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.
where \(f, g: \mathbf{R}\to\mathbf{R}\) are continuous, and \(p_{i}: \mathbf{R^{3}}\to\mathbf{R}\) (\(i=1,2\)) are continuous and 2π-periodic with respect to the first variable t.
The periodic problem for planar Hamiltonian systems is a classical topic in nonlinear analysis and ordinary differential equations, which has been widely studied in literature by using various different methods such as phase plane analysis, topological degree, fixed point theorems, variational methods (see [1–5, 10, 11, 14, 19, 20] and the references therein). For instance, using the Poincaré–Bohl fixed point theorem, Fonda and Sfecci [11] studied the existence of periodic solutions of planar Hamiltonian systems
where \(J=\bigl( {\scriptsize\begin{matrix}{}0& -1\cr 1& 0 \end{matrix}} \bigr)\) is the standard symplectic matrix, and \(H: [0, T]\times\mathbf{R}^{2}\to \mathbf{R}\) is differentiable with respect to the second variable. When \(\nabla_{z}H(t, z)\) satisfies some semilinear conditions at infinity, it was proved in [11] that (1.2) has at least one T-periodic solution. Using a generalized Poincaré–Bikhoff fixed point theorem, Boscaggin [4] studied the multiplicity of periodic solutions of (1.2), provided that \(\nabla_{z}H(t, z)\) satisfies some superlinear condition at infinity. It was pointed out that the main theorem (Theorem 2.3) in [4] applies to the forced Duffing equation
$$ x''+g(x)=p(t,x) $$
(1.3)
when g satisfies the superlinear condition
$$ \lim_{|x|\to\infty}\frac{g(x)}{x}=+\infty. $$
(1.4)
We note that Ding and Zanolin [7] proved the multiplicity of periodic solutions of Eq. (1.3) by replacing (1.4) with a weaker assumption on the time map of the autonomous equation \(x''+g(x)=0\), namely that the limit of the time map equals zero. Clearly, this condition is not included in [4].
In the present paper, we study the multiplicity of periodic solutions of (1.1) in terms of the time map. Assume that g satisfies the condition
and f satisfies the semilinear condition at infinity
Let \(G(x)=\int_{0}^{x}g(s)\,ds\). Define the function
From condition (\(h_{1}\)) we know that \(\tau(c)\) is continuous for \(|c|\) large enough; \(\tau(c)\) is usually called the time map related to the autonomous equation \(x''+g(x)=0\). The properties of \(\tau(c)\) were studied deeply in [6, 7, 18]. Assume that τ satisfies
From [18] we know that (\(h_{3}\)) holds if g satisfies the superlinear condition (1.4). Throughout the paper, we always assume that there exists a constant \(M>0\) such that
Moreover, there is a function \(\mathcal{U}: \mathbf{R}^{3}\to \mathbf{R}\) such that
where \(F(y)=\int_{0}^{y}f(s)\,ds\). Then we have that \(\nabla_{z}H(t, z)=(g(x)-p_{2}(t, x, y), f(y)+p_{1}(t, x, y))\). Since g only satisfies condition (\(h_{1}\)), we know that \(\nabla_{z}H(t, z)\) does not satisfy the semilinear condition as in [11] or the superlinear condition as in [4].
Using a generalized Poincaré–Birkhoff fixed point theorem and the phase-plane analysis method, we prove the following results.
Theorem 1.1
Assume that conditions (\(h_{i}\)) (\(i=1,\ldots,4\)) hold. Then system (1.1) has infinitely many 2π-periodic solutions\(\{(x_{j}(t),y_{j}(t))\}_{j=1}^{\infty}\)that satisfy
Assume that conditions (\(h_{i}\)) (\(i=1,\ldots,4\)) hold. Then for any given integer\(m\geq2\), system (1.1) has infinitely many\(2m\pi\)-periodic solutions\(\{(x_{j}(t), y_{j}(t))\}_{j=1}^{\infty}\)that are not\(2k\pi\)-periodic for\(1\leq k\leq m-1\)and satisfy
Assume that conditions (\(h_{2}\)), (\(h_{4}\)), and (1.4) hold. Then the conclusions of Theorems 1.1and1.2still 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.
