 Research
 Open access
 Published:
Periodic solutions of planar Hamiltonian systems with asymmetric nonlinearities
Boundary Value Problems volume 2017, Article number: 46 (2017)
Abstract
In this paper, we look for periodic solutions of planar Hamiltonian systems
By using the PoincaréBirkhoff twist theorem, we prove the existence and multiplicity of periodic solutions of the given system when f satisfies an asymmetric condition and the related time map satisfies an oscillating condition.
1 Introduction
In this paper, we are concerned with the existence and multiplicity of periodic solutions of planar Hamiltonian systems
where \(f, g: \mathbf{R}\to\mathbf{R}\) are continuous, \(p_{i}: \mathbf{R}^{2}\to\mathbf{R}\) (\(i=1,2\)) are continuous and 2πperiodic with the first variable t.
In the case when \(f(y)\equiv y\), \(p_{1}(t, y)\equiv0\) and \(p_{2}(t, x)=p(t)\), system (1.1) becomes
which is equivalent to the differential equation
The existence and multiplicity of periodic solutions of Eq. (1.2) have been widely studied in the literature (see [1–5] and the references therein). Recently, the periodic solutions of planar Hamiltonian systems have been studied with an increasing interest (see [6–11]). In [11], Fonda and Sfecci studied the periodic solutions of the planar Hamiltonian systems of the type
Assume that the following conditions hold:
and
where \(a^{\pm}\), \(a_{\pm}\), \(b^{\pm}\) and \(b_{\pm}\) are positive constants. It was proved in [11] that system (1.3) has at least one 2πperiodic solution provided that there exists an integer \(n>0\) such that
and
In the present paper, we shall deal with the periodic solutions of system (1.1) when the nonresonant conditions (1.4) and (1.5) do not hold. Assume the following conditions hold:
 \((h_{1})\) :

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
$$\lim_{y\to+\infty}\frac{f(y)}{y}=a, \qquad \lim _{y\to\infty}\frac{f(y)}{y}=b. $$
It is well known that the time map plays an important role in studying the periodic solutions of Eq. (1.2) (see [1, 2, 5] and the references therein). In this paper, we also use the time map to study the periodic solutions of system (1.1). Let us set
Under condition \((h_{1})\), we can define the time map
for \(c>0\) large enough, where \(w(c)\) and \(d(c)\) satisfy \(w(c)<0<d(c)\) and \(G(w(c))=G(d(c))=c\). Assume that the time map \(\tau(c)\) satisfies the condition:
 \((\tau)\) :

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 moreover
$$\tau(a_{k})< \frac{2\pi}{mn}\sigma, \quad\quad \tau(b_{k})>\frac{2\pi}{mn}+\sigma, $$where \(m=\frac{1}{\sqrt{a}}+\frac{1}{\sqrt{b}}\) and a, b are given in condition \((h_{4})\).
We prove the following theorem.
Theorem 1.1
Assume that conditions \((h_{i})\) (\(i=1,\ldots,4\)) and \((\tau)\) hold. Then system (1.1) has infinitely many 2πperiodic solutions \(\{(x_{k}(t), y_{k}(t))\}_{k=1}^{\infty}\) which satisfy
Moreover, for each integer \(k\geq1\), both \(x_{k}(t)\) and \(y_{k}(t)\) have exactly 2n simple zeros in \([0, 2\pi)\).
From Theorem 1.1 we can obtain the following corollary.
Corollary 1.2
Assume that a, b are two positive constants, \(e, p:\mathbf{R}\to\mathbf{R}\) are continuous and conditions \((h_{i})\) (\(i=1,2\)) and \((\tau)\) are satisfied. Then the same conclusions of Theorem 1.1 still hold for the system
Remark 1.3
From condition \((h_{4})\) we know that f can be written in the form
where \(h: \mathbf{R}\to\mathbf{R}\) is continuous and satisfies
Therefore, it suffices for us to prove the main theorem for the system
where \(p_{i}\) (\(i=1,2\)) satisfy condition \((h_{3})\). In the case \(a=b\neq1\), by introducing a rescaling of the time \(s=at\), \(u(s)=x(\frac{s}{a})\), \(v(s)=y(\frac{s}{a})\), we find the equivalent system of (1.1)′ (having the classical form)
Such a rescaling cannot be easily applied in the case \(a\neq b\) because we do not know when the solution will change its sign.
