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Boundedness of solutions for a class of second-order differential equation with singularity

Shunjun Jiang

Author Affiliations

College of Sciences, Nanjing University of Technology, Nanjing, 210009, People’s Republic of China

Boundary Value Problems 2013, 2013:84  doi:10.1186/1687-2770-2013-84


The electronic version of this article is the complete one and can be found online at: http://www.boundaryvalueproblems.com/content/2013/1/84


Received:14 September 2012
Accepted:22 March 2013
Published:10 April 2013

© 2013 Jiang; licensee Springer

This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

In this paper, we study the following second-order periodic system:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/84/mathml/M1','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/84/mathml/M1">View MathML</a>

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/84/mathml/M2','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/84/mathml/M2">View MathML</a> has a singularity. Under some assumptions on the <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/84/mathml/M2','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/84/mathml/M2">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/84/mathml/M4','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/84/mathml/M4">View MathML</a> by Ortega’s small twist theorem, we obtain the existence of quasi-periodic solutions and boundedness of all the solutions.

Keywords:
boundedness of solutions; singularity; small twist theorem

1 Introduction and main result

In 1991, Levi [1] considered the following equation:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/84/mathml/M5','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/84/mathml/M5">View MathML</a>

(1.1)

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/84/mathml/M6','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/84/mathml/M6">View MathML</a> satisfies some growth conditions and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/84/mathml/M7','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/84/mathml/M7">View MathML</a>. The author reduced the system to a normal form and then applied Moser twist theorem to prove the existence of quasi-periodic solution and the boundedness of all solutions. This result relies on the fact that the nonlinearity <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/84/mathml/M6','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/84/mathml/M6">View MathML</a> can guarantee the twist condition of KAM theorem. Later, several authors improved the Levi’s result; we refer to [2-4] and the references therein.

Recently, Capietto, Dambrosio and Liu [5] studied the following equation:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/84/mathml/M9','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/84/mathml/M9">View MathML</a>

(1.2)

with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/84/mathml/M10','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/84/mathml/M10">View MathML</a> is a π-periodic function and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/84/mathml/M11','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/84/mathml/M11">View MathML</a>, where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/84/mathml/M12','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/84/mathml/M12">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/84/mathml/M13','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/84/mathml/M13">View MathML</a> and ν is a positive integer. Under the Lazer-Leach assumption that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/84/mathml/M14','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/84/mathml/M14">View MathML</a>

(1.3)

they prove the boundedness of solutions and the existence of quasi-periodic solution by KAM theorem. It is the first time that the equation of the boundedness of all solution is treated in case of a singular potential.

We observe that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/84/mathml/M10','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/84/mathml/M10">View MathML</a> in (1.2) is smooth and bounded, so a natural question is to find sufficient conditions on <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/84/mathml/M16','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/84/mathml/M16">View MathML</a> such that all solutions of (1.2) are bounded when <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/84/mathml/M16','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/84/mathml/M16">View MathML</a> is unbounded. The purpose of this paper is to deal with this problem.

Motivated by the papers [1,5,6], we consider the following equation:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/84/mathml/M18','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/84/mathml/M18">View MathML</a>

(1.4)

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/84/mathml/M4','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/84/mathml/M4">View MathML</a> is a π-periodic function,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/84/mathml/M20','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/84/mathml/M20">View MathML</a>

(1.5)

We suppose Lazer-Leach assumption hold:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/84/mathml/M21','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/84/mathml/M21">View MathML</a>

(1.6)

Our main result is the following theorem.

Theorem 1Under the assumptions (1.5) and (1.6), all the solutions of (1.4) are bounded.

The main idea of our proof is acquired from [6]. The proof of Theorem 1 is based on a small twist theorem due to Ortega [7]. It mainly consists of two steps. The first one is to transform (1.4) into a perturbation of integrable Hamilton system. The second one is to show that Poincaré map of the equivalent system satisfies Ortega’s twist theorem, then some desired result can be obtained.

Moreover, we have the following theorem on solutions of Aubry-Mather type.