In this section, we perform some phase-plane 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 (1.1) satisfying the initial value
Assume that conditions (\(h_{i}\)) (\(i=1,2,4\)) hold. Then every solution\((x(t), y(t))\)of (1.1) exists on the wholet-axis.
Proof
In view of (\(h_{1}\)), there exists a constant \(c_{0}>0\) such that
$$ \operatorname{sgn}(x)g(x)>0 \quad\mbox{for } \vert x \vert \geq c_{0}. $$
(2.1)
Let us define two Lyapunov-like functions \(V_{\pm}:\mathbf{R^{2}}\to \mathbf{R}\):
$$V_{\pm}(x, y)=G(x)+F(y)\pm y h(x), $$
where \(h:\mathbf{R}\to\mathbf{R}\) is a continuously differentiable function such that \(h(x)=M \operatorname{sgn}(x)\) for \(|x|\geq c_{0}\) with M from (\(h_{4}\)). We will prove that \(V_{\pm}\) are coercive, that is, \(V_{\pm}(x, y)\to+\infty\) as \(|x|+|y|\to+\infty\). From (\(h_{1}\)) we know that \(\lim_{|x|\to+\infty}G(x)=+\infty\). From (\(h_{2}\)) we get that
hold uniformly with respect to \(x\in\mathbf{R}\). Therefore \(V_{\pm}(x, y)\to+\infty\) as \(|x|+|y|\to+\infty\).
Next, we show that every solution \((x(t), y(t))\) of (1.1) is defined on the interval \([0, +\infty)\). Set \(V_{+}(t)=V_{+}(x(t), y(t))\). We first prove that there exist constants \(c_{1}>0\) and \(c_{2}>0\) such that
$$ V_{+}'(t)\leq c_{1} V_{+}(t)+c_{2}. $$
(2.3)
For simplicity, we denote \(p_{1}(t)=p_{1}(t, x(t), y(t))\), \(p_{2}(t)=p_{2}(t, x(t), y(t))\). By (1.1) we have
Since \(h'(x)=0\) for \(|x|\geq c_{0}\), we know that \(h'(x)\) is bounded, and then there exists \(\beta_{0}>0\) such that \(|h'(x)|\leq\beta_{0}\), \(x\in\mathbf{R}\). It follows from (2.4) that
Since \(h(x)\) is bounded for \(x\in\mathbf{R}\), there exists \(\beta_{0}'>0\) such that \(|h(x)|\leq\beta_{0}'\), \(x\in\mathbf{R}\). Consequently, we have
$$\bigl\vert p_{2}(t) h\bigl(x(t)\bigr) \bigr\vert \leq M \beta_{0}'. $$
with \(\beta_{1}=\beta_{0} l_{0}+\frac{1}{2}\) and \(\beta_{1}'=\frac{1}{2}(\alpha_{1}\beta_{0}+M\beta_{0}+Ml_{0})^{2}+(\alpha _{0}+M\alpha_{1}+M\beta_{0}')\). From (2.2) we know that there exist \(l_{1}>0\) and \(\beta_{2}>0\) such that
Set \(c_{1}=\beta_{1}l_{1}\) and \(c_{2}=\beta_{1}l_{1}G_{0}+\beta_{1}\beta_{2}+\beta_{1}'\). We get that \(V_{+}'(t)\leq c_{1}V_{+}(t)+c_{2}\). Then, for any finite \(\omega>0\), we have
Since \(V_{+}\) is coercive, there is no blow-up for \((x(t), y(t))\) on any finite interval \([0, \omega)\). Therefore, \((x(t), y(t))\) exists on the interval \([0, +\infty)\).