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
At first, we consider the auxiliary autonomous system
The orbits of system (2.1) are curves determined by the equation
where c is an arbitrary constant. We can easily prove the following lemma.
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.
From Lemma 2.1 we know that, for \(c\geq c_{0}\), each \(\Gamma_{c}\) intersects with the xaxis at two points \((w(c), 0)\) and \((d(c), 0)\), where \(w(c)\) and \(d(c)\) are continuous and satisfy
Let \((x_{c}(t), y_{c}(t))\) be the solution of system (2.1) lying on the curve \(\Gamma_{c}\) with \(c\geq c_{0}\). Obviously, \((x_{c}(t), y_{c}(t))\) is periodic. Let us denote by \(T(c)\) the least period of \((x_{c}(t), y_{c}(t))\). From the first equation of (2.1) and (2.2) we have that
By the definition, \(T(c)\) is continuous for \(c\geq c_{0}\).
Now we perform some phaseplane analysis for system (1.1)′. Let \((x(t), y(t))=(x(t, x_{0}, y_{0}), y(t, x_{0}, y_{0}))\) be the solution of system (1.1)′ satisfying the initial condition
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. □
According to Lemma 2.2, the Poincaré map \(P: \mathbf{R}^{2}\to \mathbf{R}^{2}\) is well defined by
Clearly, the Poincaré map P is an areapreserving homeomorphism. The fixed points of P correspond to the 2π periodic solutions of system (1.1)′.
Now, we take the polar coordinates transformation \(x=r\cos\theta\), \(y=r\sin\theta\) to system (1.1)′. Under this transformation, system (1.1)′ becomes
Denote by \((r(t), \theta(t))=(r(t, r_{0}, \theta_{0}), \theta(t, r_{0}, \theta_{0}))\) the solution of (2.3) with the initial value
with \(x_{0}=r_{0}\cos\theta_{0}\), \(y_{0}=r_{0}\sin\theta_{0}\). Clearly, the Poincaré map P can be written in the polar coordinate form \(P: (r_{0}, \theta_{0})\to(r^{*}, \theta^{*})\) with
where l is an arbitrary integer.
Applying the polar coordinate transformation \(x=\rho\cos\varphi\), \(y=\rho\sin\varphi\) to system (2.1), we get
Denote by \((\rho(t), \varphi(t))=(\rho(t, \rho_{0}, \varphi_{0}), \varphi(t, \rho_{0}, \varphi_{0}))\) the solution of (2.4) satisfying the initial value
Using conditions \((h_{i})\) (\(i=1,2,3\)), it is not hard to prove the following lemma.
Lemma 2.3
Assume that conditions \((h_{i})\) (\(i=1,2,3\)) hold. Then there exist constants \(\gamma>1\) and \(R_{0}>0\) such that
In particular, under conditions \((h_{i})\) (\(i=1, 2\)), \(\rho(t)\) satisfies the inequality
Lemma 2.4
Assume that conditions \((h_{i})\) (\(i=1,2,3\)) hold, and let
Then, for any sufficiently small ε, there exists a positive constant ζ such that
Proof
Let \((\bar{x}(t), \bar{y}(t))=(\bar{x}(t, x_{0}, y_{0}), \bar{y}(t, x_{0}, y_{0}))\) be the solution of (2.1) with \((\bar{x}(0), \bar{y}(0))=(x_{0}, y_{0})\). It is noted that \((x(t), y(t))=(x(t, x_{0}, y_{0}), y(t, x_{0}, y_{0}))\) is a solution of system (1.1)′ with \((x(0), y(0))=(x_{0}, y_{0})\). Set
