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 theoremMSC
34C11 34C15 34C251 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
References
 Ding, T, Iannacci, R, Zanolin, F: Existence and multiplicity results for periodic solutions of semilinear Duffing equations. J. Differ. Equ. 105, 364409 (1993) MathSciNetView ArticleMATHGoogle Scholar
 Ding, T: An infinite class of periodic solutions of periodically perturbed Duffing equations at resonance. Proc. Am. Math. Soc. 86, 4754 (1982) MathSciNetView ArticleMATHGoogle Scholar
 Hao, D, Ma, S: Semilinear Duffing equations crossing resonance points. J. Differ. Equ. 133, 98116 (1997) MathSciNetView ArticleMATHGoogle Scholar
 Qian, D: Time maps and Duffing equations across resonant points. Sci. China Ser. A 23, 471479 (1993) Google Scholar
 Xia, J, Wang, Z: Existence and multiplicity of periodic solutions for the Duffing equation with singularity. Proc. R. Soc. Edinb., Sect. A 137, 625645 (2007) MathSciNetView ArticleMATHGoogle Scholar
 Boscaggin, A: Subharmonic solutions of planar Hamiltonian systems: a rotation number approach. Adv. Nonlinear Stud. 11, 77103 (2011) MathSciNetView ArticleMATHGoogle Scholar
 Boscaggin, A, Garrione, M: Resonance and rotation numbers for planar Hamiltonian systems: multiplicity results via the PoincaréBirkhoff theorem. Nonlinear Anal. 74, 41664185 (2011) MathSciNetView ArticleMATHGoogle Scholar
 Fabry, C, Fonda, A: Periodic solutions of perturbed isochronous Hamiltonian systems at resonance. J. Differ. Equ. 214, 299325 (2005) MathSciNetView ArticleMATHGoogle Scholar
 Fonda, A, Ghirardelli, L: Multiple periodic solutions of Hamiltonian systems in the plane. Topol. Methods Nonlinear Anal. 36, 2738 (2010) MathSciNetMATHGoogle Scholar
 Fonda, A, Sabatini, M, Zanolin, F: Periodic solutions of perturbed Hamiltonian systems in the plane by the use of the PoincaréBirkhoff theorem. Topol. Methods Nonlinear Anal. 40, 2952 (2012) MathSciNetMATHGoogle Scholar
 Fonda, A, Sfecci, A: A general method for the existence of periodic solutions of differential systems in the plane. J. Differ. Equ. 252, 13691391 (2012) MathSciNetView ArticleMATHGoogle Scholar
 Rebelo, C: A note on the PoincaréBirkhoff fixed point theorem and periodic solutions of planar systems. Nonlinear Anal. 29, 291311 (1997) MathSciNetView ArticleMATHGoogle Scholar
 Martins, R, Ureña, AJ: The starshaped condition on Ding’s version of the PoincaréBirkhoff theorem. Bull. Lond. Math. Soc. 39, 803810 (2007) MathSciNetView ArticleMATHGoogle Scholar
 Le Calvez, P, Wang, J: Some remarks on the PoincaréBirkhoff theorem. Proc. Am. Math. Soc. 138, 703715 (2010) View ArticleMATHGoogle Scholar
 Ding, W: A generalization of the PoincaréBirkhoff theorem. Proc. Am. Math. Soc. 88, 341346 (1983) MATHGoogle Scholar
 Fonda, A, Ureña, AJ: A higher dimensional PoincaréBirkhoff theorem for Hamiltonian flows. Ann. Inst. Henri Poincaré, Anal. Non Linéaire (2016). doi:https://doi.org/10.1016/j.anihpc.2016.04.002 Google Scholar
 Calamai, A, Sfecci, A: Multiplicity of periodic solutions for systems of weakly coupled parametrized second order differential equations. Nonlinear Differ. Equ. Appl. (2017). doi:https://doi.org/10.1007/s0003001604275 Google Scholar
 Fonda, A, Sfecci, A: Periodic solutions of weakly coupled superlinear systems. J. Differ. Equ. 260, 21502162 (2016) View ArticleMATHGoogle Scholar