Theorem 2Assume that<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/84/mathml/M22','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/84/mathml/M22">View MathML</a>satisfies (1.6); then, there is an<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/84/mathml/M23','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/84/mathml/M23">View MathML</a>such that, for any<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/84/mathml/M24','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/84/mathml/M24">View MathML</a>, tequation (1.4) has a solution<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/84/mathml/M25','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/84/mathml/M25">View MathML</a>of the Mather type with rotation numberω. More precisely:

Case 1: <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/84/mathml/M26','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/84/mathml/M26">View MathML</a>is rational. The solutions<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/84/mathml/M27','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/84/mathml/M27">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/84/mathml/M28','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/84/mathml/M28">View MathML</a>are independent periodic solutions of periodic; moreover, in this case,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/84/mathml/M29','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/84/mathml/M29">View MathML</a>

Case 2: ωis irrational. The solution<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/84/mathml/M30','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/84/mathml/M30">View MathML</a>is either a usual quasi-periodic solution or a generalized one.

We will apply Aubry-Mather theory, more precisely, the theorem in [8], to prove this theorem.

2 Proof of theorem

2.1 Action-angle variables and some estimates

Observe that (1.4) is equivalent to the following Hamiltonian system:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/84/mathml/M31','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/84/mathml/M31">View MathML</a>

(2.1)

with the Hamiltonian function

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/84/mathml/M32','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/84/mathml/M32">View MathML</a>

In order to introduce action and angle variables, we first consider the auxiliary autonomous equation:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/84/mathml/M33','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/84/mathml/M33">View MathML</a>

(2.2)

which is an integrable Hamiltonian system with Hamiltonian function

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/84/mathml/M34','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/84/mathml/M34">View MathML</a>

The closed curves <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/84/mathml/M35','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/84/mathml/M35">View MathML</a> are just the integral curves of (2.2).

Denote by <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/84/mathml/M36','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/84/mathml/M36">View MathML</a> the time period of the integral curve <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/84/mathml/M37','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/84/mathml/M37">View MathML</a> of (2.2) defined by <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/84/mathml/M38','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/84/mathml/M38">View MathML</a> and by I the area enclosed by the closed curve <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/84/mathml/M37','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/84/mathml/M37">View MathML</a> for every <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/84/mathml/M40','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/84/mathml/M40">View MathML</a>. Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/84/mathml/M41','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/84/mathml/M41">View MathML</a> be such that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/84/mathml/M42','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/84/mathml/M42">View MathML</a>. It is easy to see that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/84/mathml/M43','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/84/mathml/M43">View MathML</a>

and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/84/mathml/M44','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/84/mathml/M44">View MathML</a>

By direct computation, we get

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/84/mathml/M45','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/84/mathml/M45">View MathML</a>

so

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/84/mathml/M46','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/84/mathml/M46">View MathML</a>

We then have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/84/mathml/M47','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/84/mathml/M47">View MathML</a>

where

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/84/mathml/M48','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/84/mathml/M48">View MathML</a>

Similar in estimating in [5], we have the estimation of functions <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/84/mathml/M49','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/84/mathml/M49">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/84/mathml/M50','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/84/mathml/M50">View MathML</a>.

Lemma 1We have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/84/mathml/M51','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/84/mathml/M51">View MathML</a>

and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/84/mathml/M52','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/84/mathml/M52">View MathML</a>

where<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/84/mathml/M53','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/84/mathml/M53">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/84/mathml/M54','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/84/mathml/M54">View MathML</a>. Note that here and below we always useC, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/84/mathml/M55','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/84/mathml/M55">View MathML</a>or<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/84/mathml/M56','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/84/mathml/M56">View MathML</a>to indicate some constants.

Remark 1 It follows from the definitions of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/84/mathml/M57','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/84/mathml/M57">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/84/mathml/M58','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/84/mathml/M58">View MathML</a> and Lemma 1 that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/84/mathml/M59','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/84/mathml/M59">View MathML</a>

Thus the time period <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/84/mathml/M36','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/84/mathml/M36">View MathML</a> is dominated by <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/84/mathml/M57','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/84/mathml/M57">View MathML</a> when h is sufficiently large. From the relation between <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/84/mathml/M58','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/84/mathml/M58">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/84/mathml/M63','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/84/mathml/M63">View MathML</a>, we know <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/84/mathml/M64','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/84/mathml/M64">View MathML</a> is dominated by <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/84/mathml/M65','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/84/mathml/M65">View MathML</a> when h is sufficiently large.