To prove that \((x(t), y(t))\) exists on the interval \((-\infty, 0]\), we consider another Lyapunov-like function \(V_{-}(x, y)\). Set \(V_{-}(t)=V_{-}(x(t), y(t))\). Using the same methods as before, we can show that there exist two positive constants \(d_{1}\), \(d_{2}\) such that
Since \(V_{-}\) is coercive, there is also no blow-up for \((x(t), y(t))\) on any finite interval \((-\omega, 0]\). Therefore, \((x(t), y(t))\) exists on the interval \((-\infty, 0]\). The proof is complete. □
Since the uniqueness of the solutions to Cauchy problems of (1.1) is assumed, we can define the Poincaré map \(P: \mathbf{R}^{2}\to\mathbf{R}^{2}\) as follows:
It is well known that the Poincaré map P is an area-preserving homeomorphism. The fixed points of P correspond to the 2π-periodic solutions of (1.1).
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
Assume that conditions (\(h_{i}\)) (\(i=1,2,4\)) hold. Then, for any\(T>0\)and\(R_{1}>0\), there exists\(R_{2}>R_{1}\)such that:
From Lemma 2.2 we know that if \(x_{0}^{2}+y_{0}^{2}\) is large enough, then \(x^{2}(t)+y^{2}(t)\neq0\), \(t\in[0, T]\). Thus we can take the polar coordinate transformation
where \(p_{1}(t,r,\theta)=p_{1}(t, r\cos\theta,r\sin\theta)\) and \(p_{2}(t,r,\theta)=p_{2}(t,r\cos\theta,r\sin\theta)\). Let us denote by \((r(t), \theta(t))=(r(t, r_{0}, \theta_{0}), \theta(t, r_{0}, \theta_{0}))\) the solution of (2.8) satisfying the initial value
with \(x_{0}=r_{0}\cos\theta_{0}\), \(y_{0}=r_{0}\sin\theta_{0}\). We can rewrite the Poincaré map P in the polar coordinate form \(P: (r_{0}, \theta_{0})\to(r_{1}, \theta_{1})\),
Assume that conditions (\(h_{i}\)) (\(i=1,2,4\)) hold. Then, for any\(T>0\), there exists a constant\(R>0\)such that if\(r(t)\geq R\), \(t\in[0, T]\), then
$$\theta'(t)< 0, \quad t\in[0, T]. $$
Proof
It follows from (\(h_{1}\)) that there exists \(a_{1}>0\) such that
$$\operatorname{sgn}(x)g(x)>M \quad\mbox{for } \vert x \vert \geq a_{1}, $$
If \(|y(t)|\leq a_{2}\) and \(r(t)\) is large enough, then we have that \(|x(t)|\) is also large enough. Therefore we get from (\(h_{1}\)), (\(h_{4}\)), and (2.11) that, for \(r(t)\) large enough,
Combining (2.9), (2.10), and (2.12), we get the conclusion of Lemma 2.3. □
Lemma 2.4
Assume that conditions (\(h_{i}\)) (\(i=1,\ldots,4\)) hold. Letmbe a positive integer. Then, for any given integer\(n\geq1\), there exists a constant\(\varrho_{n}>0\)such that, for\(r_{0}\geq\varrho_{n}\),
Next, we will estimate the time needed for the solution \((x(t), y(t))\) to pass through the regions \(D_{i}\) (\(i=1,\ldots, 8\)), respectively. Without loss of generality, we assume that \((x_{0}, y_{0})\in D_{1}\). Then, \((x(t), y(t))\) rotates following the cycle
We first treat the case \((x(t),y(t))\in D_{1}\), \(t\in[t_{1}, t_{2}]\). It follows from (\(h_{2}\)) that there exist constants \(\beta_{0}\geq \alpha_{0}>0\) and \(M_{0}>\) such that
