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AubryMather sets in semilinear asymmetric Duffing equations
Advances in Difference Equations volume 2016, Article number: 297 (2016)
Abstract
In this paper, by introducing an appropriate actionangle variable transformation and adopting a new estimate method, we prove the existence of AubryMather sets to a class of semilinear asymmetric Duffing equations
where \(x^{\pm }=\max \{\pm x,0\}\), α and β are positive constants satisfying
with \(\omega \in \mathbb{R}^{+} \), and \(f(t,x)\in C^{0,1}(\mathbf{S}^{1}\times \mathbb{R})\) is a continuous function, 2πperiodic in the first variable and continuously differentiable in the second one, by virtue of a generalized version of AubryMather theorem on cylinder with monotone twist assumption given by Pei. It should be pointed out that the perturbation term \(f(t,x)\) satisfying some suitable growth conditions, can be allowed to be either a bounded function or an unbounded function, which differs from many existing results in the literature. Moreover, some examples are provided to illustrate the validity of the proposed results.
1 Introduction
Let us consider the following semilinear asymmetric Duffing equation:
where \(x^{\pm }=\max \{\pm x,0\}\), α and β are positive constants satisfying
with \(\omega \in \mathbb{R}^{+} \), \(f(t,x)\in C^{0,1}(\mathbf{S}^{1}\times \mathbb{R})\) is a continuous function, 2πperiodic in the first argument and has continuous derivative in the second one, where \(\mathbf{S}^{1}= \mathbb{R}/2\pi \mathbb{Z}\).
As one of the simplest but nontrivial conservative systems, equation (1.1) has been extensively and intensively studied by many researchers as regards its dynamic behavior such as periodic solutions [1–10], and the existence of bounded or unbounded solutions [11–16]. In the existing literature, a lot of work has been done to prove the existence of the periodic solutions, such as [1–10] and the references given therein, but there is little information as regards the dynamics behavior of its periodic solution. On the other hand, AubryMather theory about twist maps heavily indicates that equation (1.1) most likely possesses at least some ‘good’ periodic solutions as well as some generalized quasiperiodic solutions. This is the main objective of this paper and we will prove the existence of AubryMather sets for equation (1.1) under some suitable assumptions.
In the early 1980s, Aubry [17] and Mather [18] have proved independently that invariant curves of integral system will be broken if its perturbation increased gradually and/or the smoothness of integrable system is weakened, when they, respectively, studied a one dimensional liquid crystal model of solid state physics and the qualitative properties of the orbits of an areapreserving twist map of the annulus. They also found that when invariant curves break, they do not disappear completely, some special invariant sets still exist. Today, these invariant sets are called the AubryMather sets. For the planar differential system, AubryMather theory suggests that for its Poincaré mapping if there exist AubryMather sets \(M_{\sigma }\) with a rotation number σ, then the planar differential system possesses AubryMather type solutions \(z_{\sigma }(t)=(x_{\sigma }(t),y_{\sigma }(t))\), such that \(M_{\sigma }\equiv \overline{\{z_{\sigma }(2\pi i),i\in Z\}}\) satisfying the following geometrical properties:
\((1)\) if \(\sigma =\frac{n}{m}\in {\mathbb{Q}}\) with \((n,m)=1\), then \(z_{\sigma }(t)\) is a Birkhoff periodic solution with periodic \(2m\pi \) and \(\arg (z_{\sigma }(t)+m)=\arg (z_{\sigma }(t))+n\), the m solutions \(z_{\sigma }(t+2\pi i), 0\leq i\leq m1\), can be homotopically transformed to m distinct parallel lines;
\((2)\) if \(\sigma \in {{\mathbb{R}}\backslash {\mathbb{Q}}}\), then \(M_{\sigma }\) is either an invariant circle and its orbits are just usual quasiperiodic orbits, or an invariant Cantor set and its orbits become generalized ones. For further interpretations, we refer to the recent work of [19] and [20].
In general, we note that the existence of Birkhoff type periodic solution is very difficult to prove, see, for example, Bernstein and Katok work [21]. But AubryMather theory has provided a powerful tool for complete qualitative description of the dynamic behavior of differential equations due to their applications in many fields such as differential geometry, dynamical systems, and solid physics (see [22, 23]). In the last two decades, there has been an increasing interest in obtaining sufficient conditions for the existence of AubryMather sets for different classes of nonlinear second order differential equations. We refer to the recent papers [24–33], and references therein. Especially, in [29], Capietto and Liu have studied the following piecewise linear equation:
where \(\psi (x)\in C^{2}(\mathbb{R})\) is bounded and \(e(t)\in C^{2}(\mathbf{S}^{1})\) is 2πperiodic. Under some additional hypotheses on \(\psi (x)\) and in the resonant case
with \(\omega \in \mathbb{Q}^{+}\), they used an exchange of the role of time and angle variables and proved the existence of AubryMather sets for equation (1.3), by means of a version of the AubryMather theorem on a cylinder with the monotone twist assumption given by Pei [25], in which the author gave some sufficient conditions for the existence of AubryMather sets for a class of semilinear Duffing equations,
where \(\lambda >0\) is a constant, \(p(t)\in C^{0}(\mathbf{S}^{1})\) is a periodic function and \(\psi (x)\in C^{2}(\mathbb{R})\) is an unbounded function satisfying some other growth conditions.
Now a natural and open question is whether equation (1.3) still possesses AubryMather sets when the smoothness of \(\psi (x)\) and \(e(t)\) is further weakened. Our Theorem 1.1 in the following will answer this question and we will deal with a more general case (1.1) than that of (1.3). Owing to the appearance of weak smoothness nonlinearity, the methods in [25–31] are no longer valid. To overcome this difficulty, we first introduce a suitable action and angle variable transformation similar to [29] so that the transformed system of (1.1) is a perturbation of an integral Hamiltonian system, and then apply a new estimate approach developed by the present author (see the recent papers [32–34]) to directly prove the Poincaré map of the transformed system satisfying monotone twist property. Furthermore, the AubryMather theorem on a cylinder with monotone twist assumption by Pei [25] guarantees the existence of AubryMather sets for (1.1), which leads to our desired results. The results of this paper are new and they are natural generalizations and refinements of previously known results obtained in [25, 29].
More exactly, the following theorem is proved.
Theorem 1.1
Assume that the condition (1.2) holds true. Let \(f(t,x)\in C^{0,1}(\mathbf{S}^{1}\times \mathbb{R})\) satisfy the following conditions:
 \((A_{1})\) :

