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Existence and multiplicity of periodic solutions for some second-order Hamiltonian systems
Journal of Inequalities and Applications volume 2014, Article number: 411 (2014)
The purpose of this paper is to study the existence and multiplicity of periodic solutions for the following non-autonomous second-order Hamiltonian systems: a.e. , , where . Some new existence and multiplicity theorems are obtained by using the least action principle, and the minimax method in critical point theory, which unify and generalize some of the recent corresponding results in the literature.
1 Introduction and main results
Consider the second-order Hamiltonian systems
where and satisfies the following assumption:
is measurable in t for every and continuously differentiable in x for a.e. , and there exist , such that
for all and a.e. .
As is well known, a Hamiltonian system is a system of differential equations which can model the motion of a mechanical system. An important and interesting question is under what conditions the Hamiltonian system can support periodic solutions. During the past few years, under assumption (A) and some other suitable conditions, such as the coercivity condition, the convexity conditions, the sublinear nonlinearity conditions, the subquadratic potential conditions, the superquadratic potential conditions, the periodicity conditions, and even the type potential condition, and so on, the existence and multiplicity of periodic solutions are obtained for problem (1) in [1–20]. Inspired and motivated by the results due to Wang and Zhang , Aizmahin and An , Ye and Tang , and Ma and Tang , we obtain some new existence theorems for problem (1), which generalize some results mentioned above.
The following main results are obtained by using the least action principle and by the minimax methods.
Theorem 1.1 Suppose that satisfies assumption (A) and the following conditions:
(F1) there exist constants , , and a positive function with the properties:
(q1) , and ,
(q2) , ,
(q3) as ,
(q4) as ,
where . Moreover, there exist and such that
for all and a.e. ;
(F2) there exists a positive function which satisfies the conditions (q1)-(q4) and
(F3) there exists such that
for all .
Then problem (1) has at least one solution which minimizes the functional φ on given by
for , where
is a Hilbert space with the norm defined by
Remark 1 Theorem 1.1 extends Theorem 1.1 in , in which it is a special case of our Theorem 1.1 corresponding to . There are functions satisfying the assumptions of our Theorem 1.1 and not satisfying the corresponding assumptions in [1–20]. For example, let
with , which is bounded from below, and
where is convex in (e.g., ), , and for a.e. . Let
Then , and it is easy to see the (F1), (F2), and (F3) conditions are satisfied. Then by Theorem 1.1, we conclude that problem (1) has at least one solution which minimizes the functional φ in . We note that F does not satisfy those of the results given in [1–20] (e.g., F does not satisfy (S1) of Theorem 1.1 in [1, 3], (ii) of Theorem 2 in , (8) of Theorem 2 in  and (4) of Theorem 1 in , …).
Replacing (3) with the following condition:
we then obtain Theorem 1.2 by the Saddle Point Theorem (see Theorem 4.6 in ).
Theorem 1.2 Suppose that satisfies assumption (A), (F1), (F3), and (5). Assume that there exist , such that
for all . The problem (1) has at least one solution in .
Remark 2 We note that Theorem 1.2 generalizes Theorem 1.2 in , which is the special case of our Theorem 1.2 corresponding to . There are functions satisfying the assumptions of our Theorem 1.2, but not satisfying the corresponding assumptions in [1–20]. For example, let
with , which is bounded from above, and
where satisfies the requirement that is Lipschitz continuous and monotone in (e.g., ), and . Take
Theorem 1.3 Let the hypotheses of Theorem 1.2 be satisfied. Again, assume that there exist , and an integer such that
for all and a.e. , and
for all and a.e. , where . Then problem (1) has at least one non-trivial solution in .
Theorem 1.4 Let the hypotheses of Theorem 1.1 be satisfied. Again, assume that there exist and an integer such that
for all and a.e. . Then problem (1) has at least two non-trivial solutions in .
For , let
Then one has
It follows from assumption (A) that the corresponding function φ on given by
is continuously differentiable and weakly lower semi-continuous on (cf. , pp.12-13). Moreover, one has
for all . It is well known that the solutions to problem (1) correspond to the critical point of φ.
In order to prove our main theorems, we need the following lemmas.