with \(M_{1}=M_{0}+M\). From Lemma 2.2 we know that, for any constant l (\(>\sqrt{c_{m}^{2}+\frac{M_{1}^{2}}{\alpha_{0}^{2}}}\)), there is a constant \(l_{0}>l\) such that, for \(r_{0}\geq l_{0}\),
$$r(t)\geq l, \quad t\in[0, 2m\pi]. $$
As a result, we get that, for \(r_{0}\geq l_{0}\) and \((x(t),y(t))\in D_{1}\), \(t\in[t_{1}, t_{2}]\),
where \(\xi\in[x(t), x(t_{2})]\). Since \(x(t)\geq c_{m}\), \(t\in[t_{1}, t_{2}]\), we can take \(c_{m}\) large enough such that \(g(\xi)\geq2M\). Therefore, we obtain that, for \(t\in[t_{1}, t_{2}]\),
provided that \(x(t_{2})\) is large enough. Consequently, we have that \(t_{*}-t_{1}<\frac{\varepsilon}{2}\), provided that \(r_{0}\) is sufficiently large. We next estimate \(t_{2}-t_{*}\). If \(t\in[t_{*}, t_{2}]\), then we have
where \(\nu(x(t_{2}))=\min\{g(x):x(t_{2})-L\leq x\leq x(t_{2})\}\). Obviously, \(\nu(x(t_{2}))\to+\infty\) as \(x(t_{2})\to+\infty\). On the other hand, it follows from (2.14) that
which, together with \(\nu(x(t_{2}))\to+\infty\) as \(x(t_{2})\to+\infty\), implies that, for any sufficiently small \(\varepsilon>0\),
$$ t_{2}-t_{*}< \frac{\varepsilon}{2}, $$
(2.22)
provided that \(x(t_{2})\) is large enough or \(r_{0}\) is large enough. From (2.21) and (2.22) we know that, for any sufficiently small \(\varepsilon>0\),
$$t_{2}-t_{1}< \varepsilon, $$
provided that \(r_{0}\) is large enough.
Similarly, we have that, if \((x(t), y(t))\in D_{i}\), \(i=4, 6, 8\), \(t\in[t_{1}, t_{2}]\), then
$$t_{2}-t_{1}< \varepsilon, $$
provided that \(r_{0}\) is large enough.
We now consider the case \((x(t), y(t))\in D_{3}\), \(t\in[t_{1}, t_{2}]\). In this case, we have
where \(\mu_{*}=\min\{g(x(t)): t_{1}\leq t\leq t_{2}\}\). From (2.23), (\(h_{1}\)), and Lemma 2.2 we get that \(\mu_{*}\to+\infty\) as \(r_{0}\to\infty\). Therefore, we have that, for any sufficiently small \(\varepsilon>0\),
$$t_{2}-t_{1}< \varepsilon, $$
provided that \(r_{0}\) is large enough.
Similarly, we have that, if \((x(t), y(t))\in D_{7}\), \(t\in[t_{1}, t_{2}]\), then
$$t_{2}-t_{1}< \varepsilon, $$
provided that \(r_{0}\) is large enough.
From the previous conclusion we get that, for any sufficiently small \(\varepsilon>0\), there is \(\varrho_{1}>0\) such that if \(r_{0}\geq \varrho_{1}\), then \((x(t), y(t))\in D\), and if
$$\theta(s_{2})-\theta(s_{1})=-2\pi, $$
then
$$0< s_{2}-s_{1}< 8\varepsilon. $$
Consequently, the motion \((x(t),y(t))\) rotates clockwise a turn in a period less than 8ε. Therefore, for any integer \(n\geq 1\), there is \(\varrho_{n}>0\) such that, for \(r_{0}\geq\varrho_{n}\), the motion \((x(t), y(t))\) can rotate more than n turns during the period \(2m\pi\).
First, we recall a generalized version of the Poincaré–Birkhoff fixed point theorem by Rebelo [19].
