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A new approach to the existence of quasiperiodic solutions for a class of semilinear Duffingtype equations with timeperiodic parameters
 Xiaoming Wang^{1, 2}Email author,
 Lixia Wang^{3} and
 Hainv Tan^{1}
 Received: 23 February 2016
 Accepted: 6 July 2016
 Published: 14 July 2016
Abstract
Keywords
 Duffingtype equation
 semilinear
 AubryMather sets
 quasiperiodic solutions
MSC
 34C12
 37C55
1 Introduction
 (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.
 (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?
 (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
 (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$$\operatorname{sgn}(x) \bigl[f(t,x)xf_{x}(t,x) \bigr]>2b_{\infty}, \quad\textit{for } x\geq d. $$
 (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 setis a Denjoy minimal set (see the definition of it in [20]).$$M_{\alpha}\equiv\overline{ \bigl\{ z_{\alpha}(2\pi i),i\in\mathbb{Z} \bigr\} } $$
Remark 1.1
 (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
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
Proof
From (f_{0}) and coordinate transformation (2.2), there exist constants \(C>0\), \(K>0\), such that \(I'(t)=\frac{y(b(t)x+f(t,x))}{\gamma}\leq CI(t)+K\), \(\forall I\neq0\).
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
 (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
Similarly, the same argument as above shows that the inequality on the right side of (i) holds.
3 Monotone twist property and proof of Theorem 1.1
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
 (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
Denote \(K_{1}(\varepsilon)=\max\{f(t,x):t\in[0,2\pi],x\leq M\}\), \(K_{2}(\varepsilon)=\max\{f_{x}(t,x):t\in[0,2\pi], x\leq M\}\).
Since \(\varepsilon>0\) is arbitrary, the proof of (i) is complete.
Since \(\varepsilon>0\) is arbitrary, (ii) is proved.
Since \(\varepsilon>0\) is arbitrary, we get (iii). □
Using Lemma 2.1, Lemma 2.2, and Lemma 3.1, we have the following.
Lemma 3.2
 (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 \(I_{0}\rightarrow\infty\).
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
 (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
Thus, (i) and (ii) are proved.
Now, we further give the growth estimates on the function \(a_{1}(t)\).
Lemma 3.4
 (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
Set \(K_{d}=\frac{dM_{d}}{4}\), we have \(a_{1}(t)\leq\frac{K_{d}}{I^{2}(t)}\).
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 \(\Delta\theta\rightarrow0\).
Lemma 3.6
If \(I_{0}\gg1\), then \(\int_{0}^{2\pi}a_{1}(s)\,ds<0\).
Proof
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.
Declarations
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).
Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.
Authors’ Affiliations
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