Then we have
Let \(d(t)=\sqrt{u^{2}(t)+v^{2}(t)}\). Then we get
with \(\delta=\frac{1}{2}(\mu+L)\), \(\mu=\max\{a, b\}\). From condition \((h_{3})\) we have that, for any sufficiently small \(\eta>0\), there exists \(c_{\eta}>0\) such that
and
Therefore, we obtain
Solving this inequality, we get
where \(A_{\eta}=\frac{2c_{\eta} e^{2\pi\delta}}{\delta}\). It follows from Lemma 2.3 that, for \(t\in[0, 2\pi]\),
where \(\beta=2\sqrt{2}\pi\gamma e^{2\pi\delta}\). Write \(\psi(t)=\psi(t, r_{0}, \theta_{0})=\varphi(t, r_{0}, \theta_{0})\theta(t, r_{0},\theta_{0})\). Clearly, if \(\psi(t)<\pi\), then \(\psi(t)\) is just the angle between the vectors \((x(t), y(t))\) and \((\bar{x} (t), \bar{y}(t))\). Hence, we have
It follows that
According to Lemma 2.3, we have that if η is sufficiently small and \(r_{0}\) is large enough, then
Since \(\psi(0)=0\) and \(\psi(t)\) varies continuously as t increases from 0 to 2π, we have
Consequently, we have that there exists \(\zeta>0\) such that, for \(r_{0}\geq\zeta\),
□
Lemma 2.5
Assume that conditions \((h_{i})\) (\(i=1,2\)) and \((\tau)\) hold. Then there exists a constant \(\omega>0\) such that, for \(t\in\mathbf{R}\) and k large enough,
with \((\rho_{0}\cos\varphi_{0}, \rho_{0}\sin\varphi_{0})\in\Gamma_{a_{k}}\) or \((\rho_{0}\cos\varphi_{0}, \rho_{0}\sin\varphi_{0})\in\Gamma_{b_{k}}\).
Proof
From the definition of \(T(c)\) and condition \((\tau)\) we know that, for each \(k\in\mathbf{N}\),
In what follows, without loss of generality, we assume that the sequence \(\{T(b_{k})\}\) is bounded. Otherwise, we can replace the sequence \(\{T(b_{k})\}\) with a bounded one because \(T(c)\) is continuous for c large enough. We shall only deal with the first case, and the second one can be proved similarly. Let us set
Obviously, \(d_{k}\to+\infty\), \(w_{k}\to\infty\) as \(k\to\infty\). Next, we prove that there exist two positive constants \(\nu_{i}\) (\(i=1, 2\)) such that
Assume by contradiction that
Then there exists a subsequence of \(\{d_{k}\}\) (we still denote it by \(\{d_{k}\}\)) such that
Set
We have that \(\varepsilon_{k}\to0\) as \(k\to\infty\). From condition \((h_{2})\) we know that, for \(0< x\leq d_{k}\),
For simplicity, we assume \(g(0)=0\). Then we get that, for \(0< x\leq d_{k}\),
Consequently, we have that, for \(0\leq x\leq d_{k}\),
It follows that, for \(0\leq x\leq d_{k}\),
From the definition of T we have that
Since
we have that
This is a contradiction because \(T(a_{k})\) is a bounded sequence. Therefore, there exists a constant \(\nu_{1}>0\) such that
Similarly, there exists \(\nu_{2}>0\) such that
From condition \((h_{1})\) and (2.5), (2.6) we know that there exists sufficiently small \(\varepsilon_{0}>0\) such that, for k large enough and \(x\in[(1\varepsilon_{0})d_{k}, d_{k}]\),
