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A new approach to the existence of quasiperiodic solutions for a class of semilinear Duffingtype equations with timeperiodic parameters
Boundary Value Problems volume 2016, Article number: 132 (2016)
Abstract
Let \(q(t)\) be a continuous 2πperiodic function with \(\frac{1}{2\pi}\int_{0}^{2\pi}q(t)\,dt>0\). We propose a new approach to establish the existence of AubryMather sets and quasiperiodic solutions for the following timeperiodic parameters semilinear Duffingtype equation:
where \(f(t,x)\) is a continuous function, 2πperiodic in the first argument and continuously differentiable in the second one. Under some assumptions on the functions q and f, we prove that there are infinitely many generalized quasiperiodic solutions via a version of the AubryMather theorem given by Pei. Especially, an advantage of our approach is that it does not require any high smoothness assumptions on the functions \(q(t)\) and \(f(t,x)\).
1 Introduction
The goal of this paper is to study the existence of AubryMather sets and quasiperiodic solutions to the following timeperiodic parameters semilinear Duffingtype equation:
where \(q(t)\) is continuous and 2πperiodic function in the time t, \(f(t,x)\) is a continuous function, 2πperiodic in the first argument and continuously differentiable in the second one. As is well known such a system is one of the most important models in Hamiltonian systems due to both its physical significance and mathematical fascination despite its simple form. Here we are concerned with the socalled breakdown of stability (in the sense of the Lagrangian) by using the AubryMather theory.
In the early 1980s, Aubry [1] and Mather [2] proved independently that invariant curves of integrable system will be broken if its perturbation increased gradually and/or the smoothness of integrable system is weakened, when they, respectively, studied onedimensional 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 simply disappear, some special invariant sets still exist. Today, these sets are called AubryMather sets. For the planar differential system, AubryMather theory suggests that its Poincaré mapping has 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 \(M_{\sigma}\equiv\overline{ \{z_{\sigma}(2\pi i),i\in Z\}}\) with the following geometrical and dynamical properties:

(1)
if \(\sigma=\frac{n}{m}\in{\Bbb {Q}}\) 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 drawn to m straight lines;

(2)
if \(\sigma\in{{\Bbb {R}}\backslash{\Bbb {Q}}}\), 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.
In general, we note that the existence of Birkhoff type periodic solution is very difficult to prove, see, for example, Bernstein and Katok’s work [3]. But AubryMather theory has provided a powerful tool for the indepth study of the dynamic behavior of differential equations. Since the pioneering work of Aubry [1] and Mather [2], AubryMather sets for areapreserving monotone twist homeomorphism have been widely studied due to their applications in many fields such as onedimensional crystal model of solid state physics, differential geometry and dynamical systems (see [4, 5]). And then much work has been carried out concerning the existence of AubryMather sets for various kinds of differential equations, such as Hamiltonian systems [6–9], reversible systems [10, 11] and nonlinear asymmetric oscillator [12–15].
In the 1990s, the Duffingtype equation has been a typical model in the recent AubryMather theory for planar periodic Hamiltonian systems, there are several papers concerning this problem for the Duffing equation (see [16–18]). For example, Pei [16] and Qian [17] have proved, respectively, the existence of AubryMather sets and quasiperiodic solutions for some superlinear Duffing equations and sublinear Duffing equations by using AubryMather theory under some suitable assumptions.
When \(q(t)=\lambda^{2}\) is constant and \(f(x,t)=\varphi(x)p(t)\), equation (1.1) reduces to the semilinear Duffing equation
In 1994, based on a generalized version of AubryMather theorem, Pei [18] obtained the existence of AubryMather sets of equation (1.2) for any continuous 2πperiodic function \(p(t)\) if
 (A_{1}):

\(\frac {\varphi(x)}{x}>\varphi'(x)>0\), for \(x\geq d>0\);
 (A_{2}):

\(\varphi(x)=o(x)\), \(\varphi(x)x\varphi'(x)\to +\infty\), as \(x\to+\infty\); \(x\varphi''(x)\leq C\), where the constant \(C>0\).
An interesting question is: can the smoothness requirement of the perturbation term \(\varphi(x)\in C^{2}(\mathbb{R})\) for equation (1.2) be weakened?
Recently, the author [19] has extended such a result to the case \(\varphi(x)\in C^{1}( \mathbb{R})\) and obtained the existence of AubryMather sets and quasiperiodic solutions for equation (1.2) under the following conditions:
 (B_{1}):