A generalized form of the Poincaré–Birkhoff fixed point theorem. Let \(\mathcal{A}\) be an annular region bounded by two strictly star-shaped 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 area-preserving homeomorphism and \(F|\mathcal{A}\) admits a lifting, with the standard covering projection \(\it\Pi:(r, \theta)\to z=(r\cos\theta, r\sin\theta)\), of the form
where w and h are continuous functions of period 2π in the second variable. Correspondingly, for \(\tilde{\Gamma}_{1}={\it\Pi}^{-1}(\Gamma_{1})\) and \(\tilde{\Gamma}_{2}={\it\Pi}^{-1}(\Gamma_{2})\), assume the twist condition
$$h(r, \theta)> 0 \mbox{ on } \tilde{\Gamma}_{1}; \qquad h(r, \theta)< 0 \mbox{ on } \tilde{\Gamma}_{2}. $$
Then, F has two fixed points \(z_{1}\), \(z_{2}\) in the interior of \(\mathcal{A}\) such that
The assumption on the star-shaped boundaries of the annulus is a delicate hypothesis. Martins and Ureña [17] showed that the star-shapedness assumption on the interior boundary is not eliminable. Le Calvez and Wang [16] then proved that the star-shapedness of the exterior boundary is also necessary, although this assumption was not made in Ding’s theorem [8].
Since system (1.1) is conservative, P is an area-preserving mapping. It follows from (3.3) and (3.4) that the area-preserving map P is twisting on the annulus \(D_{1}\). According to the generalized Poincaré–Birkhoff fixed point theorem, P has at least two fixed points in \(D_{1}\). Similarly, we can find a sequence
Using the generalized Poincaré–Birkhoff fixed point theorem, we know that P has at least two fixed points in each annulus \(D_{j}\), \(j=2,3,\dots \). Thus we have obtained the existence of a sequence of 2π-periodic solutions \(\{(x_{j}(t),y_{j}(t))\}\) of system (1.1) satisfying
Then the area-preserving iteration map \(P^{m}\) is twisting on the annulus \(D_{1}'\). Using the generalized Poincaré–Birkhoff fixed point theorem, \(P^{m}\) has at least two fixed points \((x_{1i}, y_{1i})\) (\(i=1,2\)) in \(D_{1}'\). Hence
are two \(2m\pi\)-periodic solutions of system (1.1). Since \(q_{1}\) is a prime, we can further prove that \((x_{1i}(t), y_{1i}(t))\) are not \(2k\pi\)-periodic for \(1\leq k\leq m-1\) by the same method as in [7]. Similarly, we can find two sequences
It follows that \(P^{m}\) has at least two fixed points in each annulus \(D_{j}'\), \(j=2 ,3, \dots \), which correspond to two \(2m\pi\)-periodic solutions of system (1.1). In the same way, these \(2m\pi\)-periodic solutions are not \(2k\pi\)-periodic for \(1\leq k\leq m-1\). Consequently, system (1.1) has infinitely many \(2m\pi\)-periodic solutions \(\{(x_{j}(t), y_{j}(t))\}_{j=1}^{\infty }\) that are not \(2k\pi\)-periodic for \(1\leq k\leq m-1\) and satisfy
The assumption on the uniqueness of the solutions to Cauchy problems of (1.1) made in the proofs of the previous sections can be removed. In fact, Lemmas 2.3 and 2.4 guarantee the applicability of the nonuniqueness version of the Poincaré–Birkhoff theorem, which was proved by Fonda and Ureña [13]. We now state this theorem for a general Hamiltonian system in \(\mathbf {R}^{2N}\). Let us consider the 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 T-periodic 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 the rotation number of a continuous path in \(\mathbf{R}^{2}\). Let \(\sigma:[t_{1}, t_{2}]\to\mathbf{R}^{2}\) be a continuous path such that \(\sigma(t)\neq(0,0)\) for every \(t\in[t_{1}, t_{2}]\). The rotation number of σ around the origin is defined as
where \(\theta:[t_{1}, t_{2}]\to\mathbf{R}\) is a continuous determination of the argument along σ, that is, \(\sigma(t)=|\sigma(t)|(\cos \theta(t), \sin\theta(t))\).