Therefore, if \(\rho(t)\cos\varphi(t)\in[(1\varepsilon_{0})d_{k}, d_{k}]\), then we have
where \(\omega_{1}=\min\{a, b, \frac{1}{4}\nu_{1}\}\). Next, we deal with the case \(\rho(t)\cos\varphi(t)\in[0, (1\varepsilon_{0})d_{k}]\). Set \(x_{k}=(1\varepsilon_{0})d_{k}\). Assume that the line \(x=x_{k}\) intersects with the curve \(\Gamma_{a_{k}}\) at two points \((x_{k}, y_{k}^{+})\) and \((x_{k}, y_{k}^{})\) with \(y_{k}^{}<0<y_{k}^{+}\). Then we have
Therefore, we get
From (2.7) we have
and
Set
From condition \((h_{1})\) we know that there exists \(A>0\) such that \(g(x)\geq0\) for \(x\geq A\). If \(\rho(t)\cos\varphi(t)\in[A, (1\varepsilon_{0})d_{k}]\) and \(\rho(t)\sin\varphi(t)\geq0\), then we have
If \(\rho(t)\cos\varphi(t)\in[0, A]\) and \(\rho(t)\sin\varphi (t)\geq 0\), then we have that, for \(\rho_{0}\) large enough,
Similarly, we have that, if \(\rho(t)\cos\varphi(t)\in[A, (1\varepsilon_{0})d_{k}]\) and \(\rho(t)\sin\varphi(t)\leq0\), then we have
If \(\rho(t)\cos\varphi(t)\in[0, A]\) and \(\rho(t)\sin\varphi (t)\leq 0\), then we have that, for \(\rho_{0}\) large enough,
In conclusion, we have proved that there exists \(\omega_{2}>0\) such that
with \((\rho_{0}\cos\varphi_{0}, \rho_{0}\sin\varphi_{0})\in\Gamma_{a_{k}}\), \(\rho(t)\cos\varphi(t)\geq0\) and k large enough. Similarly, we can prove that there exists \(\omega_{2}'>0\) such that
with \((\rho_{0}\cos\varphi_{0}, \rho_{0}\sin\varphi_{0})\in\Gamma_{a_{k}}\), \(\rho(t)\cos\varphi(t)\leq0\) and k large enough. Let us set \(\omega=\min\{\omega_{2}, \omega_{2}'\}\). Then we have that
with \((\rho_{0}\cos\varphi_{0}, \rho_{0}\sin\varphi_{0})\in\Gamma_{a_{k}}\) and k large enough. □
Lemma 2.6
Assume that conditions \((h_{i})\) (\(i=1,2\)) and \((\tau)\) hold, and let \(\Phi(\rho_{0}, \varphi_{0})=\varphi(2\pi, \rho_{0}, \varphi_{0})\varphi_{0}\). Then there exist two positive constants δ and \(\varrho_{0}\) such that
Proof
From Lemma 2.5 we have that there exists \(\varrho_{0}>0\) such that, for \(a_{k}\geq\varrho_{0}\) or \(b_{k}\geq\varrho_{0}\),
Write \(\Phi(\rho_{0}, \varphi_{0})=2l\pi\phi\), where \(l\geq0\) is an integer, \(0\leq\phi<2\pi\). Let us denote by \(t_{\phi}\) the time for \(\varphi(t)\) to decrease from \(2l\pi\) to \(2l\pi\phi\). If \((\rho_{0}\cos\varphi_{0}, \rho_{0}\sin\varphi_{0})\in\Gamma_{a_{k}}\), then we have
Since \(t_{\phi}\leq T(a_{k})=m\tau(a_{k})\), we have
It follows that \(l\geq n\). If \(l\geq n+1\), then we have
If \(l=n\), then we have
Therefore, we get
Furthermore,
Set \(\delta=\min\{2\pi, nm\sigma\omega\}\). Then we have
Similarly, we can prove
□
3 Proof of the main theorem
At first, we recall a generalized version of the PoincaréBirkhoff fixed point theorem by Rebelo [12].
