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Abstract and Applied Analysis
Volume 2013 (2013), Article ID 610906, 11 pages
Homoclinic Orbits for a Class of Nonperiodic Hamiltonian Systems with Some Twisted Conditions
1Institute of Contemporary Mathematics, School of Mathematics and Information Science, Henan University, Kaifeng 475000, China
2Department of Mathematics, Jiangxi Normal University, Nanchang 330022, China
Received 4 January 2013; Accepted 1 April 2013
Academic Editor: Changbum Chun
Copyright © 2013 Qi Wang and Qingye Zhang. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
By the Maslov index theory, we will study the existence and multiplicity of homoclinic orbits for a class of asymptotically linear nonperiodic Hamiltonian systems with some twisted conditions on the Hamiltonian functions.
1. Introduction and Main Results
As a special case of dynamical systems, Hamiltonian systems are very important in the study of gas dynamics, fluid mechanics, relativistic mechanics and nuclear physics. However it is well known that homoclinic solutions play an important role in analyzing the chaos of Hamiltonian systems. If a system has the transversely intersected homoclinic solutions, then it must be chaotic. If it has the smoothly connected homoclinic solutions, then it cannot stand the perturbation, and its perturbed system probably produces chaotic phenomena. Therefore, it is of practical importance and mathematical significance to consider the existence of homoclinic solutions of Hamiltonian systems emanating from 0.
In the last years, the existence and multiplicity of homoclinic orbits for the first-order system were studied extensively by means of critical point theory, and many results were obtained under the assumption that depends periodically on (see, e.g., [1–12]). Without assumptions of periodicity, the problem is quite different in nature and there is not much work done so far. To the best of our knowledge, the authors in  firstly obtained the existence of homoclinic orbits for a class of first-order systems without any periodicity on the Hamiltonian function. After this, there were a few papers dealing with the existence and multiplicity of homoclinic orbits for the first-order system in this situation (see, e.g., [14–17]).
In the present paper, with the Maslov index theory of homoclinic orbits introduced by Chen and Hu in , we will study the existence and multiplicity of homoclinic orbits for without any periodicity on the Hamiltonian function. To the best of the author's knowledge, the Maslov index theory of homoclinic orbits is the first time to be used to study the existence of homoclinic solutions. We are mainly interested in the Hamiltonian functions of the form where is a symmetric matrix valued function. We assume that , and there are , and a constant matrix , satisfying where is the identity map on , and for a matrix , we say that if and only if In (), if , then () is similar to the condition () in . But the restrictions on will be different from , and we will give some examples in Remark 5. If or in condition (), for example, it is quite different from the existing results as authors known. In short, condition () means that the eigenvalues of will tend to with the speed no less than . But () does not contain all of these cases. For examples, let and , we have the eigenvalues of are , but there is no constant matrix satisfying , for all .
Denote by the self-adjoint operator on , with domain if is bounded and if is unbounded. Let be the absolute value of , and let be the square root of . is a Hilbert space equipped with the norm Let , and define on the inner product and norm by where denotes the usual inner product on . Then, is a Hilbert space. It is easy to see that is continuously embedded in , and we further have the following lemma.
Lemma 1. Suppose that satisfies . Then is compactly embedded in with the usual norm for any .
This lemma is similar to Lemmas 2.1–2.3 in , and we will prove it in Section 3. Define the quadratic form on by It is easy to check that is a bounded quadratic form on , and, hence, there exists a unique bounded self-adjoint operator such that Besides, define a linear operator by In view of Lemma 1, we know that is a Fredholm operator and is a compact operator.
Denote by the set of all uniformly bounded symmetric matric functions. That is to say, if and only if for all , and is uniformly bounded in as the operator on . For any , it is easy to see that determines a bounded self-adjoint operator on , by , for any , we still denote this operator by and then is a self-adjoint compact operator on and satisfies
Before presenting the conditions on , we need the concept of Maslov index for homoclinic orbits introduced by Chen and Hu in  which is equivalent to the relative Morse index. We will give a brief introduction of it by Definition 7, where for any , we denote the associated index pair by (, ).