the limit \(\lim_{\vert x \vert \to +\infty }f_{x}(t,x)=0\), uniformly in \(t\in [0,2\pi ]\);
 \((A_{2})\) :

there exist constants \(d\geq 0, \mu >0\), such that
$$ \operatorname{sgn}(x) \bigl[f(t,x)xf_{x}(t,x) \bigr]>\mu ,\quad\textit{for } \vert x \vert \geq d. $$
Then there exists \(\varepsilon _{0}>0\), such that, for any \(\sigma \in (2\omega \pi ,2\omega \pi +\varepsilon _{0})\), equation (1.1) possesses an AubryMather type solution \(z_{\sigma }(t)=(x_{\sigma }(t),x'_{\sigma }(t))\) with rotation number σ, namely,

(i)
when \(\sigma =\frac{n}{m}\) is rational, and \((n,m)=1\), the solutions \(z_{\sigma }^{i}(t)=z_{\sigma }(t+2\pi i) , 0\leq i\leq m1\), are mutually unlinked periodic solutions of period m;

(ii)
when σ is irrational, the solution \(z_{\sigma }(t)\) is either a usual quasiperiodic solution or a generalized one exhibiting a Denjoy minimal set (see the definition in [35])
$$ M_{\sigma }\equiv \overline{ \bigl\{ z_{\sigma }(2\pi i),i\in { \mathbb{Z}} \bigr\} }. $$
Remark 1.1
Applying the rule of L’Hospital to condition \((A_{1})\), it is easy to see that
The rest of the paper is organized as follows: The proof of Theorem 1.1 will be given in Section 4. Section 2 introduces some basic results which are necessary for the proof of Theorem 1.1. In Section 2.1, we introduce a polar coordinate type actionangle variable which transform equation (1.1) into a perturbation of an integral Hamiltonian system, and then in Section 2.2, we will give some estimates on the corresponding action and angle variables functions. Section 3 deals with the proof of monotone twist property of the Poincaré map P of the new system around infinity. Finally, Section 5 presents some examples and remarks on Theorem 1.1.
2 Some basic results
2.1 Actionangle variables
Introducing a new variable y as \(x'=y\), then equation (1.1) is equivalent to the following system:
Let \((C(t),S(t))\) be the solution of the autonomous system
satisfying the initial condition \((C(0),S(0))=(1,0)\). Then it is well known that \(S(t)\in C^{1}(\mathbb{R})\) and \(C(t)\in C^{2}(\mathbb{R})\) are \(\frac{2\pi }{\omega }\)periodic and the even function \(C(t)\) can be expressed as
It is easy to see that the functions \(C(t)\) and \(S(t)\) are very similar to the cosine and the sine. Moreover, it is not difficult to prove that

