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:
where . In this paper, denotes the derivative with respect to x. Our main result is the following theorem.
Theorem 1.1 Assume that in , 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 ,
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
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   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
In , an index for second order linear Hamiltonian systems was defined. And in , an index for more general linear self-adjoint operator equations was developed. In    , by Conley, Zehnder and Long, an index theory for sympletic paths was defined. More applications about these index theories can be found in  - . As in , throughout this paper, for , we write , if , for ; we write , if , and holds on a subset of with nonzero measure.
It is well known  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   :
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   . These three papers    are about existence of solutions.
In , 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 . First order asymptotical linear Hamiltonian systems satisfying Sturm-Liouville boundary vale with the nonresanonce are studied in . In  , , 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:
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  .
2. Index Theory for Linear Duffing Equations
For any , consider the following equation:
where . Define a Hilbert space . Here as ; as ; as and
. With norm and a bilinear form as as follows
From proposition 2.1.1 and 2.3.3 in , 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.
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
By Proposition 2.1 (1) and (3), we know that with respect to , the following decomposition holds,
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 , 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 , 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
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
Next, we will prove is bounded. Indeed, it suffices to prove that is bounded. By a contradiction, we assume that , as .
from (H1), (H2) and is continuous, we have
where is a constant. Then we get
By (3.1), it follows that
Assuming , by (3.4), and multiplying on both sides of (3.5), we can get that
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 ,
From (3.7) and , we have that is a solution of the following problem:
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
So, by (H3), and (3.9), the following holds
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
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
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
where will be defined later. By assumption, for any , we have
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
So by (15), we have
By (H1), we have
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
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
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
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.
then is continuous, and is a deformation from to and is a strong deformation retract. Hence,
Recall that for any topological spaces , we have exact sequences
From (3.20), in order to prove
we only to prove
And from (3.21), it suffices to verify
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).
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.
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.
This research work was partially supported by the National Science Foundation of China (11501178).
 Dong, Y. (2000) On Equivalent Conditions for the Solvability of Equation Satisfying Linear Boundary Conditions with f Restricted by Linear Growth Conditions. Journal of Mathematical Analysis and Applications, 245, 204-220.
 Mawhin, J. and Willem, M. (1989) Critical Point Theory and Hamiltonian System. In: John, F., Marsden, J.E. and Sirovich, L., Eds., Applied Mathematical Sciences, Springer, Berlin, 153-166.
 Dong, Y. (2010) Index for Linear Selfadjoint Operator Equations and Nontrivial Solutions for Asymptotically Linear Operator Equations. Calculus of Variations and Partial Differential Equations, 38, 75-109.
 Conley, C. and Zehnder, E. (1984) Morse-Type Index Theory for Flows and Periodic Solutions for Hamiltonian Equations. Communications on Pure and Applied Mathematics, 37, 207-253.
 Long, Y. and Zehnder, E. (1990) Morse Theory for Forced Oscillations of Asymptotically Linear Hamiltonian Systems. In: Alberverio, S., et al., Eds., Stochastic Processes, Phiscs and Geometry, World Scientific, Teaneck, 528-563.
 Ekeland, I. (1990) Convexity Methods in Hamiltonian Mechanics. In: Bombieri, E., Feferman, S., et al., Eds., Ergebnisse der Mathematik und ihrer Grenzgebiete, Springer, Berlin, 110-186.
 Long, Y. (2002) Index Theory for Symplectic Paths with Applications. In: Chambert-Loir, A., Lu, J.-H., Ruzhansky, M. and Tschinkel, Y., Eds., Progress in Mathematics, Birkhauser, Basel, 132-173.
 Dong, Y. (2005) Index Theory, Nontrivial Solutions, and Asymptotically Linear Second-Order Hamiltonian Systems. Journal of Differential Equations, 214, 233-255.
 Dong, D. and Long, Y. (1997) The Iteration Formula of Maslov-Type Theory Applications to Nonliear Haniltonian Systems. Transactions of the American Mathematical Society, 349, 2619-2661.
 Fabry, C. (1995) Landesman-Lazer Conditions for Periodic Boundary Value Problems with Asymmetric Nonliearities. Journal of Differential Equations, 116, 405-418.
 Iannacci, R. and Nkashama, M.N. (1987) Unbounded Perturbations of Forced Second Order Ordinary Differential Equations at Resonance. Journal of Differential Equations, 69, 289-309.
 Chang, K.C. (1993) Infinite Dimensinal Morse Theory and Mutiple Solition Problems. In: Brezis, H., Ed., Progress in Nonlinear Differential Equations and Their Applications, Birkhauser, Basel, 179-228.
 Fabry, C. and Fonda, A. (1990) Periodic Solutions of Nonlinear Differential Equations with Double Resonance. Annali di Matematica Pura ed Applicata, 157, 99-116.
 Shan, Y. (2011) Multiple Solutions of Generalized Asymptotical Linear Hamiltonian Systems Satisfying Sturm-Liouville Boundary Conditions. Nonlinear Analysis: Theory, Methods and Applications, 74, 4809-4819.
 Li, K. (2011) Multiple Solutions for an Asymptotically Linear Duffing Equations with Neumann Boundary Value Conditions. Nonlinear Analysis: Theory, Methods and Applications, 74, 2819-2830.
 Li, K., Li, J. and Mao, W. (2013) Multiple Solutions for Asymptotically Linear Duffing Equations with Neumann Boundary Value Conditions (II). Journal of Mathematical Analysis and Applications, 401, 548-553.