Now we can present the conditions on as follows. For notational simplicity, we set , and in what follows, the letter will be repeatedly used to denote various positive constants whose exact value is irrelevant. Besides, for two symmetric matrices and , means that is semipositive definite., and there exists a constant such that and .There exists some and continuous symmetric matrix functions with and such that Then, we have our first result.
Condition is a two-side pinching condition near the infinity, and we can relax to condition as follows. There exist some and a continuous symmetric matrix function with such that Then, we have the following results.
Remark 5. Lemma 1 shows that , the spectrum of , consists of eigenvalues numbered by (counted in their multiplicities): with as . Let and , with the constants , satisfying , and for some and (or ). Define where is a smooth cutoff function satisfying By Proposition 12 that, it is easy to verify satisfies all the conditions in Theorem 2. Furthermore, let the constant satisfy for some and (or ). Define Then satisfies all the conditions in Theorems 3 and 4. However, it is easy to see that some conditions of the main results in [13–15, 17] do not hold for these examples.
Remark 6. Note that the assumption in is not essential for our main results. For the case of with , let with small enough, where is the identity map on ; then, and , and, hence, holds for . Therefore, Theorems 3 and 4 still hold in this case. While for the case of with , if we replace by in Theorems 3 and 4, then similar results hold. Indeed, let with small enough such that and , then this case is also reduced to the case of for with .
In this section, we recall the definition of relative Morse index and saddle point reduction and give the relationship between them. For this propose, the notion of spectral flow will be used.
2.1. Relative Morse Index
Let be a separable Hilbert space; for any self-adjoint operator on , there is a unique -invariant orthogonal splitting where is the null space of , is positive definite on and negative definite on , and denotes the orthogonal projection from to . For any bounded self-adjoint Fredholm operator and a compact self-adjoint operator on , is compact (see [19, Lemma 2.7]), where and are the respective projections. Then, by Fredholm operator theory, is a Fredholm operator. Here and in the sequel, we denote by the Fredholm index of a Fredholm operator.
Definition 7. For any bounded self-adjoint Fredholm operator and a compact self-adjoint operator on , the relative Morse index pair is defined by
2.2. Saddle Point Reduction
In this subsection, we describe the saddle point reduction in [20–22]. Recall that is a real Hilbert space, and is a self-adjoint operator with domain . Let , with . Assume that (1)there exist real numbers such that , and that consists of at most finitely many eigenvalues of finite multiplicities;(2) is Gateaux differentiable in , which satisfies and without loss of generality, we may assume that ;(3), , with the norm where small and .
Consider the solutions of the following equation: Let where is the spectral resolution of , and let Decompose the space as follows: where , and .
For each , we have the decomposition where and ; let , with Define a functional on as follows: where , , and .
The Euler equation of this functional is the system Thus, is a solution of (21) if and only if is a critical point of . The implicit function can be applied, yielding a solution for fixed , such that . Since is finite, all topologies on are equivalent, and we choose as it norm. We have which solves the system (28).
Theorem 8. Under assumptions (1), (2), and (3), there is a one-one correspondence between the critical points of the -function with the solutions of the operator equation Moreover, the functional satisfies
Since is a finite dimensional space, for every critical point of in , the Morse index and nullity are finite, and we denote them by .
Now, let the Hilbert space be , and the operator be , . Then we have . For and , for all , let and ; we have and satisfying the previous conditions. Thus, from Theorem 8, we can solve our problems on the finite dimensional space. Similar to Lemma 2.2 and Remark 2.3 in , we have the following estimates.
Lemma 9. Assume that , , for all and ; then one has Moreover, one has
Proof. Note that From , , we have and . Since , we have Therefore, Next, since where is the identity map on , we have
Remark 10. For , we also have that there is a constant dependent of , but independent of , such that
If satisfies the condition , then for any homoclinic orbit of , , and, hence, we have the associated index pair (, ). For notation simplicity, in what follows, we set
This theorem shows the relations between the relative Morse index and the Morse index of the saddle point reduction, and it will play an important role in the proof of our main results. The proof of this theorem will be postponed to the next subsection where the notion of spectral flow will be used.
