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Infinitely many periodic solutions of planar Hamiltonian systems via the Poincaré–Birkhoff theorem
 Zaihong Wang^{1}Email author and
 Tiantian Ma^{2}
 Received: 26 December 2017
 Accepted: 23 June 2018
 Published: 4 July 2018
Abstract
In this paper, we study the multiplicity of periodic solutions of one kind of planar Hamiltonian systems with a nonlinear term satisfying semilinear conditions. Using a generalized Poincaré–Birkhoff fixed point theorem, we prove that the system has infinitely many periodic solutions, provided that the time map tends to zero.
Keywords
 Planar Hamiltonian system
 Time map
 Periodic solution
MSC
 34C25
 34C15
1 Introduction
Using a generalized Poincaré–Birkhoff fixed point theorem and the phaseplane analysis method, we prove the following results.
Theorem 1.1
Theorem 1.2
Corollary 1.3
Assume that conditions (\(h_{2}\)), (\(h_{4}\)), and (1.4) hold. Then the conclusions of Theorems 1.1 and 1.2 still hold.
Remark 1.4
Ding and Zanolin [7] proved the multiplicity of periodic solutions of Eq. (1.3) when conditions (\(h_{1}\)) and (\(h_{3}\)) hold and \(p(t,x)\) is bounded. Note that Eq. (1.3) is equivalent to the planar Hamiltonian system \(x'=y\), \(y'=g(x)+p(t,x)\), which is a particular form of (1.1). Therefore our conclusions generalize the main results in [7].
Remark 1.5
We will prove the above results under the additional assumption that the solutions to Cauchy problems of (1.1) are unique. It is shown in Sect. 4 that this requirement is not restrictive and that our results are valid when the uniqueness of the solutions to Cauchy problems is not satisfied.
Throughout this paper, by R and N we denote the sets of real and natural numbers, respectively.
2 Several lemmas
Lemma 2.1
Assume that conditions (\(h_{i}\)) (\(i=1,2,4\)) hold. Then every solution \((x(t), y(t))\) of (1.1) exists on the whole taxis.
Proof
On the basis of the global existence of solutions of (1.1), we can get the elasticity property of solutions of (1.1) by using a classical result (Theorem 6.5 in [15]).
Lemma 2.2
 (1)
If \(x_{0}^{2}+y_{0}^{2}\leq R_{1}^{2}\), then \(x(t)^{2}+y(t)^{2}\leq R_{2}^{2}\), \(t\in[0, T]\).
 (2)
If \(x_{0}^{2}+y_{0}^{2}\geq R_{2}^{2}\), then \(x(t)^{2}+y(t)^{2}\geq R_{1}^{2}\), \(t\in[0, T]\).
Lemma 2.3
Proof
Lemma 2.4
Proof
The proof of Lemma 2.4 is thus complete. □
3 Proof of main theorems
First, we recall a generalized version of the Poincaré–Birkhoff fixed point theorem by Rebelo [19].
Remark 3.1
The assumption on the starshaped boundaries of the annulus is a delicate hypothesis. Martins and Ureña [17] showed that the starshapedness assumption on the interior boundary is not eliminable. Le Calvez and Wang [16] then proved that the starshapedness of the exterior boundary is also necessary, although this assumption was not made in Ding’s theorem [8].
Proof of Theorem 1.1
Proof of Theorem 1.2
4 Remarks
Theorem 4.1
([13])
Remark 4.2
 (\(h_{1}'\)):

\(\lim_{s\to\infty}\operatorname{sgn}(s)g_{i}(s)=+\infty\) (\(i=1,\ldots, N\)).
 (\(h_{2}'\)):

There are positive constants \(\alpha_{i}\), \(\beta_{i}\) such that$$0< \alpha_{i}=\liminf_{s\to+\infty}\frac{f_{i}(s)}{s}\leq \limsup_{s\to+\infty}\frac{f_{i}(s)}{s}=\beta_{i}< +\infty. $$
 (\(h_{3}'\)):

There are positive constants \(M_{i}\) such that$$\bigl\vert p_{1i}(t, x, y) \bigr\vert \leq M_{i}, \qquad \bigl\vert p_{2i}(t, x, y) \bigr\vert \leq M_{i}\quad \mbox{for all } t, x, y\in\mathbf{R} \mbox{ and } i=1, \ldots, N. $$
 (\(h_{4}'\)):

The time maps \(\tau_{i}(c)\) satisfy \(\lim_{c\to \infty}\tau_{i}(c)=0\), where \(\tau_{i}(c)\) are defined like \(\tau(c)\) in Sect. 1.
Using Theorem 4.1, we can prove that (4.3) has infinitely many 2πperiodic solutions and, for any integer \(m\geq2\), (4.3) has infinitely many \(2m\pi\)periodic solutions that are not \(2k\pi \)periodic for \(1\leq k\leq m1\), provided that conditions (\(h_{i}'\)) (\(i=1, \ldots, 4\)) hold. For brevity, we omit the details.
Declarations
Acknowledgements
The authors are grateful to the referees for many valuable suggestions to make the paper more readable.
Availability of data and materials
Not applicable.
Funding
Research supported by the National Nature Science Foundation of China, No. 11501381.
Authors’ contributions
ZW proved the global existence of the solution of any Cauchy problem. TM proved the other conclusions and helped to draft the manuscript. Both authors read and approved the final manuscript.
Competing interests
The authors declare that they have no competing interests.
Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.
Authors’ Affiliations
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