A generalized form of the PoincaréBirkhoff fixed point theorem. Let \(\mathcal{A}\subset\mathbf{R}^{2}\) be an annular region bounded by two strictly starshaped curves around the origin, \(\Gamma_{1}\) and \(\Gamma_{2}\), \(\Gamma_{1}\subset\operatorname{int}(\Gamma_{2})\), where \(\operatorname{int}(\Gamma_{2})\) denotes the interior domain bounded by \(\Gamma_{2}\). Suppose that \(F:\overline{\operatorname{int}(\Gamma_{2})}\to R^{2}\) is an areapreserving homeomorphism and \(F\mathcal{A}\) admits a lifting, with the standard covering projection \(\Pi :(r, \theta)\to z=(r\cos\theta, r\sin\theta)\), of the form
where w and h are continuous functions of period 2π in the first variable. Correspondingly, for \(\tilde{\Gamma}_{1}=\Pi^{1}(\Gamma_{1})\) and \(\tilde{\Gamma}_{2}=\Pi^{1}(\Gamma_{2})\), assume the twist condition
Then, F has two fixed points \(z_{1}\), \(z_{2}\) in the interior of \(\mathcal{A}\) such that
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 Lemmas 2.4 and 2.6 we know that there exists an integer \(k_{0}>0\) such that, for any \(k\geq k_{0}\),
Without loss of generality, we assume that \(a_{k}< b_{k}< a_{k+1}\) for \(k\geq k_{0}\). Let \(D_{k}\) be an annular region with boundary \(\Gamma_{a_{k}}\) and \(\Gamma_{b_{k}}\). Consider the Poincaré map \(P: D_{k}\to\mathbf{R}^{2}\). Write the Poincaré map in the form
where \(\Theta^{*}(r_{0}, \theta_{0})=\theta(2\pi, r_{0}, \theta_{0})\theta_{0}+2n\pi\). From (3.1) and (3.2) we know that, for \(k\geq k_{0}\),
Therefore, the areapreserving homeomorphism P is twisting on the annulus \(D_{k}\). On the other hand, by Lemma 2.1, we have that both \(\Gamma_{a_{k}}\) and \(\Gamma_{b_{k}}\) are starshaped with respect to the origin O for k large enough. Consequently, all assumptions of the generalized form of the PoincaréBirkhoff fixed point theorem are satisfied. Therefore, the Poincaré map P has at least two fixed points in annulus \(D_{k}\). Furthermore, system (1.1)′ has infinitely many 2πperiodic solutions \(\{(x_{k}(t), y_{k}(t))\}_{k=1}^{\infty}\) which satisfy
Clearly, each 2πperiodic solution \((x_{k}(t), y_{k}(t))\) rotates clockwise strictly n turns around the origin in the interval \([0, 2\pi]\). It follows that both \(x_{k}(t)\) and \(y_{k}(t)\) have exactly 2n simple zeros in \([0, 2\pi)\). Hence, the conclusion of Theorem 1.1 holds. □
From Theorem 1.1 we know that the existence of periodic solutions of system (1.1) has tight relation with the asymptotic property of time map \(\tau(c)\). In case g is odd, we can easily check condition \((\tau)\). Set
From Theorem 3.1 in [1], we have the following lemma.
Lemma 3.2
Assume that condition \((h_{1})\) holds, g is odd and \(G_{+}< G^{+}\). Then
where \(\tau_{+}=\liminf_{c\to+\infty}\tau(c)\), \(\tau^{+}=\limsup_{c\to+\infty}\tau(c)\).
Applying Theorem 1.1 and Lemma 3.2, we can obtain the following corollary.
Corollary 3.3
Assume that conditions \((h_{i})\) (\(i=1, \ldots, 4\)) hold. Let \(g(x)\) be an odd function and \(G_{+}< G^{+}\). If
then system (1.1) has infinitely many 2πperiodic solutions \(\{(x_{k}(t), y_{k}(t))\}_{k=1}^{\infty}\) which satisfy
Moreover, for each integer \(k\geq1\), both \(x_{k}(t)\) and \(y_{k}(t)\) have exactly 2n simple zeros in \([0, 2\pi)\).
4 Concluding remarks
The restrictions on the local Lipschitz conditions of f and \(p_{i}\) (\(i=1, 2\)) made in the proofs of the above sections can be removed. Indeed, Lemmas 2.4 and 2.6 guarantee the applicability of the following nonuniqueness version of the PoincaréBirkhoff theorem which was proved by Fonda and Ureña in [16]. We now state this theorem for a general Hamiltonian system in \(\mathbf{R}^{2N}\). Let us consider the (timedependent) Hamiltonian system
where the continuous function \(H: \mathbf{R}\times \mathbf{R}^{N}\times\mathbf{R}^{N}\to\mathbf{R}\), \(H=H(t, x, y)\) is Tperiodic in its first variable t and continuously differentiable with respect to \((x, y)\), \(x=(x_{1}, \ldots, x_{N})\), \(y=(y_{1}, \ldots, y_{N})\).