\(\lim_{x\to+\infty}\varphi'(x)=0\);
 (B_{2}):

\(\operatorname{sgn}(x)(\varphi(x)x\varphi'(x))>2p_{\infty }\), for \(x\geq d>0\), where \(p_{\infty}= \max_{t\in[0,2\pi]}p(t)\).
In this paper, we will continue the study of the existence of AubryMather sets and quasiperiodic solutions of equation (1.1) initiated in [18, 19]. In our case, we cannot apply the estimation method used in [18]. Instead, we seek a suitable action and angle variable transformation so that the transformed system of (1.1) is a perturbation of an integral Hamiltonian system, and then propose a new estimate approach and apply a kind of analytical techniques developed by the present author (see the recent papers [11, 14, 15]) to directly prove the Poincaré map of the transformed system satisfying monotone twist property, which leads to our desired results. Especially, an advantage of our approach is that it does not require any high smoothness assumptions on the functions \(q(t)\) and \(f(t,x)\). The results obtained in this paper are natural generalizations and refinements of the results obtained in [18, 19].
Our main result is the following.
Theorem 1.1
Assume that equation (1.1) satisfies:
 (D_{1}):

\(q(t)\in C(\mathbf{S}^{1})\) and \(\frac{1}{2\pi}\int_{0}^{2\pi}q(t)\,dt=\gamma^{2}>2b_{\infty}\), where \(\gamma>0\),
$$b_{\infty}= \max_{t\in[0,2\pi]}\biglb(t)\bigr,\qquad b(t)=q(t) \gamma^{2}; $$  (D_{2}):

\(f(t,x)\in C^{0,1}(\mathbf{S}^{1}\times\mathbb{R})\) and has limits
$$\lim_{x\to+\infty}f_{x}(t,x)=0,\quad \textit{uniformly in } t \in[0,2\pi]; $$  (D_{3}):

there is a constant \(d> 0\), such that
$$\operatorname{sgn}(x) \bigl[f(t,x)xf_{x}(t,x) \bigr]>2b_{\infty}, \quad\textit{for } x\geq d. $$
Then there exists \(\varepsilon_{0}>0\), such that for any \(\alpha\in(2\gamma\pi,2\gamma\pi+\varepsilon_{0})\), equation (1.1) possesses an AubryMather type solution \(z_{\alpha}(t)=(x_{\alpha}(t),x'_{\alpha}(t))\) with rotation number α, that is:

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

(ii)
if α is irrational, the solution \(z_{\alpha}(t)\) either a usual quasiperiodic solution or a bounded solution such that the closed set
$$M_{\alpha}\equiv\overline{ \bigl\{ z_{\alpha}(2\pi i),i\in\mathbb{Z} \bigr\} } $$is a Denjoy minimal set (see the definition of it in [20]).
Remark 1.1
Applying the rule of L’Hospital to condition (D_{2}), it is easy to see that
 (f_{0}):