Assume that, for each \(i=1,\ldots,N\), there are two strictly star-shaped Jordan closed curves around the origin \(\Gamma^{i}_{1}, \Gamma ^{i}_{2}\subset\mathbf{R}^{2}\) such that
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
$$z_{i}(t)\neq(0, 0) \quad \textit{for all } t\in[0, T] \textit{ and } i=1, \ldots, N. $$
Assume that there are integer numbers\(v_{1}, \ldots, v_{N}\in\mathbf {Z}\)such that, for each\(i=1, \ldots, N\), either
Then Hamiltonian system (4.1) has at least\(N+1\)distinctT-periodic solutions\(z^{0}(t), \ldots, z^{N}(t)\), with\(z^{0}(0), \ldots, z^{N}(0)\in \mathcal{A}\), such that
$$\operatorname{Rot}\bigl(z_{i}^{k}(t); [0, T] \bigr)=v_{i} \quad \textit{for all } k=0, \ldots, N \textit{ and } i=1, \ldots, N. $$
Remark 4.2
Note that there is no requirement of uniqueness of the solutions to Cauchy problems of (4.1) in this higher-dimensional Poincaré–Birkhoff theorem for Hamiltonian flows. Theorem 4.1 can be applied to deal with the multiplicity of periodic solutions of higher-dimensional Hamiltonian systems [9, 12]. Fonda and Sfecci [12] studied the multiplicity of periodic solutions of weakly coupled Hamiltonian systems
where all the functions \(h_{i}: \mathbf{R}\times\mathbf{R}\to\mathbf {R}\) are continuous, T-periodic in the first variable t, and such that
$$\lim_{|x|\to\infty}h_{i}(t, x)=+\infty\quad\mbox{uniformly in } t\in[0, T]. $$
The functions \(p_{i}: \mathbf{R}\times\mathbf{R}^{N}\) are continuous, T-periodic in the first variable t, and bounded. Moreover, there is a function \(\mathcal{U}:\mathbf{R}\times\mathbf{R}^{N}\to\mathbf {R}\) such that
$$\frac{\partial\mathcal{U}}{\partial x_{i}}=x_{i}p_{i}(t, x_{1}, \ldots, x_{N})\quad\mbox{for all } (t, x_{1}, \ldots, x_{N})\in[0, T]\times \mathbf{R}^{N} \mbox{ and } i=1, \ldots, N. $$
In this case, (4.2) is a superlinear Hamiltonian system. Under these conditions, it was proved in [12] that (4.2) has infinitely many periodic solutions by using Theorem 4.1. Theorems 1.1 and 1.2 in the present paper can also be extended to a weakly coupled Hamiltonian system of the type
where \(x=(x_{1}, \ldots, x_{N})\), \(y=(y_{1}, \ldots, y_{N})\), \(f_{i}, g_{i}:\mathbf{R}\to\mathbf{R}\) are continuous, \(p_{ji}:\mathbf {R}\times\mathbf{R}^{N}\times\mathbf{R}^{N}\to\mathbf{R}\) (\(j=1,2\), \(i=1,\ldots, N\)) are continuous and 2π-periodic in the first variable t. Assume that there is a function \(\mathcal{W}: \mathbf {R}\times\mathbf{R}^{N}\times\mathbf{R}^{N}\to\mathbf{R}\) 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 2π-periodic solutions and, for any integer \(m\geq2\), (4.3) has infinitely many \(2m\pi\)-periodic solutions that are not \(2k\pi \)-periodic for \(1\leq k\leq m-1\), provided that conditions (\(h_{i}'\)) (\(i=1, \ldots, 4\)) hold. For brevity, we omit the details.
