Abstract
In this paper, some existence theorems are obtained for subharmonic solutions of secondorder Hamiltonian systems with linear part under nonquadratic conditions. The approach is the minimax principle. We consider some new cases and obtain some new existence results.
MSC: 34C25, 58E50, 70H05.
Keywords:
secondorder Hamiltonian systems; subharmonic solution; critical point; linking theorem1 Introduction and main results
Consider the secondorder Hamiltonian system
where A is an symmetric matrix and is Tperiodic in t and satisfies the following assumption:
Assumption (A)′is measurable intfor everyand continuously differentiable inxfor a.e. , and there existandwhich isTperiodic andwithsuch that
When , system (1.1) reduces to the secondorder Hamiltonian system
There have been many existence results for system (1.2) (for example, see [17] and references therein). In 1978, Rabinowitz [6] obtained the nonconstant periodic solutions for system (1.2) under the following ARcondition: there exist and such that
From then on, the condition has been used extensively in the literature; see [812] and the references therein. In [13], Fei also obtained the existence of nonconstant solutions for system (1.2) under a kind of new superquadratic condition. Subsequently, Tao and Tang [14] gave the following more general one than Fei’s: there exist and such that
They also considered the existence of subharmonic solutions and obtained the following result.
Theorem A (See [14], Theorem 2)
Suppose thatFsatisfies
(A) is measurable intfor everyand continuously differentiable inxfor a.e. , and there existandsuch that
for alland a.e. . Assume that (1.3), (1.4) and the following conditions hold:
Then system (1.2) has a sequence of distinct periodic solutions with periodsatisfyingandas.
Recently, Ma and Zhang [15] considered the following pLaplacian system:
where . By using some techniques, they obtained the following more general result than Theorem A.
Theorem B (See [15], Theorem 1)
Suppose thatFsatisfies (A), (1.3) and (1.4) with 2 replaced byp, (1.5) and the following condition:
Then system (1.8) has a sequence of distinct periodic solutions with periodsatisfyingandas.
When , where and is the unit matrix of order N. Ye and Tang [16] obtained the following result.
Theorem C (See [16], Theorem 2)
Suppose that, Fsatisfies (A), (1.3), (1.4), (1.5), (1.6) and the following conditions:
Then system (1.1) has a sequence of distinct periodic solutions with periodsatisfyingandas.
Recently, in [17], we considered a more general case than that in [16]. We considered the case that A only has 0 or as its eigenvalues, where , , and . In [17], we used the following condition which presents some advantages over (1.3) and (1.4):
(H) there exist positive constantsm,ζ,ηandsuch that
In this paper, we consider some new cases which can be seen as a continuance of our work in [17].
Next, we state our main results. Assume that and . Let () and () be the positive and negative eigenvalues of A, respectively, where r and s denote the number of positive eigenvalues and zero eigenvalues of A (counted by multiplicity), respectively. Moreover, we denote by q the number of negative eigenvalues of A (counted by multiplicity). We make the following assumption:
Assumption (A0)Ahas at least one nonzero eigenvalue and all positive eigenvalues are not equal tofor all, where, that is, () for all.
The Assumption (A0) implies that one can find such that
For the sake of convenience, we set
Then
Corresponding to (1.10), we know that there exist such that
Moreover, set
and let . Then . Corresponding to (1.10), there exists such that
Theorem 1.1Assume that (A0) holds andFsatisfies (A)′, (1.5) and the following conditions.
(H1) For some, assume thatksatisfies
(H2) There exist positive constantsm, ζ, ηandsuch that
(H3) Assume that one of the following cases holds:
(1) when, and, there existandsuch that
whereandare defined by (1.11);
(2) when, and, there existandsuch that (1.13) holds;
(3) when, and, there existandsuch that (1.13) holds;
(4) when, and, there existandsuch that (1.13) holds;
(5) when, and, there existandsuch that (1.13) holds;
(6) when, and, there existandsuch that (1.13) holds;
where
whereσimplies thatis independent ofk. Then system (1.1) has a nonzerokTperiodic solution. Especially, for cases (H3)(1) and (H3)(4), system (1.1) has a nonconstantkTperiodic solution.