Lemma 2.1 (Lemma 2.1 of )
Suppose that there exists a positive function q which satisfies the conditions (q1), (q3), (q4) of (F1), then we have the following estimates:
, , , ,
Lemma 2.2 (Theorem 4 of )
Let X be a Banach space with a direct sum decomposition with , and let φ be a function on X with , satisfying . Assume that for some
Again, assume that φ is bounded from below and . Then φ has at least two non-zero critical points.
3 The proof of main results
For the sake of convenience, we will denote various positive constants as , . Now, we are ready to prove our main result, Theorem 1.1.
Proof of Theorem 1.1 It follows from (F1), Lemma 2.1, and Sobolev’s inequality that
for . From (F3) and Wirtinger’s inequality we obtain
for . Hence we have
Taking into account Lemma 2.1 and (F2), one has
As if and only if , for ε small enough, by (12) and (13) one deduces that
Hence, by the least action principle, the problem (1) has at least one solution which minimizes the functional φ in . □
Proof of Theorem 1.2 First we prove that φ satisfies the condition. Suppose that is a sequence of φ, that is,
and is bounded. In a way similar to that the proof of Theorem 1.1 above, we have
for all n. Hence one has
for large n. On the other hand, it follows from Wirtinger’s inequality that
for large n. Combining (14) with (15), we obtain
for all large n and ε small enough. It follows from (6), Cauchy-Schwarz’s inequality, and Wirtinger’s inequality that
for all n. By the proof Theorem 1.1 we have
for all n. By (16), (17), (18), Lemma 2.1, and (5), one has
as . This contradicts the boundedness of .
Thus is bounded. Notice (16) and (a) of Lemma 2.1, is bounded, and by following the same arguments used as Proposition 4.1 in  we conclude that the condition is satisfied.
We now prove that φ satisfies the other conditions of the Saddle Point Theorem. Let be the subspace of given by
Then one has
as in . In fact, by the proof of Theorem 1.1 we have
for all . In addition, by (F1), Sobolev’s inequality and Lemma 2.1, we have
Hence one has
By Wirtinger’s inequality, one has if and only if on . Hence, for ε small enough, this implies (19) by (15) and (20).
On the other hand, by (5) and Lemma 2.1, we get
as in .
Now, Theorem 1.2 is proved by (19), (21), and the application of the Saddle Point Theorem. □
Proof of Theorem 1.3 Let ,
Then satisfies the condition by the proof of Theorem 1.2. In view of Theorem 5.29 and Example 5.26 in  (e.g., the Generalized Mountain Pass Theorem), we only need to prove that
() as in ,
() for all , and
() as in .
By (F1) and Lemma 2.1, one has
for all and a.e. . From (6), it follows that
for all and a.e. . From (23) and (24), we obtain
for all , a.e. and some given by
Now, it follows from (8) that
for all and a.e. . Hence, we obtain
for all . Then () follows from the above inequality. For , by (7), one has
which is (). Finally, () follows from (21). Hence the proof of Theorem 1.3 is completed. □
Proof of Theorem 1.4 From the proof of Theorem 1.1 we know that φ is coercive, which implies that φ satisfies the condition. In a manner similar to ones used by the literature of [10, 11], we can get the multiplicity results. For convenience of the readers, we give details.
Let be the finite-dimensional subspace given by (22) and let . Then by (9) we have
for all with and
for all with , where is the positive constant such that for all .
The case that
for some , implies . Now our Theorem 1.4 follows from Lemma 2.2.
On the contrary, we have
for all . Then it follows from (9) that for every given one has
for a.e. . Let
Then for all . Given we have
for , where is the canonical basis of . Thus we obtain
for all , which implies that for a.e. , that is, is a solution to problem (1). Hence all are the solutions to problem (1). Therefore, Theorem 1.4 is proved. □
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All authors were supported by Science and Technology Foundation of the Guizhou Province (no. LKM31; no. LKB19; no. 2141). The second author was also supported by the Six Talent Peaks Project of Jiangsu Province (No. DZXX-028). The authors would like also to thank the referees for their valuable suggestions and comments which improve the exposition of this paper.
The authors declare that they have no competing interests.
All authors contributed equally to the writing of this paper. All authors read and approved the final manuscript.
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Suo, H., Di, L., An, Y. et al. Existence and multiplicity of periodic solutions for some second-order Hamiltonian systems. J Inequal Appl 2014, 411 (2014). https://doi.org/10.1186/1029-242X-2014-411
- second-order systems
- periodic solution
- Sobolev’s inequality
- Wirtinger’s inequality