 Research
 Open Access
Periodic solutions of planar Hamiltonian systems with asymmetric nonlinearities
 Zaihong Wang^{1}Email author and
 Tiantian Ma^{2}
 Received: 19 November 2016
 Accepted: 23 March 2017
 Published: 4 April 2017
Abstract
Keywords
 asymmetric nonlinearity
 periodic solution
 PoincaréBirkhoff twist theorem
MSC
 34C11
 34C15
 34C25
1 Introduction
 \((h_{1})\) :

g satisfies \(\lim_{x\to+\infty}\operatorname{sgn}(x)g(x)=+\infty\);
 \((h_{2})\) :

there exists a constant \(L>0\) such that, for all \(x, y\in\mathbf{R}\), \(g(x)g(y)\leq Lxy\);
 \((h_{3})\) :

the limits \(\lim_{y\to+\infty}\frac{p_{i}(t,y)}{y}=0\) (\(i=1,2\)) hold uniformly with respect to \(t\in[0, 2\pi]\);
 \((h_{4})\) :

there are two positive constants a and b such that$$\lim_{y\to+\infty}\frac{f(y)}{y}=a, \qquad \lim _{y\to\infty}\frac{f(y)}{y}=b. $$
 \((\tau)\) :

There exist a constant \(\sigma>0\), an integer \(n>0\), and two sequences \(\{a_{k}\}\) and \(\{b_{k}\}\) such that \(\lim_{k\to\infty}a_{k}=+\infty\), \(\lim_{k\to\infty}b_{k}=+\infty\); and moreoverwhere \(m=\frac{1}{\sqrt{a}}+\frac{1}{\sqrt{b}}\) and a, b are given in condition \((h_{4})\).$$\tau(a_{k})< \frac{2\pi}{mn}\sigma, \quad\quad \tau(b_{k})>\frac{2\pi}{mn}+\sigma, $$
We prove the following theorem.
Theorem 1.1
From Theorem 1.1 we can obtain the following corollary.
Corollary 1.2
Remark 1.3
We finally stress the fact that the proofs of the above results will be given under the additional assumptions that f and \(p_{i}\) (\(i=1,2\)) are locally Lipschitz continuous with variables y or x. It is shown in Section 4 that this requirement is not restrictive and that our results are valid for any continuous functions f and \(p_{i}\) (\(i=1,2\)).
2 Basic lemmas
Lemma 2.1
Assume that condition \((h_{1})\) holds. Then there exists a constant \(c_{0}>0\) such that, for any \(c>c_{0}\), \(\Gamma_{c}\) is a closed curve which is starshaped around the origin O.
Lemma 2.2
Assume that conditions \((h_{i})\) (\(i=1,2,3\)) hold. Then each solution \((x(t), y(t))\) of system (1.1)′ exists uniquely on the whole taxis.
Proof
The proof follows directly from the fact that the nonlinearities are locally Lipschitz continuous and all have at most linear growth. □
Using conditions \((h_{i})\) (\(i=1,2,3\)), it is not hard to prove the following lemma.
Lemma 2.3
Lemma 2.4
Proof
Lemma 2.5
Proof
Lemma 2.6
Proof
3 Proof of the main theorem
At first, we recall a generalized version of the PoincaréBirkhoff fixed point theorem by Rebelo [12].
Remark 3.1
The assumption on the starshaped boundaries of the annulus is a delicate hypothesis. Martins and Ureña [13] showed that the starshapedness assumption on the interior boundary is not eliminable. Le Calvez and Wang [14] then proved that starshapedness of the exterior boundary should also be imposed, while this assumption was not made in Ding’s theorem [15].
Proof of Theorem 1.1
From Theorem 3.1 in [1], we have the following lemma.
Lemma 3.2
Applying Theorem 1.1 and Lemma 3.2, we can obtain the following corollary.
Corollary 3.3
4 Concluding remarks
Theorem 4.1
[16]
Remark 4.2
 \((h_{1}')\) :

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

there exist constants \(L_{i}>0\) such that, for all \(u, v\in\mathbf{R}\), \(g_{i}(u)g_{i}(v)\leq L_{i}uv\), \(i=1, \ldots, N\);
 \((h_{3}')\) :

there are constants \(M_{1}>0\), \(M_{2}>0\) and \(0<\gamma_{i}<1\) such that \(p_{ji}(t, x, y)\leq M_{1}(x_{i}+y_{i})^{\gamma_{i}}+M_{2}\) for all \(t\in[0, 2\pi]\), \((x, y)\in\mathbf{R}^{2N}\), \(j=1,2\), \(i=1, \ldots, N\);
 \((h_{4}')\) :

there are positive constants \(a_{i}\) and \(b_{i}\) such that$$\lim_{v\to+\infty}\frac{f_{i}(v)}{v}=a_{i}, \qquad \lim _{v\to\infty}\frac{f_{i}(v)}{v}=b_{i}. $$
 \((\tau')\) :

There exist constants \(\sigma_{i}>0\), integers \(n_{i}>0\), and sequences \(\{a_{k}^{i}\}\) and \(\{b_{k}^{i}\}\) such that \(\lim_{k\to\infty}a_{k}^{i}=+\infty\), \(\lim_{k\to\infty}b_{k}^{i}=+\infty\); and moreoverwhere \(m_{i}=\frac{1}{\sqrt{a_{i}}}+\frac{1}{\sqrt{b_{i}}}\) and \(a_{i}\), \(b_{i}\) are given in condition \((h_{4}')\).$$\tau_{i}\bigl(a_{k}^{i}\bigr)< \frac{2\pi}{m_{i}n_{i}} \sigma_{i}, \quad\quad \tau_{i}\bigl(b_{k}^{i} \bigr)>\frac{2\pi}{m_{i}n_{i}}+\sigma_{i}, \quad i=1, \ldots, N, $$
With a slight modification of the proof of Theorem 1.1 and using the higher dimensional PoincaréBirkhoff Theorem 4.1, we can prove the result.
Theorem 4.3
Declarations
Acknowledgements
Research supported by the National Nature Science Foundation of China, No. 11501381 and the Grant of Beijing Education Committee Key Project, No. KZ201310028031.
The authors are grateful to the referees for many valuable suggestions to make the paper more readable.
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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