Remark 1.1 For cases (H3)(1)(H3)(4), from (1.10) and (1.12), it is easy to see that the number of satisfying (1.12) is finite. Let be the maximum integer satisfying (1.12), where
Then . Hence, Theorem 1.1 implies that system (1.1) has nonzero kTperiodic solutions (). For cases (H3)(5) and (H3)(6), since , (1.12) holds for every . Hence, Theorem 1.1 implies that system (1.11) has nonzero kTperiodic solutions for every .
Remark 1.2 In [18], Costa and Magalhães studied the firstorder Hamiltonian system
They obtained that system (1.14) has a periodic solution under the following nonquadraticity conditions:
and the socalled asymptotic noncrossing conditions
where are consecutive eigenvalues of the operator . Moreover, they also obtained system (1.14) has a nonzero periodic solution under (1.15) and the called crossing conditions
One can also establish the similar results for the secondorder Hamiltonian system (1.1). Some related contents can be seen in [19]. It is worth noting that in [18] and [19], are consecutive eigenvalues of the operator or . In our Theorem 1.1 and Theorem 1.2, we study the existence of subharmonic solutions for system (1.1) from a different perspective. () in our theorems are the eigenvalues of the matrix A. Obviously, it is much easier to seek the eigenvalue of a matrix. In Section 4, we present an interesting example satisfying our Theorem 1.1 but not satisfying the theorem in [19].
Theorem 1.2Suppose that (A0) holds andFsatisfies (A)′, (1.5), (H2) and the following conditions:
(H3)′ when, and, there existandsuch that
(H4)′
Then system (1.1) has a sequence of distinct periodic solutions with periodsatisfyingandas.
In the final theorem, we present a result about the existence of subharmonic solutions for system (1.8). Using a condition like (H2) and similar to the argument of Remark 1.1 in [17], we can improve Theorem B.
Theorem 1.3Suppose thatFsatisfies (A), (1.5) and the following conditions:
(H5) there exist positive constantsm, ζ, ηandsuch that
(H6)
Then system (1.8) has a sequence of distinct nonconstant periodic solutions with periodsatisfyingandas.
2 Some preliminaries
Let
Then is a Hilbert space with the inner product and the norm defined by
and
Then one has
(see Proposition 1.3 in [1]).
Proof Fix . For every , we have
Set
Integrating (2.1) over and using the Hölder inequality, we obtain
Hence, we have
The proof is complete. □
Lemma 2.2 (see [[17], Lemma 2.2])
Assume thatisTperiodic in t, is measurable intfor everyand continuously differentiable inxfor a.e. . If there existand () such that
then
is weakly continuous and uniformly differentiable on bounded subsets of.
Remark 2.1 In [[17], Lemma 2.2], . In fact, in its proof, it is not essential that F is continuously differentiable in t.
We use Lemma 2.3 below due to Benci and Rabinowitz [20] to prove our results.
Lemma 2.3 (see [20] or [[5], Theorem 5.29])
LetEbe a real Hilbert space withand. Suppose thatsatisfies (PS)condition, and
(I_{1}) , whereandbounded and selfadjoint, ;
(I_{3}) there exists a subspaceand sets, and constantssuch that
(iii) Sand∂Qlink.
Thenφpossesses a critical valuewhich can be characterized as
where
Remark 2.2 As shown in [21], a deformation lemma can be proved with replacing the usual (PS)condition with condition (C), and it turns out that Lemma 2.3 holds true under condition (C). We say φ satisfies condition (C), i.e., for every sequence , has a convergent subsequence if is bounded and as .
