A Note on the Minimal Period Problem for Second Order Hamiltonian Systems

Abstract

We study periodic solutions of second order Hamiltonian systems with even potential. By making use of generalized Nehari manifold, some sufficient conditions are obtained to guarantee the multiplicity and minimality of periodic solutions for second order Hamiltonian systems. Our results generalize the outcome in the literature.

1. Introduction

Denote by the sets of all natural numbers, integers, nonnegative real numbers, and real numbers, respectively. For , denote by the -dimensional Euclidean space with the usual inner product and norm .

Consider the second order Hamiltonian systems Assume that (A1) is a symmetric, negative semidefinite matrix; (V1) and for all ; (V2) as ; (V3);  (V4)for any with , is strictly increasing on ; (V5)there exist and such that ; (V6) is even; that is, for all .

When , in his pioneering work [1] of 1978, Rabinowitz established the existence of periodic solutions of (1) when satisfies (V1), (V2), and the well-known AR-condition: (V7) He conjectured that (1) possesses a nonconstant solution with any prescribed minimal period under the same assumptions. Since then, many authors devoted themselves to the study of periodic solutions with prescribed minimal period of (1).

When , in 1993, Long (cf. [2]) proved the existence of -periodic solutions with minimal period or of (1) under the assumptions that satisfies (V2), (V6), and (V7). Releasing assumption (V6), Long showed that (1) possesses periodic solutions with minimal periodic for some integer satisfying (cf. [3]) or (cf. [4]). In 2001, Fei et al. (cf. [5]) proved that (1) has a nonconstant periodic solution with any prescribed minimal period when satisfies (V2), (V6), and (V7), and is positive semidefinite.

When is positive semidefinite, satisfies (V2) and (V7); in 1997, Fei and Wang (cf. [6]) showed that (1) possesses nonconstant -periodic solutions with minimal periodic for some odd interger satisfying , where , , and . If is even, replacing (V7) by (V3), and the following condition Fei and Wang (cf. [7]) estimated that the minimal period is not smaller than .

The above results on minimal period problem were obtained by making use of index theory. Another method being used to study such a problem is Nehari manifold. As is well known, the Nehari manifold, introduced by Nehari (cf. [8, 9]), has been used widely to study the existence of ground state solutions of partial differential equations (cf. [10–14]) and that of (1) (cf. [13, 15, 16]). A ground state solution of a system is a solution which possesses the minimal energy of all solutions of the system. However, such kind of solutions may not have prescribed minimal periods. In 1981, Ambrosetti and Mancini made use of Nehari manifold to study the existence of periodic solutions with any prescribed minimal period of first order Hamiltonian systems with convex potential (cf. [17]). In 2010, Xiao (cf. [18]) proved that (1) possesses a periodic solution with any prescribed minimal period and when , satisfies (V6) and the following assumption: (V8) where . For more results on this direction, we refer to [19, 20].

Motivated by [13, 17, 18], in this paper, we consider the multiplicity and minimality of periodic solutions of (1) under the assumption (A1), (V1)–(V6). Our main result reads as follows.

Theorem 1. Assume that (A1), (V1)–(V6) hold. Then, for any , (1) possesses a periodic solution with minimal period .

Corollary 2. If and satisfies (V6) and (V8), then, for any , (1) possesses a periodic solution with minimal period .

Theorem 3. Assume that (A1), (V1)–(V6) hold. Then, for any , (1) possesses infinitely many pairs of -periodic solutions.

The rest of this paper is divided into two parts. In Section 2, we establish the variational functional and state some useful lemmas. In Section 3, we introduce Nehari Manifold and prove our main results.

2. Preliminary

Denote by . is a Hilbert space equipping with the usual norm and inner product The variational functional defined on , corresponding to (1), is If (A1), (V1)–(V5) hold, then is continuous differentiable on and Moreover, is weakly continuous and is compact.

Define a subspace of as follows: Then is a closed subspace of . If , it has a Fourier expansion Obviously, . We can define an equivalent inner product on

Let us state a useful lemma and omit the proof. One can find the details in [21].

Lemma 4. If is a critical point of on , then is a critical point of on .

Define an operator on by extending bilinear form It is easy to verify that is a linear self-adjoint operator. Since is a negative semidefinite matrix, then has a sequence of eigenvalues with as . Thus for any , , where denotes the norm induced by . Define another equivalent inner product on Then, for all , there exist such that , where denotes the norm induced by .

Functional (6) can be rewritten as

In the end of this section, we introduce two useful lemmas.

Lemma 5 (see [22]). If and , then where .

If , . Thus elements of satisfy the above two inequalities.

Let be a Banach space such that the unit sphere in is a submanifold of class (at least) and let . A sequence is called sequence for if it satisfies is bounded and as . We say that satisfies the condition if every sequence contains a convergent subsequence.

Lemma 6 (see [13]). If is infinite-dimensional and   is bounded below and satisfies the condition on , then has infinitely many pairs of critical points.

3. Proofs of the Main Results

Denote by . We define the Nehari manifold

Denote by . Fixing , define for . On the one hand, if , then and On the other hand, if , then Thus, for any , if and only if . The following lemma shows that .