$$\lim_{x\to+\infty}\frac{f(t,x)}{x}=0, \quad\mbox{uniformly in } t\in[0,2 \pi]. $$
Example 1.1
Let \(q(t)=1+\frac{1}{4}\sin t\) and \(f(t,x)=\operatorname {sgn}(x)\ln(1+x)\cdot(1+p(t))\), where \(p(t)\) is a continuous function with \(p(t +2\pi) = p(t)\). Then \(q(t)\in C(\mathbf{S}^{1})\) and \(f(t,x)\) meet the conditions of (D_{1})(D_{3}) in Theorem 1.1. We can check it as follows: (i) By simple calculation, we have \(\gamma=1\), \(b_{\infty}= \max_{t\in[0,2\pi]}\frac{ 1}{4}\sin t=\frac{1}{4}\), then \(1=\gamma^{2}>2b_{\infty}=\frac{1}{2}\); (ii) since \(f_{x}(t,0)=1+p(t)\) and \(f_{xx}(t,0)=\infty\), we have \(f(t,x)\in C^{0,1}(\mathbf {S}^{1}\times\mathbb{R})\) and it is obvious that \(f(t,x)\) has limits \(\lim_{x\to+\infty}f_{x}(t,x)=0\), uniformly in \(t\in[0,2\pi]\); (iii) choose \(d=e^{2}\), then we have \(\operatorname {sgn}(x)[f(t,x)xf_{x}(t,x)]\geq1>\frac{1}{2}=2b_{\infty}\), for \(x\geq d\). Thus, according to (i)(iii), the assumptions (D_{1})(D_{3}) in Theorem 1.1 hold.
Remark 1.2
It is easy to verify that the results in [18, 19] cannot be applied to Example 1.1 to obtain the existence of AubryMather sets and quasiperiodic solutions. Therefore, the results obtained in this paper can be viewed as natural generalizations and refinements of the results in [18, 19].
Remark 1.3
It seems that the breakdown of stability (in the sense of the Lagrangian) is related to the smoothness of \(f(t,x)\). And we do not know whether or not \(f(t,x)\in C^{0,0}(\mathbf{S}^{1}\times\mathbb{R})\) is sufficient to guarantee the existence of AubryMather sets and quasiperiodic solutions of equation (1.1).
The main idea of our proof is acquired from [21]. The proof of Theorem 1.1 is based on a version of AubryMather theorem due to Pei [18]. The rest of this paper is organized as follows. In Section 2, we introduce the actionangle variables which transform equation (1.1) into a perturbation of an integral Hamiltonian system. In Section 3, we will show that the Poincaré map of the equivalent system satisfies the monotone twist property around infinity, then some results can be obtained.
2 Actionangle variables and some properties
Let \(q(t)=\gamma^{2}+b(t)\). Then \(b(t)\) is a 2π periodic function and \(\frac{1}{2\pi}\int_{0}^{2\pi}b(t)\,dt=0\). Hence, equation (1.1) is equivalent to the system
Now we introduce the action and angle variables \((I,\theta)\) as follows:
where \(I>0\) and \(\theta\in\mathbf{S}^{1}=\mathbb{R}/2\pi\mathbb{Z}\), then it is not difficult to prove that the mapping \(\Psi:\mathbf{S}^{1}\times(0,\infty)\rightarrow \mathbb{R}^{2}\{0\}\), \((\theta,I)\mapsto(x,y)\) is a canonical transformation, such that (2.1) is transformed into
where
Under the assumptions of (D_{2}) and (D_{3}), it is easy to prove the existence and uniqueness of the solution of the initial value problem associated with (2.3). Moreover, this solution has continuous derivatives with respect to initial data.
Let \((\theta(t;\theta_{0},I_{0}),I(t;\theta_{0},I_{0}))\) be the solution of (2.3) with initial value \(\theta(0)=\theta_{0}\) and \(I(0)=I_{0}\). Then \(x(t;\theta_{0},I_{0})=x(\theta(t;\theta_{0},I_{0}),I(t;\theta _{0},I_{0}))=\sqrt{\frac{2I(t;\theta_{0},I_{0})}{\gamma}}\cos\theta (t;\theta_{0},I_{0})\) is the solution of (1.1).
First, we give the following growth estimates as regards \(I(t;\theta_{0},I_{0})\) and \(\theta(t;\theta_{0},I_{0})\).
For the sake of convenience and simplicity, in the following, we let \(\theta=\theta(t;\theta_{0},I_{0})\), \(I=I(t;\theta_{0},I_{0})\) and \(x=x(\theta,I)=x(\theta(t;\theta_{0},I_{0}),I(t;\theta_{0},I_{0}))\), \(y=y( \theta,I)=y(\theta(t;\theta_{0},I_{0}),I(t;\theta_{0},I_{0}))\).
Lemma 2.1
The limit
holds uniformly on \(t\in[0,2\pi]\).
Proof
From (f_{0}) and coordinate transformation (2.2), there exist constants \(C>0\), \(K>0\), such that \(I'(t)=\frac{y(b(t)x+f(t,x))}{\gamma}\leq CI(t)+K\), \(\forall I\neq0\).
Then, by the Gronwall inequality, one has
for all \(t\in[0,2\pi]\).
So, by (2.4), \(I(t;\theta_{0},I _{0})\to+\infty\) as \(I_{0}\to+\infty\) uniformly for \(t\in[0,2\pi]\). □
Lemma 2.2
There exist constants \(k_{2}>k_{1}>0\) and \(\bar{I}>0\), such that for any \(I_{0}\geq\bar{I}\), we have:

(i)
\(\frac{\gamma}{2}\leq\theta'(t;\theta_{0},I_{0})\leq2\gamma\), \(\forall \theta_{0}\in\mathbb{R}\) and \(\forall t\in [0,2\pi]\);