2.3. The Relationship between , Spectral Flow, and the Morse Index of Saddle Point Reduction
It is well known that the concept of spectral flow was first introduced by Atiyah et al. in  and then extensively studied in [19, 25–28]. Here, we give a brief introduction of the spectral flow as introduced in . Let be a separable Hilbert space as defined before, and be a continuous path of self-adjoint Fredholm operators on the Hilbert space . The spectral flow of represents the net change in the number of negative eigenvalues of as runs from 0 to 1, where the counting follows from the rule that each negative eigenvalue crossing to the positive axis contributes and each positive eigenvalue crossing to the negative axis contributes , and for each crossing, the multiplicity of eigenvalue is taken into account. In the calculation of spectral flow, a crossing operator introduced in  will be used. Take a path and let be the projection from to . When eigenvalue crossing occurs at , the operator is called a crossing operator, denoted by . As mentioned in , an eigenvalue crossing at is said to be regular if the null space of is trivial. In this case, we define A crossing occurring at is called simple crossing if .
As indicated in , the spectral flow will remain the same after a small disturbance of , that is, for and small enough, where is the identity map on . Furthermore, we can choose suitable such that all the eigenvalue crossings occurring in , are regular . Thus, without loss of generality, we may assume that all the crossings are regular. Let be the set containing all the points in at which the crossing occurs. The set contains only finitely many points. The spectral flow of is where . In what follows, the spectral flow of will be simply denoted by when the starting and end points of the flow are clear from the contents. And will be simply denoted by .
Proposition 12 (see [18, Proposition 3]). Suppose that, for each , is a compact operator on then
Thus, from Definition 7, where , is a compact operator. Moreover, if and , from the definition of spectral flow, we have The proof of Theorem 11 is the direct consequence of the aformentioned Proposition 12 and Theorem 3.2 in , so we omit it here.
Remark 13. The case of can be transformed into the case of . More concretely, the case follows from the case by applying to the function . If is a homoclinic solution of , let , it is easy to check that is a homoclinic solution of , and this is a one-one correspondence between the two systems. By the definition of spectral flow and it is catenation property , we have . Thus, we only consider the case of from now on.
3. Proof of Our Main Results
Proof of Lemma 1. Recall the operator , with domain if is bounded and if is unbounded. is a Hilbert space equipped with the norm , for all . Recall the Hilbert space , with the inner product and norm by
where denotes the usual inner product on . From (), there is a matrix such that for all . We have , with and . Thus, for any ,
Let be a bounded set. We will show that is precompact in for . We divide the proof into three steps.
Step 1 (the case of ). For , from () and (53) we have For any , from (54), we can choose large enough, such that On the other hand, by the definition of , we have Thus, by the Sobolev compact embedding theorem, there exist , such that for any , there is satisfying From (55) and (57), we have ; thus, has a finite net in , and so the embedding is compact.
Step 2 (the case of ). Since is continuously embedded in , hence, by the Sobolev embedding theorem, is continuously embedded in , for all . For any , by the Hölder inequality we have thus, the embedding is compact, for all .
Step 3 (the case of ). First, we have ; so we can choose satisfying and . Denote that . For and , denote that and . Then, from (53), and so Let be a bounded set. For any , from (60), choose large enough, such that On the other hand, by the Sobolev compact embedding theorem, there are , such that for any , there exists satisfying From (61) and (62), we have that is to say, has a finite -net in , and the embedding is compact. The proof of the lemma is complete.
Consider the homoclinic orbits of the linear Hamiltonian systems where , , and is a continuous symmetric matrix function. Denote by the set of homoclinic orbits of linear systems (64); then, is a linear subspace of and we have the following lemma.
Proof. As usual, we define the symplectic groups on by where is the set of all real matrices, and denotes the transpose of . Let be the fundamental solution of (64); then, is a path in . Let be a nontrivial homoclinic orbit of (64); that is to say, and satisfies Denote by the subset of satisfying Then, we have . We claim that if and . We prove it indirectly. Assume that with ; that is to say, Since is a path in , , for all , thus which contradicts . Since is an isomorphism on , we have . And from the definition of in the last part of Section 2.2, we have completed the proof.