3 Proofs of theorems
Proof of Theorem 1.1 It follows from Assumption (A)′ that the functional on given by
is continuously differentiable. Moreover, one has
for and the solutions of system (1.1) correspond to the critical points of (see [1]).
Obviously, there exists an orthogonal matrix Q such that
Furthermore
that is,
Then the critical points of correspond to solutions of system (3.2). It is easy to verify that and G satisfies all the conditions of Theorem 1.1 and Theorem 1.2 if F satisfies them. Hence, w is the critical point of if and only if is the critical point of . Therefore, we only need to consider the special case that is the diagonal matrix defined by (3.1). We divide the proof into six steps.
Step 1: Decompose the space . Let
Note that
Define
Then , and are closed subsets of and
(1)
(2)
where
Let
Then
and
Remark 3.1 When , it is easy to see .
Step 2: Let
Next we consider the relationship between and on those subspaces defined above. We only consider the case that (H3)(2) holds. For others, the conclusions are easy to be seen from the argument of this case.
then
and
Let
Then
Remark 3.2 Obviously, if one of (H3)(5) and (H3)(6) holds, then . Hence,
where
Then
and
and
Moreover, it is easy to verify that
Let
Then
Remark 3.3 From the above discussion, it is easy to see the following conclusions:
(i) if (H3)(1) holds, then (3.4) holds with
(ii) if (H3)(2) holds, then (3.4) holds with
(iii) if (H3)(3) holds, then (3.4) holds with
(iv) if (H3)(4) holds, then (3.4) holds with
(v) if (H3)(5) holds, then (3.4) holds with
(vi) if (H3)(6) holds, then (3.4) holds with
and
Obviously, when , . So . When , it follows from
that
Step 3: Assume that (H3)(2) holds. We prove that there exist and such that
Let
Choosing and , by Lemma 2.1, (H4) and (3.4), we have, for all ,
For cases (H3)(1) and (H3)(3)(H3)(6), correspondingly, by (H4) and Remark 3.3, similar to the above argument, we can also obtain that
Step 4: Let
where , and will be determined later. In this step, we prove . We only consider the case that F satisfies (H3)(2). For other cases, the results can be seen easily from the argument of case (H3)(2).
Assume that F satisfies (H3)(2). Let
Case (i): if
then we choose
By (H3)(2), (1.5) and the periodicity of F, we have
where and . Since is the finite dimensional space, there exists a constant such that
By (3.3), (3.5), (3.6), (3.7) and (3.8), we know that for all and ,
Hence,
where
Case (ii): if , then we choose
Then
and
By (H3)(2), (1.5) and the periodicity of F, we have
where and . By (3.3), (3.5), (3.6), (3.8) and (3.11), we know that for all and ,
Hence,
where
Case (iii): if , then we choose
Then
By (H3)(2), (1.5) and the periodicity of F, we have
where and . By (3.3), (3.5), (3.6), (3.8) and (3.12), for all and , we have
Hence,
where
Combining cases (i), (ii) and (iii), if we let
then
By (1.5), (3.3), (3.5) and (3.6), for all , we have
Thus, it follows from (3.13) and (3.14) that .
Step 5: We prove that satisfies (C)condition in . The proof is similar to that in Theorem 1.1 in [17]. We omit it.
Step 6: We claim that has a nontrivial critical point such that . Especially, we claim that, for cases (H3)(1) and (H3)(4), since A is a positive semidefinite matrix, (1.5) implies that is nonconstant.