(i)
\(C'(t)=S(t), S'(t)=\alpha C^{+}(t)\beta C^{}(t)\);

(ii)
\((S(t))^{2}+\alpha (C^{+}(t))^{2}+\beta (C^{}(t))^{2}\equiv \alpha \);

(iii)
\(\vert C(t) \vert \leq \max \{1,\sqrt{\frac{\alpha }{\beta }}\}:=C_{\infty }, \vert S(t) \vert \leq \sqrt{\alpha }\).
For \(r>0\) and \(\theta (\operatorname{mod} 2\pi )\), we introduce the action and angle variables transformation \(T:(r,\theta )\rightarrow (x,y)\) as
with \(\lambda =\frac{\omega }{\alpha }\). Then its Jacobian
where \(x_{r}=\frac{\partial x}{\partial r}=\frac{x}{2r}, x_{\theta }=\frac{\partial x}{\partial \theta }=\frac{y}{\omega }, y_{r}=\frac{\partial y}{\partial r}=\frac{y}{2r}, y_{\theta }=\frac{\partial y}{\partial \theta }=\frac{\alpha x^{+}}{\omega }\frac{\beta x^{}}{\omega }\). Therefore the map T is symplectic.
2.2 Some estimates on action and angle variables functions
where \(\Phi _{1}(t,\theta ,r)=\omega +\frac{x(\theta ,r)f(t,x(\theta ,r) )}{2r},\Phi _{2}(t,\theta ,r)=\frac{y(\theta ,r)f(t,x(\theta ,r) )}{\omega }\).
According to the assumptions of \((A_{1})\) and \((A_{2})\), it is easy to prove the existence and uniqueness of the solution of the initial value problem associated with (2.4). Moreover, this solution has continuous derivatives with respect to initial data.
Let \(\theta (t)=\theta (t;\theta _{0},r_{0}),r(t)=r(t;\theta _{0},r_{0})\) be the solution of initial value problem \(\theta (0)=\theta _{0},r(0)=r_{0}\), then
and
are the solutions of (2.1) with initial data \(x(0)=x(0;\theta _{0},r_{0}),y(0)=y(0;\theta _{0},r_{0})\). So the Poincaré map P of (2.4) is of the form
For notional convenience, hereinafter, we write x, y, θ, r instead of \(x(\theta (t;\theta _{0},r_{0}),r(t;\theta _{0}, r_{0}))\), \(y(\theta (t;\theta _{0},r_{0}),r(t;\theta _{0},r_{0})),\theta (t;\theta _{0},r_{0}), r(t;\theta _{0},r_{0})\), respectively.
Now in the following, we will give some growth estimation properties with respect to the action and angle variables functions \(r(t;\theta _{0},r_{0})\) and \(\theta (t;\theta _{0},r_{0})\).
Proposition 2.1
Assume that \((A_{1})\) holds. Then the limit
holds uniformly with respect to \(t\in [0,2\pi ]\).
Proof
By \((f_{0})\) and the action and angle variables transformation (2.3), there exist constants \(B>0, F>0\), such that
Then, by the Gronwall inequality, we obtain
for all \(t\in [0,2\pi ]\).
Consequently, by (2.6), \(r(t;\theta _{0},r _{0})\to +\infty \) as \(r_{0}\to +\infty \) uniformly for \(t\in [0,2\pi ]\). □
By (2.6), we can easily verify the following.
Proposition 2.2
Suppose that \((A_{1})\) holds. Then there exist constants \(\beta _{2}>\beta _{1}>0\) and \(\bar{r}>0\), such that, for any \(r_{0}\geq \bar{r}\), we have
for \(\forall \theta _{0}\in \mathbb{R}\) and \(\forall t\in[0,2\pi ]\).
Proposition 2.3
Let \((A_{1})\) hold. Then, for any \(\theta _{0}\in \mathbb{R}\) and \(t\in [0,2\pi ]\), there exists \(\bar{r}>0\), such that
as long as \(r_{0}\geq \bar{r}\).
Proof
Since \((f_{0})\) holds, for every \(\varepsilon >0\), there exists \(X=X(\varepsilon )>0\), such that
if \(\vert x \vert \geq X\) and \(\forall t\in [0,2\pi ]\). Hence,
Thus, in view of (2.3) and Proposition 2.1, there exists \(\bar{r_{1}}>0\) such that
if \(r_{0}\geq \bar{r_{1}}\).
If \(\vert x \vert \leq X\), we may assume that \(\vert f(t,x) \vert \leq f_{\infty }\), where \(f_{\infty }= \max \{\vert f(t,x) \vert :t\in [0,2\pi ],\vert x \vert \leq X\}\), then
Hence, by (2.3) and Proposition 2.1, there exists a constant \(\bar{r_{2}}>0\), such that \(\frac{d\theta }{dt}\geq \frac{\omega }{2}\) if \(r_{0}\geq \bar{r_{2}}\).
If we choose \(\bar{r}=\max \{\bar{r_{1}},\bar{r_{2}}\}\), then \(r_{0}\geq \bar{r}\) implies \(\frac{d\theta }{dt}\geq \frac{\omega }{2}\).
Using the same arguments as above, one can prove that the inequality on the right side of (2.7) holds. □
3 Monotone twist property
In this section, we will prove
if \(r_{0}\gg 1\). This implies the Poincaré map P given by (2.5) has monotone twists around infinity.
Now in the following, we will start to examine the behavior of \(\frac{\partial \theta (2\pi ;\theta _{0},r_{0}) }{\partial r_{0}} \) when \(r_{0}\gg 1\).
Let us consider the variational equation of (2.4) with respect to the initial value \(r_{0}\). One can verify that, for any \(t\in [0, 2\pi ]\),
where
Then we have the following.
Lemma 3.1
For \(r_{0}\gg 1\) and \(\forall t,s \in [0,2\pi ]\), the following conclusions hold:

(i)
\(a_{1}(t)=o(\frac{1}{r_{0}})\);

(ii)
\(a_{2}(t)=o(1)\);

(iii)
\(a_{1}(t)\cdot a_{3}(s)=o(1)\).
Proof
By using Lemma 3.2, the proof can be obtained immediately. □
Next we are going to give a detailed proof of Lemma 3.2.
Lemma 3.2
As \(r_{0}\rightarrow +\infty \), the following convergences hold uniformly on \(t\in [0,2\pi ]\):

(i)
\(\frac{xf(t,x)}{r}\rightarrow 0\), \(\frac{x^{2}f_{x}(t,x)}{r}\rightarrow 0\);

(ii)
\(\frac{yf(t,x)}{r}\rightarrow 0\), \(\frac{yxf_{x}(t,x)}{r}\rightarrow 0\), \(\frac{y^{2}xf(t,x)f_{x}(t,x)}{r^{2}}\rightarrow 0\).
Proof
If \((A_{1})\) and \((f_{0})\) hold, then to each \(\varepsilon >0\) there corresponds a positive number \(X=X(\varepsilon )>0\), such that
and
if \(\vert x \vert \geq X\) and \(t\in[0,2\pi ]\), here \(C_{\infty }\) is defined in Section 2.1.
Set \(F_{1}(\varepsilon )=\max \{\vert f(t,x) \vert :t\in [0,2\pi ],\vert x \vert \leq X\}, F_{2}(\varepsilon )=\max \{\vert f_{x}(t,x) \vert :t\in [0,2\pi ],\vert x \vert \leq X\}\).
(i) According to the action and angle variables transformation (2.3), one has
Then, given \(\bar{r}>0\), choose \(r_{0}\) so that \(r_{0}\geq \bar{r}\), by using Proposition 2.2, provided
we have
Since \(\varepsilon >0\) is arbitrary, the proof of (i) is complete.
(ii) Corresponding to (2.3), one can see that
Then, given \(\bar{r}>0\), choose \(r_{0}\) so that \(r_{0}\geq\bar{r}\), by using Proposition 2.2, provided
we have
Since \(\varepsilon >0\) is arbitrary, (ii) is proved. □
Combining the previous lemmas, we have the following.
Lemma 3.3
For all \(t \in (0,2\pi ]\), as \(r_{0}\rightarrow +\infty \), we have

(i)
\(\theta _{r_{0}}(t;\theta _{0}, r_{0})\rightarrow 0\);

(ii)
\(r_{r_{0}}(t;\theta _{0}, r_{0})=1+o(1)\);

