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# Existence of subharmonic solutions for non-quadratic second-order Hamiltonian systems

Xingyong Zhang1* and Xianhua Tang2

Author Affiliations

1 Department of Mathematics, Faculty of Science, Kunming University of Science and Technology, Kunming, Yunnan, 650500, P.R. China

2 School of Mathematical Sciences and Computing Technology, Central South University, Changsha, Hunan, 410083, P.R. China

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Boundary Value Problems 2013, 2013:139  doi:10.1186/1687-2770-2013-139

 Received: 23 October 2012 Accepted: 10 May 2013 Published: 30 May 2013

This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

### Abstract

In this paper, some existence theorems are obtained for subharmonic solutions of second-order Hamiltonian systems with linear part under non-quadratic conditions. The approach is the minimax principle. We consider some new cases and obtain some new existence results.

MSC: 34C25, 58E50, 70H05.

##### Keywords:
second-order Hamiltonian systems; subharmonic solution; critical point; linking theorem

### 1 Introduction and main results

Consider the second-order Hamiltonian system

(1.1)

where A is an symmetric matrix and is T-periodic in t and satisfies the following assumption:

Assumption (A)′is measurable intfor everyand continuously differentiable inxfor a.e. , and there existandwhich isT-periodic andwithsuch that

for alland a.e. .

When , system (1.1) reduces to the second-order Hamiltonian system

(1.2)

There have been many existence results for system (1.2) (for example, see [1-7] and references therein). In 1978, Rabinowitz [6] obtained the nonconstant periodic solutions for system (1.2) under the following AR-condition: there exist and such that

From then on, the condition has been used extensively in the literature; see [8-12] 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

(1.3)

(1.4)

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:

(1.5)

(1.6)

(1.7)

Then system (1.2) has a sequence of distinct periodic solutions with periodsatisfyingandas.

Recently, Ma and Zhang [15] considered the following p-Laplacian system:

(1.8)

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:

(1.9)

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

(1.10)

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

(1.11)

Theorem 1.1Assume that (A0) holds andFsatisfies (A)′, (1.5) and the following conditions.

(H1) For some, assume thatksatisfies

(1.12)

(H2) There exist positive constantsm, ζ, ηandsuch that

(H3) Assume that one of the following cases holds:

(1) when, and, there existandsuch that

(1.13)

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;

(H4) there existandsuch that

where

whereσimplies thatis independent ofk. Then system (1.1) has a nonzerokT-periodic solution. Especially, for cases (H3)(1) and (H3)(4), system (1.1) has a nonconstantkT-periodic 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 kT-periodic 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 kT-periodic solutions for every .

Remark 1.2 In [18], Costa and Magalhães studied the first-order Hamiltonian system

(1.14)

They obtained that system (1.14) has a periodic solution under the following non-quadraticity conditions:

(1.15)

and the so-called 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 second-order 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

(1.16)

(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

for each . Let

Then one has

(see Proposition 1.3 in [1]).

Lemma 2.1If, then

where.

Proof Fix . For every , we have

(2.1)

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 thatisT-periodic in t, is measurable intfor everyand continuously differentiable inxfor a.e. . If there existand () such that

(2.2)

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

(I1) , whereandbounded and self-adjoint, ;

(I2) is compact, and

(I3) there exists a subspaceand sets, and constantssuch that

(i) and,

(ii) Qis bounded and,

Thenφpossesses a critical valuewhich can be characterized as

where

,

for, and

, whereandKis compact.

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

(3.1)

Let . Then by (1.1),

Furthermore

that is,

(3.2)

Let and then . Let

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.

(a) For , since

then

and

Let

Then

(3.3)

Remark 3.2 Obviously, if one of (H3)(5) and (H3)(6) holds, then . Hence,

(b) For , let

where

Then

and

Since for fixed ,

are strictly increasing on ,

and

Moreover, it is easy to verify that

Let

Then

(3.4)

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

(c) For , since

and

Obviously, when , . So . When , it follows from

that

(3.5)

(d) Obviously, for , we have

(3.6)

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

Obviously, and , . Then

By (H3)(2), (1.5) and the periodicity of F, we have

(3.7)

where and . Since is the finite dimensional space, there exists a constant such that

(3.8)

By (3.3), (3.5), (3.6), (3.7) and (3.8), we know that for all and ,

(3.9)

Hence,

where

Case (ii): if , then we choose

Then

and

(3.10)

By (H3)(2), (1.5) and the periodicity of F, we have

(3.11)

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

(3.12)

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

(3.13)

By (1.5), (3.3), (3.5) and (3.6), for all , we have

(3.14)

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 self-adjoint 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 self-adjoint compact operator, it is easy to see that () are bounded and self-adjoint. 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 1-Step 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 kT-periodic solution for every .