In fact, it is easy to see that
where is the linear selfadjoint operator defined, using the Riesz representation theorem, by
The compact imbedding of into implies that K is compact. In order to use Lemma 2.3, we let and define , by
where and . Since K is a selfadjoint compact operator, it is easy to see that () are bounded and selfadjoint. Let
Assumption (A)′ and Lemma 2.2 imply that b is weakly continuous and is uniformly differentiable on bounded subsets of . Furthermore, by standard theorems in [22], we conclude that is compact. Let . Then and link. Hence, by Step 1Step 5, Lemma 2.3 and Remark 2.2, there exists a critical point such that , which implies that is nonzero. For cases (H3)(1) and (H3)(4), since A is a positive semidefinite matrix, it follows from (1.5) that is nonconstant. The proof is complete. □
Proof of Theorem 1.2 Obviously, when , and , (H1) holds for any . Moreover, since (H3)′ implies that (H3)(5) and (H4)′ implies that (H4), system (1.1) has kTperiodic solution for every .
Let . Like the argument of case (ii) in the proof of Theorem 1.1, choose
By (H3)′, (1.5) and the Tperiodicity of F, we have
where . In the proof of Theorem 1.1, if we replace (3.15) with (3.11), then we obtain
where
Note that is independent of k. Hence, if is the critical point of , then it follows from (3.3), (3.5), (3.6), the definitions of critical value c in Lemma 2.3 and that
Obviously, we can find such that , then we claim that is distinct from for all . In fact, if for some , it is easy to check that
Then by (3.16), we have , a contradiction. We also can find such that for all . Otherwise, if for some , we have . Then by (3.16), we have , a contradiction. In the same way, we can obtain that system (1.1) has a sequence of distinct periodic solutions with period satisfying and as . The proof is complete. □
Proof of Theorem 1.3 Except for verifying (C) condition, the proof is the same as in Theorem B (that is Theorem 1 in [15]). To verify (C) condition, we only need to prove the sequence is bounded if is bounded and as . Other proofs are the same as in [15]. The proof of boundedness of is essentially the same as in Theorem 1.1 in [17] except that 2 is replaced by p, by
equipped with the norm
and
by
4 Examples
Then , , , , , , , and . Obviously, the matrix A satisfies Assumption (A0) and such that
It is easy to verify that (H1) holds with . Let
It is easy to verify that
Choose , and . Moreover, obviously, there exists such that . Then
Hence, (H2) holds.
By (4.2), we can find such that
Let . Then (H3)(2) holds with . Moreover, by (4.1), we can find such that
Let . Then (H4) holds. By Theorem 1.1, we obtain that system (1.1) has a Tperiodic solution.
By (4.2), we can find such that
Let . Then (H3)(2) holds with . Moreover, by (4.1), we can find such that
Let . Then (H4) holds. Note that . So, when , by Theorem 1.1, we cannot judge that system (1.1) has a Tperiodic solution. However, we can obtain that system (1.1) has a 2Tperiodic solution.
By (4.2), we can find such that
Let . Then (H3)(2) holds with . Moreover, by (4.1), we can find such that
Let . Then (H4) holds. Note that . So, when , by Theorem 1.1, we cannot judge that system (1.1) has Tperiodic solution and 2Tperiodic solution. However, we can obtain that system (1.1) has a 3Tperiodic solution. It is easy to verify that Example 4.1 does not satisfy the theorem in [19] even if .
Example 4.2 Let
and
Then
Obviously, (A0), (A)′, (1.5), (H3)′ and (H4)′ hold. Let , and . Similar to the argument in Example 4.1, we obtain (H2) also holds. Then by Theorem 1.2, system (1.1) has a sequence of distinct periodic solutions with period satisfying and as .
Then (1.5) holds and
Let , and . Then it is easy to obtain that there exists such that (H5) holds. By Theorem 1.3, system (1.8) has a sequence of distinct periodic solutions with period satisfying and as . It is easy to see that Example 4.3 does not satisfy (1.3). Hence, Theorem 1.3 improved Theorem B.
Competing interests
The authors declare that they have no competing interests.
Authors’ contributions
XZ proposed the idea of the paper and finished the main proofs. XT provided some important techniques in the process of proofs.
Acknowledgements
The authors would like to thank the anonymous referees for their valuable suggestions.
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