(ii)
\(k_{1}I_{0}\leq I(t;\theta_{0},I_{0})\leq k_{2}I_{0}\), \(\forall \theta _{0}\in\mathbb{R}\) and \(\forall t\in [0,2\pi]\).
Proof
(i) Since (f_{0}) holds, for every \(\varepsilon>0\) (we may restrict \(0<\varepsilon<\frac{\gamma^{2}}{2}b_{\infty}\)), there exists \(M=M(\varepsilon)>0\), such that
if \(x\geq M\) and \(\forall t\in[0,2\pi]\). Hence, by (2.3), we have
By (D_{1}), we have \(\gamma^{2}>2b_{\infty}\). Thus, we have
In the case \(x\leq M\), we may assume that \(f(t,x)\leq f_{\infty}\), where \(f_{\infty}= \max\{f(t,x):t\in[0,2\pi],x\leq M\}\), then by (2.2), we have
So, by (D_{1}) and Lemma 2.1, there exists a constant \(\bar{I}>0\), such that \(\frac{d\theta}{dt}\geq\frac{\gamma}{2}\) if \(I_{0}\geq\bar{I}\).
Similarly, the same argument as above shows that the inequality on the right side of (i) holds.
(ii) By (2.4), we can easily find constants \(k_{2}>k_{1}>0\) and \(\bar{I}>0\), such that
for any \(I_{0}\geq\bar{I}\) and \(\forall t\in [0,2\pi]\). □
3 Monotone twist property and proof of Theorem 1.1
Let the Poincaré map P of equation (2.3) be
To complete the proof of Theorem 1.1, one can see that it suffices to show that the Poincaré map P is a monotone twist map around infinity, that is, we only need to prove \(\frac{\partial \theta(2\pi;\theta_{0},I_{0}) }{\partial I_{0}}<0\) if \(I_{0}\gg1\). Then the existence of AubryMather sets and quasiperiodic solutions is guaranteed by a generalized AubryMather theorem given by Pei [18].
In the following, we will investigate the behavior of \(\frac{\partial \theta(2\pi;\theta_{0},I_{0}) }{\partial I_{0}} \) when \(I_{0}\gg1\) by some lemmas. For the sake of convenience, in later discussions we also write x, y, θ, I instead of \(x(\theta(t;\theta_{0},I_{0}),I(t;\theta_{0},I_{0}))\), \(y(\theta(t;\theta_{0},I_{0}),I(t;\theta_{0},I_{0}))\), \(\theta(t;\theta_{0},I_{0})\), \(I(t;\theta_{0},I_{0})\), respectively.
Lemma 3.1
The following convergences hold uniformly on \(t\in [0,2\pi]\):

(i)
\(\frac{xf(t,x)}{I}\rightarrow0\), \(\frac{x^{2}f_{x}(t,x)}{I}\rightarrow0\), as \(I_{0}\rightarrow+\infty\);

(ii)
\(\frac{yf(t,x)}{I}\rightarrow0\), \(\frac{yxf_{x}(t,x)}{I}\rightarrow0\), as \(I_{0}\rightarrow+\infty\);

(iii)
\(\frac{f^{2}(t,x)}{I}\rightarrow0\), \(\frac{xf(t,x)f_{x}(t,x)}{I}\rightarrow0\), \(\frac{x^{2}f^{2}_{x}(t,x)}{I}\rightarrow0\), as \(I_{0}\rightarrow+\infty\).
Proof
If (D_{2}) and (f_{0}) hold, then to each \(\varepsilon>0\) there corresponds a positive number \(M=M(\varepsilon)>0\), if \(x\geq M\) and \(\forall t\in [0,2\pi]\), and we have
and
Denote \(K_{1}(\varepsilon)=\max\{f(t,x):t\in[0,2\pi],x\leq M\}\), \(K_{2}(\varepsilon)=\max\{f_{x}(t,x):t\in[0,2\pi], x\leq M\}\).
(i) According to the coordinate transformation (2.2), one has
Then, given \(\bar{I}>0\), choose \(I_{0}\) so that \(I_{0}\geq\bar{I}\), by using Lemma 2.2(ii), provided
we have
Since \(\varepsilon>0\) is arbitrary, the proof of (i) is complete.
(ii) Corresponding to (2.2), one can see that
Then, given \(\bar{I}>0\), choose \(I_{0}\) so that \(I_{0}\geq \bar{I}\), by using Lemma 2.2(ii), provided
we have
Since \(\varepsilon>0\) is arbitrary, (ii) is proved.
(iii) From (2.2) we deduce that
Then, given \(\bar{I}>0\), choose \(I_{0}\) so that \(I_{0}\geq \bar{I}\), by using Lemma 2.2(ii), provided
we have
Since \(\varepsilon>0\) is arbitrary, we get (iii). □
For \(t\in[0,2\pi]\), set
Using Lemma 2.1, Lemma 2.2, and Lemma 3.1, we have the following.
Lemma 3.2
For \(t,s \in[0,2\pi]\), the following conclusions hold:

(i)
\(a_{1}(t)=o(\frac{1}{I_{0}})\), as \(I_{0}\rightarrow+\infty\);

(ii)
\(a_{21}(t)=o(1)\); \(a_{22}(t)\leq\frac{b_{\infty}}{\gamma}\) as \(I_{0}\rightarrow+\infty\);

(iii)
\(a_{1}(t)\cdot a_{3}(s)=o(1)\), as \(I_{0}\rightarrow+\infty\).
Here and below \(o(1)\) denotes a generic infinitesimal as \(I_{0}\rightarrow\infty\).
Let us consider the variational equation (2.3) with respect to the initial value \(I_{0}\). One can verify that
For convenience, we set \(\sigma_{1}=e^{\frac{2\pi b_{\infty}}{\gamma }}\), \(\sigma_{2}=e^{\frac{2\pi b_{\infty}}{\gamma}}\). Then we have the following.
Lemma 3.3
For all \(t \in(0,2\pi]\), \(I_{0}\rightarrow +\infty\), we have:

(i)
\(\theta_{I_{0}}(t;\theta_{0}, I_{0})\rightarrow0\);

(ii)
\(\sigma_{1}(1+o(1))\leq I_{I_{0}}(t;\theta_{0}, I_{0})\leq\sigma_{2}(1+o(1))\);

(iii)
\(\sigma_{1}(1+o(1))\leq\theta_{\theta_{0}}(t;\theta_{0},I_{0})\leq \sigma_{2}(1+o(1))\).
Proof
From the variational equations (3.1), one has
here we have used \(\theta_{I_{0}}(0)=0\). Then, if \(\int_{0}^{t} a_{1}(s)\cdot I_{I_{0}}(s)\,ds\geq0\) for \(t\in(0,2\pi ]\), by Lemma 3.2(ii), we have
and if \(\int_{0}^{t} a_{1}(s)\cdot I_{I_{0}}(s)\,ds\leq0\) for \(t\in(0,2\pi ]\), by Lemma 3.2(ii), we have
On the other hand, by the variational equations (3.1), we have
here we have used \(I_{I_{0}}(0)=1\). Now we will discuss it in the following two cases:
Case 1. If \(\int_{0}^{t} a_{1}(s)\cdot I_{I_{0}}(s)\,ds\geq0\) and \(a_{3}(t)\geq0\) for \(t\in(0, 2\pi]\), or \(\int_{0}^{t} a_{1}(s)\cdot I_{I_{0}}(s)\,ds\leq0\) and \(a_{3}(t)\leq0\) for \(t\in(0, 2\pi]\), then by Lemma 3.2(ii) and (iii), we have
and
Case 2. If \(\int_{0}^{t} a_{1}(s)\cdot I_{I_{0}}(s)\,ds\geq0\) and \(a_{3}(t)\leq0\) for \(t\in(0, 2\pi]\), or \(\int_{0}^{t} a_{1}(s)\cdot I_{I_{0}}(s)\,ds\leq0\) and \(a_{3}(t)\geq0\) for \(t\in(0, 2\pi]\), by Lemma 3.2(ii) and (iii), we get
and
Hence, for \(t\in(0,2\pi]\), \(I_{0}\rightarrow+\infty\), by Lemma 3.2(i) and (3.2)(3.7), we have
and
Thus, (i) and (ii) are proved.
To prove (iii), we consider the variational equation of (2.3) about \(\theta_{0}\),
Similar to the proof of (ii), one can see that
for \(t\in (0,2\pi]\), \(I_{0}\rightarrow+\infty\). This proves the statement of Lemma 3.3. □
Now, we further give the growth estimates on the function \(a_{1}(t)\).
Lemma 3.4
Let d satisfy (D_{3}).

(i)
If \(x(t;\theta_{0},I_{0})\leq d\) for \(t \in[0,2\pi]\), \(\theta_{0}\in\mathbb{R} \), and \(I_{0}\in\mathbb {R}^{+}\), then there exists a constant \(K_{d}>0\), such that \(a_{1}(t)\leq\frac{K_{d}}{I^{2}(t)}\).