We next introduce the definition of rotation number of a continuous path in \(\mathbf{R}^{2}\). Let \(w: [t_{1}, t_{2}]\to \mathbf{R}^{2}\) be a continuous path such that \(w(t)\neq(0, 0)\) for every \(t\in[t_{1}, t_{2}]\). The rotation number of w around the origin is defined as
where \(\theta:[t_{1}, t_{2}]\to\mathbf{R}\) is a continuous determination of the argument along w, i.e., \(w(t)=w(t)(\cos\theta(t), \sin\theta(t))\).
Assume that for each \(i=1, \ldots, N\), there are two strictly starshaped Jordan closed curves around the origin \(\Gamma_{1}^{i}, \Gamma_{2}^{i}\subset\mathbf{R^{2}}\) such that
Here we denote by \(\mathcal{D}(\Gamma)\) the open bounded region bounded by the Jordan closed curve Γ. Consider the generalized annular region
Theorem 4.1
[16]
Under the framework above, denoting \(z_{i}(t)=(x_{i}(t), y_{i}(t))\), assume that every solution \(z(t)=(z_{1}(t),\ldots, z_{N}(t))\) of (4.1) departing from \(z(0)\in \partial\mathcal{A}\) is defined on \([0, T]\) and satisfies
Assume that there are integer numbers \(\nu_{1}, \ldots, \nu_{N}\in \mathbf{Z}\) such that, for each \(i=1, \ldots, N\), either
or
Then Hamiltonian system (4.1) has at least \(N+1\) distinct Tperiodic solutions \(z^{0}(t), \ldots, z^{N}(t)\), with \(z^{0}(0), \ldots, z^{N}(0)\in\mathcal{A}\), such that
Remark 4.2
It is noted that there is no requirement of uniqueness of Cauchy problems associated to system (4.1) in this higher dimensional PoincaréBirkhoff theorem for Hamiltonian flows. In [17, 18], Theorem 4.1 is applied to prove the multiplicity of periodic solutions of weakly coupled Hamiltonian systems. Theorem 1.1 in the present paper can also be extended to a weakly coupled system of the type
where \(x=(x_{1}, \ldots, x_{N})\), \(y=(y_{1}, \ldots, y_{N})\), \(f_{i}, g_{i}:\mathbf{R}\to\mathbf{R}\) (\(i=1, \ldots, N\)) are continuous, \(p_{ji}: \mathbf{R}^{2N+1}\to\mathbf{R}\) (\(j=1, 2\), \(i=1, \ldots, N\)) are continuous and 2πperiodic with the variable t. Assume that there is a function \(U: \mathbf{R}^{2N+1}\to \mathbf{R}\) such that
In this case system (4.2) is a Hamiltonian system. Simple examples of such functions can be given. For example,
Assume that the following conditions hold:
 \((h_{1}')\) :

\(g_{i}\) 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
$$\lim_{v\to+\infty}\frac{f_{i}(v)}{v}=a_{i}, \qquad \lim _{v\to\infty}\frac{f_{i}(v)}{v}=b_{i}. $$
Set \(G_{i}(u)=\int_{0}^{u}g_{i}(s)\,ds\), \(i=1, \ldots, N\). Let us define the time maps
for \(c>0\) large enough, where \(w_{i}(c)\) and \(d_{i}(c)\) satisfy \(w_{i}(c)<0<d_{i}(c)\) and \(G_{i}(w_{i}(c))=G_{i}(d_{i}(c))=c\). Assume that the time map \(\tau_{i}(c)\) satisfies the condition:
 \((\tau')\) :

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 moreover
$$\tau_{i}\bigl(a_{k}^{i}\bigr)< \frac{2\pi}{m_{i}n_{i}} \sigma_{i}, \quad\quad \tau_{i}\bigl(b_{k}^{i} \bigr)>\frac{2\pi}{m_{i}n_{i}}+\sigma_{i}, \quad i=1, \ldots, N, $$where \(m_{i}=\frac{1}{\sqrt{a_{i}}}+\frac{1}{\sqrt{b_{i}}}\) and \(a_{i}\), \(b_{i}\) are given in condition \((h_{4}')\).