Let . Like the argument of case (ii) in the proof of Theorem 1.1, choose

By (H3)′, (1.5) and the T-periodicity of F, we have

(3.15)

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

(3.16)

Hence, is bounded for any .

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

for some . So, we omit the details. □

### 4 Examples

Example 4.1 Let and

Then , , , , , , , and . Obviously, the matrix A satisfies Assumption (A0) and such that

It is easy to verify that (H1) holds with . Let

Then for all and a.e. and

(4.1)

(4.2)

It is easy to verify that

Choose , and . Moreover, obviously, there exists such that . Then

Hence, (H2) holds.

When ,

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 T-periodic solution.

When ,

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 T-periodic solution. However, we can obtain that system (1.1) has a 2T-periodic solution.

When ,

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 T-periodic solution and 2T-periodic solution. However, we can obtain that system (1.1) has a 3T-periodic 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 .

Example 4.3 Let and

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.

### References

1. Mawhin, J, Willem, M: Critical Point Theory and Hamiltonian Systems, Springer, New York (1989)

2. Tang, CL: Periodic solutions of nonautonomous second order systems with sublinear nonlinearity. Proc. Amer. Math. Soc.. 126, 3263–3270 (1998). Publisher Full Text

3. Ailva, EAB: Subharmonic solutions for subquadratic Hamiltonian systems. J. Differ. Equ.. 115, 120–145 (1995). Publisher Full Text

4. Schechter, M: Periodic non-autonomous second-order dynamical systems. J. Differ. Equ.. 223, 290–302 (2006). Publisher Full Text

5. Rabinowitz, PH: Minimax Methods in Critical Point Theory with Applications to Differential Equations, Am. Math. Soc., Providence (1986)

6. Rabinowitz, PH: Periodic solutions of Hamiltonian systems. Commun. Pure Appl. Math.. 31, 157–184 (1978). Publisher Full Text

7. Tao, ZL, Yan, S, Wu, SL: Periodic solutions for a class of superquadratic Hamiltonian systems. J. Math. Anal. Appl.. 331, 152–158 (2007). Publisher Full Text

8. Chang, KC: Infinite Dimensional Morse Theory and Multiple Solution Problems (1993)

9. Ekeland, I: Convexity Method in Hamiltonian Mechanics, Springer, Berlin (1990)

10. Ekeland, I, Hofer, H: Periodic solutions with prescribed period for convex autonomous Hamiltonian systems. Invent. Math.. 81, 155–188 (1985). Publisher Full Text

11. Fei, G, Qiu, Q: Minimal periodic solutions of nonlinear Hamiltonian systems. Nonlinear Anal.. 27, 821–839 (1996). Publisher Full Text

12. Fei, G, Kim, S, Wang, T: Minimal period estimates of periodic solutions for superquadratic Hamiltonian systems. J. Math. Anal. Appl.. 238, 216–233 (1999). PubMed Abstract | Publisher Full Text

13. Fei, G: On periodic solutions of superquadratic Hamiltonian systems. Electron. J. Differ. Equ.. 2002, 1–12 (2002)

14. Tao, ZL, Tang, CL: Periodic and subharmonic solutions of second order Hamiltonian systems. J. Math. Anal. Appl.. 293, 435–445 (2004). Publisher Full Text

15. Ma, S, Zhang, Y: Existence of infinitely many periodic solutions for ordinary p-Laplacian systems. J. Math. Anal. Appl.. 351, 469–479 (2009). Publisher Full Text

16. Ye, YW, Tang, CL: Periodic and subharmonic solutions for a class of superquadratic second order Hamiltonian systems. Nonlinear Anal.. 71, 2298–2307 (2009). Publisher Full Text

17. Zhang, X, Tang, X: Subharmonic solutions for a class of non-quadratic second order Hamiltonian systems. Nonlinear Anal., Real World Appl.. 13, 113–130 (2012). Publisher Full Text

18. Costa, DG, Magalhães, CA: A unified approach to a class of strongly indefinite functions. J. Differ. Equ.. 125, 521–547 (1996). Publisher Full Text

19. Kyristi, ST, Papageorgiou, NS: On superquadratic periodic systems with indefinite linear part. Nonlinear Anal.. 72, 946–954 (2010). Publisher Full Text

20. Benci, V, Rabinowitz, PH: Critical point theorems for indefinite functions. Invent. Math.. 52, 241–273 (1979). Publisher Full Text

21. Bartolo, P, Benci, V, Fortunato, D: Abstract critical point theorems and applications to some nonlinear problems with strong resonance at infinity. Nonlinear Anal.. 7, 241–273 (1983)

22. Krosnoselski, MA: Topological Methods in the Theory of Nonlinear Integral Equations, Macmillan Co., New York (1964)