(iii)
\(\theta _{\theta _{0}}(t;\theta _{0}, r_{0})=1+o(1)\).
Proof
By the variational equations (3.1) and Lemma 3.2, we have
where we have used \(\theta _{r_{0}}(0)=0\) and \(r_{r_{0}}(0)=1\).
Therefore, for all \(t\in (0,2\pi ]\), sending \(r_{0}\rightarrow +\infty \), we have \(r_{r_{0}}(t)=1+o(1)\) and \(\theta _{r_{0}}(t)=(1+o(1))\int _{0}^{t}a_{1}(s)\,ds\rightarrow 0\). This completes the proof of (i) and (ii).
To prove (iii), we consider the variational equation of (2.4) about \(\theta _{0}\)
Similar to the proof of (ii), one can see that \(\theta _{\theta _{0}}(t;\theta _{0}, r_{0})=1+o(1)\) for \(\forall t\in (0,2\pi ]\), as \(r_{0}\rightarrow +\infty \). This completes the proof of Lemma 3.3. □
Now, we will give an estimate of upper bound and lower bound for \(a_{1}(t)\).
Lemma 3.4
Let \(d\geq 0\) satisfy \((A_{2})\).

(i)
If \(\vert x(t;\theta _{0},r_{0}) \vert \leq d\) for all \(t \in [0,2\pi ]\), then there exists a constant \(K_{d}>0\), such that \(\vert a_{1}(t) \vert \leq \frac{K_{d}}{r^{2}(t)}\).

