, if
, we assume
, and if
, we assume
. Firstly, we discuss the situation that
. If
, i.e.
, then for
such that for
,
holds. Here, we take the
such that
, i.e. when
,
belong to the neighborhood of
,
, for all
. This means
, as
, for all
.
So by
, we can get that for any
,
for
. Then
for all
, as
. By the assumption that
, as
, taking the limits on both sides of (3.1) and letting
, we can obtain
(3.9)
So, by (H3), and (3.9), the following holds
(3.10)
Furthermore, by the Fatou’s Lemma and (3.10), we have
a contradiction to assumption (H3). Hence, if
, this leads to a contradiction. Secondly, in a similar way, we can show that if
, there also be a contradiction. Therefore, the sequence
is a bounded sequence. By the equality
and the fact that
is bounded, we can get that
is bounded in E. Furthermore,
has a weak convergent subsequence in E, without loss of generality, still denoted by
. So we have
in E and
in
. In addition, by (3.1), we also have
(3.11)
At last, we only need to finish the mission that
in E. Indeed, by (3.5), (3.11) and
, we obtain the fact that
The (PS) condition is verified.
After the preliminary work, we can prove Theorem 1.1.
Proof of Theorem 1.1. Since
for any
, by Lemma 2.4, we only need to prove
(3.12)
for
large enough, where
. By Lemma 2.3, we know that E can be split into two subspaces
and
, i.e.
Next, we will take two steps to obtain the proof of (3.12).
First step: For
large enough, we have
(3.13)
where
will be defined later. By assumption, for any
, we have
(3.14)
We will consider the behavior of f in two subintervals of
. One is
, the other is
. Since f is continuous on
, it is obvious that
is bounded on
. So there exists a constant
such that
when
.
By Lemma 2.3, we have a decomposition with respect to
, i.e. there exist
, and
such that
. When
, we have
. Furthermore, we get
(3.15)
So by (15), we have
(3.16)
By (H1), we have
(3.17)
where
is a constant. By (3.14), (3.16), (3.17), Proposition 2.1 (5) and Lemma 2.3, we obtain
where
and
are constants. And hence, there exists
such that
Set
, where
. We want to define a deformation from
to
. Since for every
, f is decreasing along vector field
, we can define the flow
and
, which is the first time that
arrives at
. Then the deformation is
One can verify that
is continuous and satisfy
Then,
is a deformation retract of
. So (3.13) is verified.
Second step: we will prove the following
(3.18)
for any
large enough. In fact, assuming that
, by (H1), we will have two cases: one is
as
, another is
as
. Firstly, we analyze the situation that
as
. Since
and
is a monotonically increasing nonnegative function with respect to
, by (H3), we have
Then
, for all
,
holds. What’s more, since
for all
. So there exists a
, such that
, i.e.
.
So letting
, where
, l is fixed and
, we have
(3.19)
Furthermore, by (3.19), we obtain
So we get
, as
, uniformly in
. Secondly, we analyze the situation that
as
. In a similar way, we also get
, as
, uniformly in
. So we obtain that
Thus, there exist
such that
(3.20)
where
by remark. For the sake of convenience, we set
. Then (3.20) can also be denoted as
We now begin to define a deformation from
to
. For every
, since the flow is defined by
,
is continuous with respect to t,
and
as
, so the time
arriving at
exists uniquely and is defined by
. Since
as
, the continuity of
comes from the implicit function theorem.
Define
then
is continuous, and is a deformation from
to
and
is a strong deformation retract. Hence,
(3.21)
Recall that for any topological spaces
, we have exact sequences
From (3.20), in order to prove
(3.22)
we only to prove
And from (3.21), it suffices to verify
Let
satisfy
We can verify that
is continuous, where
,
,
, and satisfies
for any
. So
. And
for any
. So
. We can also see that
satisfy
,
,
. Then
is a deformation retract of
. This means (3.22) and hence (3.21) holds. Finally from (3.21) we have
Here in the second
we used the deformation
defined by
, and excision property. So (3.18) is proved. And by (3.13) and (3.18), (3.12) is obtained. The proof is completed.
In our theorem, we get one nontrivial solution of Equations (1.1)-(1.3). By adding assumption (H4), we get two nontrivial solutions of Equations (1.1)-(1.3).
4. Conclusion
By index theories established in this paper, and Morse theory, we study the functional corresponding to the problem to obtain more nontrivial solutions of asymptotically linear ordinary differential equations satisfying Sturm-Liouville BVPs with resonance. It’s better than the results obtained by topological degree method.
Acknowledgements
The authors would like to express their sincere thanks to the editors and reviewers for their remarkable comments, suggestions, and ideas that helped to improve this paper.
Funding
This research work was partially supported by the National Science Foundation of China (11501178).