Before the proof of Theorem 2, we need the following lemma. Since satisfies condition (), performing on the saddle point reduction, choose a suitable number , which is used in the projection for the saddle point reduction in Section 2.2. Let By Theorem 8, we have a functional with , whose critical points give rise to solutions of .
(1) satisfies (PS) condition;
(2) for large enough, where .
Proof. Assume that there is a sequence , satisfying . That is, where defined in Section 2.1. Since is a finite dimensional space, and from the definition of , it is enough to prove that is bounded in . For each , define by It is easy to verify that satisfies where is the constant in condition and is the identity map on . Since , and , we can choose small enough, such that for each , and . Thus is reversible on , and there is a constant , such that On the other hand, for , there is a constant depending on , such that for each , Choose in (76); that is, , we have As we claimed in the introduction, in (76) and (77), the letter denotes different positive constants whose exact value is irrelevant. Thus, from (72), (75), (77), and Lemma 1, we have that is bounded in , and satisfies the (PS) conditions. And by Lemma 5.1 in Chapter II of , we have for large enough.
Lemma 16. Assume that , , and hold; then, there exists a sequence of functions , satisfying the following properties. (1)There exists an increasing sequence of real numbers such that (2)For each , there is a independent of , such that (3)For each , there exist some and a constant with , , such that where is the identity map on .
For each , we consider the following problem where is given in Lemma 16. Performing on the saddle point reduction, we choose the number which is used in the projection for the saddle point reduction in Section 2.2. First, we choose Let Thus, for each and such a fixed, by Theorem 8, we have a functional whose critical points give rise to solutions of . Similarly we have a functional whose critical points give rise to solutions of the following systems For notational simplicity, we denote for and . Define For the functional , similar to Lemma 15, we have the following lemma.
(1) satisfies (PS) condition, and the critical point set of is compact;
(2) for large enough, where .
Proof. The proof is similar to Lemma 15. From Theorem 11, we have Similar to (76), for , there is some , such that Choose , similar to (77), and we have From Lemma 1, Remark 10, and (91), From (89) and (92), we have Now, for each , we assume that satisfying . By , we have that is invertible on , since the sequence must be bounded. Thus the (PS) condition for holds. From the same reason, we have the compactness of the critical point set of . And by Lemma 5.1 in Chapter II of , we have for large enough.
Proof. We prove it indirectly. Assume there exist and satisfying the conditions, and ; that is, . Since , for all , we have . Denote that ; then, we have in for some with , and
Then, for any , there exists , satisfying
where . Since , there is a , such that
Then, from the similar argument in , there is a subsequence throught which we may assume that converges in uniform norm to , and , for all . Therefore uniformly on , and there is depending on , such that , for any and .
Performing the saddle point reduction on the following systems: for large enough, we have the functional (denote by for simplicity) and the function ; since , we have the following decomposition: where is positive definite on and negative definite on . From Remark 10, and , there exists , such that for large enough, From the uniform boundary of and Remark 10, we can choose large enough, such that where , defined in Theorem 8. Choose small enough and , such that . Since is finite dimensional space, choose large enough, such that where , and from the definition of , that is, From (101) and (103), for large enough. Thus we have That is, , from Theorem 11, , , and thus , which contradicts the assumption.
Proof of Theorem 3. As claimed in Remark 13, we can only consider the case of . Note that is a critical point of , and the morse index of for is , since , we have From proposition (2) in Lemma 17, use the and Morse inequalities, and has a nontrivial critical point with its morse index ; that is, , then from Lemma 18, we have that is bounded in . Thus is a nontrivial solution of for large enough.
Let be a Hilbert space and assume is an even functional, satisfying the (PS) condition and . Denote .
Lemma 19 (see [30, Corollary 10.19]). Assume that and are subspaces of satisfying . If there exist and such that then has pairs of nontrivial critical points , so that , for .
First, we consider the case of , since is even, we have that is also even and satisfies Lemma 16. Let , and the positive space of in , and we have and , , . So, has pairs of nontrivial critical points and pairs of them satisfy Then, we can complete the proof. In order to prove the case of , we need the following lemma.
Lemma 20 (see [21, Corollary II 4.1]). Assume that and are subspaces of satisfying