(ii)
If \(\vert x(t;\theta _{0},r_{0}) \vert \geq d\) for all \(t \in [0,2\pi ]\), then there exists a constant \(L_{d}>0\), such that \(\vert a_{1}(t) \vert \geq \frac{L_{d}}{r^{2}(t)}\). Moreover, \(\vert x(t;\theta _{0},r_{0}) \vert \geq d\) implies that \(a_{1}(t)<0\).
Proof
(i) If \(\vert x(t;\theta _{0},r_{0}) \vert \leq d\), we take \(f_{d}=\max_{\vert x \vert \leq d, t\in [0,2\pi ]}\vert f(x,t)xf_{x}(x,t) \vert \).
Then
Set \(K_{d}= \frac{df_{d}}{4}\), we get \(\vert a_{1}(t) \vert \leq \frac{K_{d}}{r^{2}(t)}\).
(ii) If \(\vert x(t;\theta _{0},r_{0}) \vert \geq d\), then from the condition of \((A_{2})\), one can see that \(xf(x,t)x^{2}f_{x}(x,t)>\mu x\) when \(x\geq d\), and \(xf(x,t)x^{2}f_{x}(x,t)>\mu x\) when \(x\leq d\). Hence, \(a_{1}(t)<0\). Further,
Choosing \(L_{d}=\frac{d\mu }{4}\), it yields
Denote \(a_{1}(t)=a_{1}^{+}(t)a_{1}^{}(t)\), where \(a_{1}^{\pm }(t)=\max \{\pm a_{1}(t),0\}\). To estimate the integral of \(a_{1}^{+}(t)\) on \([0,2\pi ]\) is smaller than the integral of \(a_{1}^{}(t)\) on \([0,2\pi ]\), we need the following lemma. □
Lemma 3.5
Let \(d\geq 0\) be as in Theorem 1.1. Define \(\Delta t=\{t\in [0,2\pi ]\mid \vert x(t;\theta _{0},r_{0}) \vert \leq d\}\). Then there exist \(\bar{r_{0}}>0,D>0\), such that
for all \(r_{0}\geq \bar{r_{0}}\).
Proof
By Proposition 2.3, we know that \(\Delta t\rightarrow 0\) if and only if \(\Delta \theta \rightarrow 0\).
According to the action and angle variables transformation (2.3), we see that there exists \(\gamma >0\) such that \(\vert \tan \Delta \theta \vert \leq \frac{\gamma d}{\sqrt{2\lambda r(t)}}\) when \(\Delta \theta \rightarrow 0 \). Therefore, by using Proposition 2.2, we know that there exist \(\bar{r_{0}}>0, D>0\), such that
for all \(r_{0}\geq \bar{r_{0}}\). Lemma 3.5 follows. □
In the following, we will give an estimate of \(\frac{\partial \theta (2\pi ;\theta _{0},r_{0}) }{\partial r_{0}}\) if \(r_{0}\gg 1\).
Lemma 3.6
For \(r_{0}\gg 1\), we have \(\theta _{r_{0}}(2\pi )<0\).
Proof
Combining Lemma 3.4, Lemma 3.5 and Proposition 2.2, we can make it clear that
Consequently, if \(\sqrt{r_{0}}>\frac{(\beta _{1}^{2} L_{d} +\beta _{2}^{2}K_{d})D}{2\pi L_{d}\beta _{1}^{2}}\), then \(\theta _{r_{0}}(2\pi )<0\). □
4 Proof of Theorem 1.1
Now we are in a position to prove Theorem 1.1.
Proof of Theorem 1.1.
According to Lemma 3.6 and the AubryMather theory framework developed by Pei [25], we know that the Poincaré map P of system (2.4) is a monotone twist map if \(r_{0}\gg 1\). At last, using similar arguments as in [25], one may extend the Poincaré map P to a new map P̂ which is a global monotone twist homeomorphism of the cylinder \(\mathbf{S}^{1}\times \mathbb{R}\) and coincides with P on \(\mathbf{S}^{1}\times [r_{0},+\infty )\) with a fixed constant \(r_{0}\gg 1\). Therefore, the existence of AubryMather sets \(M_{\sigma }\) of P̂ is guaranteed by the AubryMather theorem by Pei [25]. Moreover, for some small \(\varepsilon _{0}>0\), all those AubryMather sets with rotation number \(\sigma \in (2\omega \pi ,2\omega \pi +\varepsilon _{0}) \) lie in the domain \(\mathbf{S}^{1}\times [r_{0},+\infty )\). Hence they are just the AubryMather sets of the Poincaré map of P. From the above discussions, we have showed the existence of AubryMather sets, this implies that equation (1.1) has a solution \(z_{\sigma }(t)=(x_{\sigma }(t),x'_{\sigma }(t))\) with rotation number σ. Thus, the proof of Theorem 1.1 is completed. □
5 Examples and remarks
As an application, let us consider the following equation:
where \(x^{\pm }=\max \{\pm x,0\}\), α and β are positive constants satisfying
with \(\omega \in \mathbb{R}^{+} \), \(f(t,x)\in C^{0,1}(\mathbf{S}^{1}\times \mathbb{R})\) and \(\mathbf{S}^{1}= \mathbb{R}/2\pi \mathbb{Z}\).
Example 5.1
Bounded perturbation
Consider \(f(t, x) =\operatorname{sgn}(x)\arctan x\cdot (1+\vert p(t) \vert )\), where \(p(t)\) is a continuous function satisfying \(p(t +2\pi ) = p(t)\). We can easily check \(f(t, x)\in C^{0,1}(\mathbf{S}^{1}\times \mathbb{R})\) is a bounded function satisfying conditions \((A_{1})\) and \((A_{2})\) in Theorem 1.1 when \(d=1\) and \(\mu =\frac{\pi }{4}\frac{1}{2}\) for any \(\alpha \neq \beta \) in (5.2). But \(f(t, x)\) does not satisfy the conditions \((H_{1})\)\((H_{4})\) of Theorem 1 in [29]. Therefore, the results obtained in this paper can be viewed as natural generalizations and refinements of the results in [29].
Example 5.2
Unbounded perturbation
Let \(\alpha =\beta =\lambda \) in (5.2) and \(f(t,x)=\operatorname{sgn}(x)\ln (1+\vert x \vert )\cdot (1+\vert p(t) \vert )\), where \(p(t)\) is a continuous function satisfying \(p(t +2\pi ) = p(t)\). It is obvious that \(f(t, x)\in C^{0,1}(\mathbf{S}^{1}\times \mathbb{R})\) is an unbounded function satisfying conditions \((A_{1})\) and \((A_{2})\) in Theorem 1.1 if \(d=e^{2}\) and \(\mu =1\). It is clear that the conditions \((D_{1})\) and \((D_{2})\) of Theorem B in [25] are not satisfied. Thus our situation is more general than the results obtained in [25].
Remark 5.1
Comparing the estimation method in this paper with [25, 29], it seems that our estimation processes are much simpler than those used in [25, 29] to some degree. Moreover, it is easy to see that the results of this paper generalize and refine the early results published so far, our results are new and not covered by any known work in the literature.
Remark 5.2
Finally, let us note that our method could be useful in more general situations such as equations depending nonlinearly on the derivative, Hamiltonian or more general second order differential equations. These extensions should be developed elsewhere.
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This research was supported by the National Natural Science Foundation of China (11461056).
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Wang, X. AubryMather sets in semilinear asymmetric Duffing equations. Adv Differ Equ 2016, 297 (2016). https://doi.org/10.1186/s136620161024y
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DOI: https://doi.org/10.1186/s136620161024y