Lemma 7. For any , there exists a unique dependent on such that

Proof. By the definition of , we have
By (V2), for any , there exists such that for . If , then Lemma 5 implies that and for . Fixing , there exists such that for all and
Let . Then . For any , denote Then meas, where meas denotes the measure of . Thus (V1) and (V3) yield that Consequently, there exists dependent on such that for . By the mean value theorem, there exists dependent on such that and .
By the definition, is continuous differentiable on . Differentiating , we have where . Because of (V4), is strictly decreasing on . Thus and hence has a unique zero point on . It follows that has a unique critical point, which is a maximum.

Remark 8. Fixing , then . Obviously, and . If we extend the definition of to , then Lemma 7 implies that there exists a unique such that . Also, for and for .

Now, we study the properties of restricted on .

Lemma 9. Consider the following:
.

Proof. For any , both Lemma 7 and Remark 8 imply that . Hence .

Lemma 10. satisfies the condition on .

Proof. Assume that is a sequence of . Then there exists such that and as . We show that is bounded. Suppose, to the opposite, that as . Set . Then . Passing to a subsequence, . If , then uniformly for . Hence, for any , we have This is a contradiction if . So . Arguing similarly to [23], there exist , such that Denote . But (V3) implies that This is a contradiction. Hence is bounded. Since is compact, passing to a subsequence, converges strongly. Since , then contains a convergent subsequence.

Define the maps as follows: where is defined on Remark 8.

Lemma 11. Assume that (V1)–(V4) hold. Then the following statements hold: (B1)there exists a normalization function such that   is bounded on bounded sets and for all ; (B2)for each there exists such that if , then for and for ; (B3)there exists such that for all and for each compact subset there exists a constant such that   for all .

Proof. It follows from the discussion in [13] and Remark 8 that (B1), (B2) hold, and for all . Next, we show that the last part of (B3) holds. Arguing by contradiction, we suppose that there exists a compact subset and a sequence , where and , such that as . Since is compact and , passing to a subsequence, . Since the embedding of into is compact and , then uniformly for . Denote by . Since , if , then Fatou’s lemma yields Thus , which contradicts with the fact that . Consequently, there exists such that for all and (B3) holds.

Lemma 12 (see [13]). The mapping is continuous and is a homeomorphism between and .

Lemma 13 (see [13]). Consider the following.(1) and where is the tangent space of at .(2)If is a (PS) sequence for , then is a (PS) sequence for . If is a bounded (PS) sequence for , then is a (PS) sequence for .(3) is a critical point of if and only if is a nontrivial critical point of . Moreover, the corresponding values of and coincide and .(4)If is even, then so is .

Lemma 14. If satisfies the condition, so does .

Proof. Assume that is a sequence for . According to Lemma 13, is a sequence for . Since satisfies the condition, passing to a subsequence, . Thus contains a convergent subsequence, which converges to . Hence satisfies the condition.

Proof of Theorem 1. According to Lemmas 9 and 13, . Let be a minimizing sequence for restricted to . By Ekeland’s variational principle, we may assume that as . The condition implies that contains a converging subsequence, whose limit is denoted by . Thus is a critical point of . According to Lemma 13 again, is a critical point of , which is also a nonconstant -periodic solution of (1).
Claim. has minimal period .
Suppose, to the opposite, that has minimal period , where is an integer. Let . Obviously, . It follows from Lemma 7 that there exists such that . It follows that which is a contradiction. Hence has minimal period .

Proof of Corollary 2. For with , set . Then (V8) implies that that is, is increasing strictly on . Thus (V4) is satisfied. It is easy to check that satisfies (V1)–(V3) when (V8) is available.
Without assumption (V5), may not be continuous differentiable on . However, we can use the method introduced in [1] to handle such a situation. Let and such that if , if , and if . Set where . Since satisfies (V8), it follows from Lemma 2.3 in [18] that Arguing similarly to Lemma 2.9 in [1], we have where . Let and . Since , then and . Obviously, are independent of . Arguing similarly to Lemma 4.1 of [22], we can prove the following inequality: Hence satisfies (V1)–(V6).
Consider the disturbed second order Hamiltonian systems whose variational function is Since satisfies (V1)–(V6), applying Theorem 1, (38) possesses a periodic solution with minimal period .
Claim. There exists independent of such that .
Let with , where (′) denotes the transposition of a vector. Computing directly, we have So . It follows from (37) that where denotes the constant such that . Since , as . Then there exists independent of such that . Thus Hence Consequently, we have It follows that there exists independent of such that . If , by the definition of , and hence is a nonconstant -periodic solution of (1) with .

Proof of Theorem 3. Since is even, so do and . Since and satisfies the condition, then Lemma 6 yields that has infinitely many pairs of critical points. According to Lemma 13, has infinitely many pairs of critical points. Thus (1) has infinitely many pairs of periodic solutions.

Example 15. Let and let be It is easy to check that does not satisfy (V8) or even (V7) (cf. [23]). However, satisfies (V1)–(V6). Applying Theorems 1 and 3, for any , (1) possesses infinitely many pairs of -periodic solutions and has at least one solution with as its minimal period.

Conflict of Interests

The author declares that there is no conflict of interests regarding the publication of this paper.

Acknowledgments

This project is supported by NSFC (no. 11301102) and by PCSIRT (no. IRT1226).