Remark 2 It also follow from the definition of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/84/mathml/M66','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/84/mathml/M66">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/84/mathml/M63','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/84/mathml/M63">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/84/mathml/M65','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/84/mathml/M65">View MathML</a> and Remark 1 that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/84/mathml/M69','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/84/mathml/M69">View MathML</a>

Remark 3 Note that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/84/mathml/M70','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/84/mathml/M70">View MathML</a> is the inverse function of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/84/mathml/M71','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/84/mathml/M71">View MathML</a>. By Remark 2, we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/84/mathml/M72','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/84/mathml/M72">View MathML</a>

We now carry out the standard reduction to the action-angle variables. For this purpose, we define the generating function <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/84/mathml/M73','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/84/mathml/M73">View MathML</a>, where C is the part of the closed curve <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/84/mathml/M37','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/84/mathml/M37">View MathML</a> connecting the point on the y-axis and point <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/84/mathml/M75','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/84/mathml/M75">View MathML</a>.

We define the well-know map <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/84/mathml/M76','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/84/mathml/M76">View MathML</a> by

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/84/mathml/M77','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/84/mathml/M77">View MathML</a>

which is symplectic since

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/84/mathml/M78','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/84/mathml/M78">View MathML</a>

From the above discussion, we can easily get

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/84/mathml/M79','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/84/mathml/M79">View MathML</a>

(2.3)

and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/84/mathml/M80','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/84/mathml/M80">View MathML</a>

(2.4)

In the new variables <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/84/mathml/M81','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/84/mathml/M81">View MathML</a>, the system (2.1) is

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/84/mathml/M82','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/84/mathml/M82">View MathML</a>

(2.5)

where

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/84/mathml/M83','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/84/mathml/M83">View MathML</a>

(2.6)

In order to estimate <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/84/mathml/M84','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/84/mathml/M84">View MathML</a>, we need the following lemma.

Lemma 2 [[5], Lemma 2.2]

ForIsufficient large and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/84/mathml/M85','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/84/mathml/M85">View MathML</a>, the following estimates hold:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/84/mathml/M86','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/84/mathml/M86">View MathML</a>

2.2 New action and angle variables

Now we are concerned with the Hamiltonian system (2.5) with Hamiltonian function <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/84/mathml/M87','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/84/mathml/M87">View MathML</a> given by (2.6). Note that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/84/mathml/M88','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/84/mathml/M88">View MathML</a>

This means that if one can solve I from (2.6) as a function of H (θ and t as parameters), then

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/84/mathml/M89','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/84/mathml/M89">View MathML</a>

(2.7)

is also a Hamiltonian system with Hamiltonian function I and now the action, angle and time variables are H, t and θ.

From (2.6) and Lemma 1, we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/84/mathml/M90','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/84/mathml/M90">View MathML</a>

So, we assume that I can be written as

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/84/mathml/M91','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/84/mathml/M91">View MathML</a>

where R satisfies <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/84/mathml/M92','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/84/mathml/M92">View MathML</a>. Recalling that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/84/mathml/M93','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/84/mathml/M93">View MathML</a> is the inverse function of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/84/mathml/M71','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/84/mathml/M71">View MathML</a>, we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/84/mathml/M95','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/84/mathml/M95">View MathML</a>

which implies that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/84/mathml/M96','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/84/mathml/M96">View MathML</a>

As a consequence, R is implicitly defined by

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/84/mathml/M97','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/84/mathml/M97">View MathML</a>

(2.8)

Lemma 3The function<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/84/mathml/M98','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/84/mathml/M98">View MathML</a>satisfies the following estimates:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/84/mathml/M99','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/84/mathml/M99">View MathML</a>

Proof Case <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/84/mathml/M100','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/84/mathml/M100">View MathML</a>. By (2.8), Lemma 2 and noticing that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/84/mathml/M101','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/84/mathml/M101">View MathML</a> as <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/84/mathml/M102','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/84/mathml/M102">View MathML</a>, we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/84/mathml/M103','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/84/mathml/M103">View MathML</a>

Case <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/84/mathml/M104','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/84/mathml/M104">View MathML</a>. Derivative both sides of (2.8) with respect to H, we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/84/mathml/M105','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/84/mathml/M105">View MathML</a>

By Remark 2, Lemma 2 and the estimate of R, we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/84/mathml/M106','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/84/mathml/M106">View MathML</a>

Since

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/84/mathml/M107','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/84/mathml/M107">View MathML</a>

we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/84/mathml/M108','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/84/mathml/M108">View MathML</a>

We suppose that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/84/mathml/M109','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/84/mathml/M109">View MathML</a>

(2.9)

holds where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/84/mathml/M110','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/84/mathml/M110">View MathML</a>. We will prove (2.9) also holds where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/84/mathml/M111','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/84/mathml/M111">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/84/mathml/M112','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/84/mathml/M112">View MathML</a>.