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
Assume that conditions \((h_{i}')\) (\(i=1, \ldots, 4\)) and \((\tau')\) hold. Then Hamiltonian system (4.2) has infinitely many 2πperiodic solutions \(\{(x^{k}(t), y^{k}(t))\}_{k=1}^{\infty}\) which satisfy
where \(x^{k}(t)=\sum_{i=1}^{i=N}x_{i}^{k}(t)\), \(y^{k}(t)=\sum_{i=1}^{i=N}y_{i}^{k}(t)\). Moreover, for each index i, both \(x^{k}_{i}(t)\) and \(y^{k}_{i}(t)\) have exactly \(2n_{i}\) simple zeros in the interval \([0, 2\pi)\).
References
Ding, T, Iannacci, R, Zanolin, F: Existence and multiplicity results for periodic solutions of semilinear Duffing equations. J. Differ. Equ. 105, 364409 (1993)
Ding, T: An infinite class of periodic solutions of periodically perturbed Duffing equations at resonance. Proc. Am. Math. Soc. 86, 4754 (1982)
Hao, D, Ma, S: Semilinear Duffing equations crossing resonance points. J. Differ. Equ. 133, 98116 (1997)
Qian, D: Time maps and Duffing equations across resonant points. Sci. China Ser. A 23, 471479 (1993)
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)
Boscaggin, A: Subharmonic solutions of planar Hamiltonian systems: a rotation number approach. Adv. Nonlinear Stud. 11, 77103 (2011)
Boscaggin, A, Garrione, M: Resonance and rotation numbers for planar Hamiltonian systems: multiplicity results via the PoincaréBirkhoff theorem. Nonlinear Anal. 74, 41664185 (2011)
Fabry, C, Fonda, A: Periodic solutions of perturbed isochronous Hamiltonian systems at resonance. J. Differ. Equ. 214, 299325 (2005)
Fonda, A, Ghirardelli, L: Multiple periodic solutions of Hamiltonian systems in the plane. Topol. Methods Nonlinear Anal. 36, 2738 (2010)
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)
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)
Rebelo, C: A note on the PoincaréBirkhoff fixed point theorem and periodic solutions of planar systems. Nonlinear Anal. 29, 291311 (1997)
Martins, R, Ureña, AJ: The starshaped condition on Ding’s version of the PoincaréBirkhoff theorem. Bull. Lond. Math. Soc. 39, 803810 (2007)
Le Calvez, P, Wang, J: Some remarks on the PoincaréBirkhoff theorem. Proc. Am. Math. Soc. 138, 703715 (2010)
Ding, W: A generalization of the PoincaréBirkhoff theorem. Proc. Am. Math. Soc. 88, 341346 (1983)
Fonda, A, Ureña, AJ: A higher dimensional PoincaréBirkhoff theorem for Hamiltonian flows. Ann. Inst. Henri Poincaré, Anal. Non Linéaire (2016). doi:10.1016/j.anihpc.2016.04.002
Calamai, A, Sfecci, A: Multiplicity of periodic solutions for systems of weakly coupled parametrized second order differential equations. Nonlinear Differ. Equ. Appl. (2017). doi:10.1007/s0003001604275
Fonda, A, Sfecci, A: Periodic solutions of weakly coupled superlinear systems. J. Differ. Equ. 260, 21502162 (2016)
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.
Author information
Authors and Affiliations
Corresponding author
Additional information
Competing interests
The authors declare that they have no competing interests.
Authors’ contributions
ZW proved Lemma 2.5 and gave the concluding remarks. TM proved the other conclusions and helped to draft the manuscript. All authors read and approved the final manuscript.
Publisher’s Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Rights and permissions
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.
About this article
Cite this article
Wang, Z., Ma, T. Periodic solutions of planar Hamiltonian systems with asymmetric nonlinearities. Bound Value Probl 2017, 46 (2017). https://doi.org/10.1186/s1366101707802
Received:
Accepted:
Published:
DOI: https://doi.org/10.1186/s1366101707802