(ii)
If \(x(t;\theta_{0},I_{0})\geq d\) for \(t \in[0,2\pi]\), \(\theta_{0}\in\mathbb{R} \), and \(I_{0}\in\mathbb{R}^{+}\), then there exists a constant \(L_{d}>0\), such that \(a_{1}(t)\geq \frac{L_{d}}{I^{2}(t)}\). Moreover, \(x(t;\theta_{0},I_{0})\geq d\) implies that \(a_{1}(t)<0\).
Proof
(i) If \(x(t;\theta_{0},I_{0})\leq d\) for \(t \in[0,2\pi]\), \(\theta_{0}\in\mathbb{R} \), and \(I_{0}\in\mathbb {R}^{+}\), we write
Then
Set \(K_{d}=\frac{dM_{d}}{4}\), we have \(a_{1}(t)\leq\frac{K_{d}}{I^{2}(t)}\).
(ii) If \(x(t;\theta_{0},I_{0})\geq d\) for \(t \in[0,2\pi]\), \(\theta_{0}\in\mathbb{R} \) and \(I_{0}\in\mathbb{R}^{+}\), then from condition (D_{3}), one can see that \(x[f(t,x)xf_{x}(t,x)]>0\), \(\operatorname{sgn}(x)[f(t,x)xf_{x}(t,x)]> b_{\infty}\). Hence, \(a_{1}(t)<0\) and
Thus, putting \(L_{d}=\frac{db_{\infty}}{4}\), we have \(a_{1}(t)\geq \frac{L_{d}}{I^{2}(t)}\). This completes the proof. □
Write \(a_{1}(t)=a_{1}^{+}(t)a_{1}^{}(t)\), where \(a_{1}^{\pm}(t)=\max\{\pm a_{1}(t),0\}\). To see that 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 simple lemma.
Lemma 3.5
Let d be as in Theorem 1.1. Define the set \(\Delta t=\{t\in[0,2\pi] x(t;\theta_{0},I_{0})\leq d\}\). Then there exist \(\bar{I_{0}}>0\), \(K_{3}>0\), such that \(\Delta t\leq \frac{K_{3}}{\sqrt{I_{0}}}\), for all \(I_{0}\geq\bar{I_{0}}\).
Proof
By Lemma 2.2(i), we have \(\Delta t\rightarrow0\) when and only when \(\Delta\theta\rightarrow0\).
According to the coordinate transformation (2.2), we see that \(\tan\Delta\theta\leq\frac{d}{\sqrt{2\gamma I(t)}}\) when \(\Delta\theta\rightarrow0 \). Therefore, by using Lemma 2.2(ii), we know that there exist \(\bar{I_{0}}>0\), \(K_{3}>0\), such that
for all \(I_{0}\geq\bar{I_{0}}\). Lemma 3.5 follows. □
Lemma 3.6
If \(I_{0}\gg1\), then \(\int_{0}^{2\pi}a_{1}(s)\,ds<0\).
Proof
According to the above discussions and Lemma 2.2(ii), we have
So, if \(\sqrt{I_{0}}>\frac{(k_{2}^{2}K_{d}+k_{1}^{2}L_{d})K_{3}}{2\pi k_{1}^{2}L_{d}}\), then \(\int_{0}^{2\pi}a_{1}(s)\,ds<0\). □
Therefore, in view of Lemma 3.3(ii) and Lemma 3.6, when \(I_{0} \gg1\), the following cannot occur:
then by (3.2), the case of
does not happen. Hence, by (3.3), Lemma 3.3(ii), and Lemma 3.6, we have
if \(I_{0} \gg1\).
Using similar arguments to [18], one may extend the Poincaré map P to a new one which is a globally monotone twist map and which is guaranteed by the AubryMather theorem [18]. From the above discussion, we come to the conclusion that there exists \(\varepsilon_{0}>0\), such that for any \(\alpha\in(2\gamma\pi,2\gamma\pi+\varepsilon_{0})\), equation (1.1) has a solution \(z_{\alpha}(t)=(x_{\alpha}(t),x'_{\alpha}(t))\) of AubryMather type with rotation number α. Thus, the proof of Theorem 1.1 is completed.
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Acknowledgements
Xiaoming Wang ang Hainv Tan are partially supported by NSFC (No. 11461056). Lixia Wang is partially supported by Tian Yuan Special Foundation (No. 11526148).
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Wang, X., Wang, L. & Tan, H. A new approach to the existence of quasiperiodic solutions for a class of semilinear Duffingtype equations with timeperiodic parameters. Bound Value Probl 2016, 132 (2016). https://doi.org/10.1186/s1366101606405
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DOI: https://doi.org/10.1186/s1366101606405