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.
Abbondandolo, A., Asselle, L., Benedetti, G., Mazzucchelli, M., Taimanov, I.: The multiplicity problem for periodic orbits of magnetic flows on the 2-sphere. Adv. Nonlinear Stud. 17, 17–30 (2017)
MathSciNetView ArticleMATHGoogle Scholar
Boscaggin, A.: Subharmonic solutions of planar Hamiltonian systems: a rotation number approach. Adv. Nonlinear Stud. 11, 77–103 (2011)
MathSciNetView ArticleMATHGoogle Scholar
Dalbono, F., Rebelo, C.: Poincaré–Birkhoff fixed point theorem and periodic solutions of asymptotically linear planar Hamiltonian systems. Rend. Semin. Mat. (Torino) 60, 233–263 (2002)
MATHGoogle Scholar
Ding, T., Iannacci, R., Zanolin, F.: Existence and multiplicity results for periodic solutions of semilinear Duffing equations. J. Differ. Equ. 105, 364–409 (1993)
MathSciNetView ArticleMATHGoogle Scholar
Ding, T., Zanolin, F.: Periodic solutions of Duffing’s equations with super-quadratic potential. J. Differ. Equ. 97, 328–378 (1992)
View ArticleMATHGoogle Scholar
Ding, W.: A generalization of the Poincaré–Birkhoff theorem. Proc. Am. Math. Soc. 88, 341–346 (1983)
MATHGoogle Scholar
Fonda, A., Garrione, M., Gidoni, P.: Periodic perturbations of Hamiltonian systems. Adv. Nonlinear Anal. 5, 367–382 (2016)
MathSciNetMATHGoogle Scholar
Fonda, A., Ghirardelli, L.: Multiple periodic solutions of Hamiltonian systems in the plane. Topol. Methods Nonlinear Anal. 36, 27–38 (2010)
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, 1369–1391 (2012)
MathSciNetView ArticleMATHGoogle Scholar
Fonda, A., Ureña, A.J.: A higher dimensional Poincaré–Birkhoff theorem for Hamiltonian flows. Ann. Inst. Henri Poincaré, Anal. Non Linéaire 34, 679–698 (2017)
MathSciNetView ArticleMATHGoogle Scholar
Garrione, M.: Resonance at the first eigenvalue for first order systems in the plane: vanishing Hamiltonian and the Landesman–lazer condition. Differ. Integral Equ. 25, 505–526 (2012)
MathSciNetMATHGoogle Scholar
Krasnosel’skii, M.A.: The Operator of Translation Along the Trajectories of Differential Equations. Amer. Math. Soc., Providence (1968)
Google Scholar
Le Calvez, P., Wang, J.: Some remarks on the Poincaré–Birkhoff theorem. Proc. Am. Math. Soc. 138, 703–715 (2010)
View ArticleMATHGoogle Scholar
Martins, R., Ureña, A.J.: The star-shaped condition on Ding’s version of the Poincaré–Birkhoff theorem. Bull. Lond. Math. Soc. 39, 803–810 (2007)
MathSciNetView ArticleMATHGoogle Scholar
Opial, Z.: Sur les périodes des solutions de l’équation différentielle \(x''+g(x) = 0\). Ann. Pol. Math. 10, 49–72 (1961)
View ArticleMATHGoogle Scholar
Rebelo, C.: A note on the Poincaré–Birkhoff fixed point theorem and periodic solutions of planar systems. Nonlinear Anal. 29, 291–311 (1997)
MathSciNetView ArticleMATHGoogle Scholar
Wang, Z., Ma, T.: Periodic solutions of planar Hamiltonian systems with asymmetric nonlinearities. Bound. Value Probl. 2017, Article ID 46 (2017)
MathSciNetView ArticleMATHGoogle Scholar