By direct calculation, we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/84/mathml/M113','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/84/mathml/M113">View MathML</a>

(2.10)

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/84/mathml/M114','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/84/mathml/M114">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/84/mathml/M115','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/84/mathml/M115">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/84/mathml/M116','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/84/mathml/M116">View MathML</a>.

Since

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/84/mathml/M117','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/84/mathml/M117">View MathML</a>

by Lemma 1 and (2.9), when <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/84/mathml/M118','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/84/mathml/M118">View MathML</a>, we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/84/mathml/M119','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/84/mathml/M119">View MathML</a>

(2.11)

When <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/84/mathml/M120','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/84/mathml/M120">View MathML</a>, we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/84/mathml/M121','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/84/mathml/M121">View MathML</a>

(2.12)

By (2.11) and (2.12), we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/84/mathml/M122','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/84/mathml/M122">View MathML</a>

(2.13)

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/84/mathml/M123','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/84/mathml/M123">View MathML</a>.

By (2.13), we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/84/mathml/M124','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/84/mathml/M124">View MathML</a>

(2.14)

By (2.10), (2.14) and Lemma 2, we have (2.9) holds where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/84/mathml/M111','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/84/mathml/M111">View MathML</a>. Thus, we prove Lemma 3. □

Analogously, one may obtain, by a direct but cumbersome commutation, the following estimates.

Lemma 4The function<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/84/mathml/M98','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/84/mathml/M98">View MathML</a>satisfies the following estimates:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/84/mathml/M127','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/84/mathml/M127">View MathML</a>

Moreover, by the implicit function theorem, there exists a function <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/84/mathml/M128','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/84/mathml/M128">View MathML</a> such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/84/mathml/M129','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/84/mathml/M129">View MathML</a>

Since

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/84/mathml/M130','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/84/mathml/M130">View MathML</a>

for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/84/mathml/M131','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/84/mathml/M131">View MathML</a>, we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/84/mathml/M132','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/84/mathml/M132">View MathML</a>

For <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/84/mathml/M133','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/84/mathml/M133">View MathML</a>, we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/84/mathml/M134','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/84/mathml/M134">View MathML</a>

By Lemmas 1 and 4, we have the estimates on <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/84/mathml/M135','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/84/mathml/M135">View MathML</a>.

For concision, in the estimates and the calculation below, we only consider the case <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/84/mathml/M131','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/84/mathml/M131">View MathML</a>, since the case <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/84/mathml/M133','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/84/mathml/M133">View MathML</a> have the similar result.

Lemma 5<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/84/mathml/M138','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/84/mathml/M138">View MathML</a>for<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/84/mathml/M139','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/84/mathml/M139">View MathML</a>.

For the estimates of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/84/mathml/M140','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/84/mathml/M140">View MathML</a>, we need the estimates on <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/84/mathml/M141','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/84/mathml/M141">View MathML</a>. By Lemmas 1 and 5, noticing that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/84/mathml/M142','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/84/mathml/M142">View MathML</a>, we have the following lemma.

Lemma 6<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/84/mathml/M143','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/84/mathml/M143">View MathML</a>for<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/84/mathml/M139','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/84/mathml/M139">View MathML</a>.

Now the new Hamiltonian function <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/84/mathml/M145','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/84/mathml/M145">View MathML</a> is written in the form

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/84/mathml/M146','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/84/mathml/M146">View MathML</a>

The system (2.7) is of the form

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/84/mathml/M147','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/84/mathml/M147">View MathML</a>

(2.15)

Introduce a new action variable <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/84/mathml/M148','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/84/mathml/M148">View MathML</a> and a parameter <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/84/mathml/M149','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/84/mathml/M149">View MathML</a> by <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/84/mathml/M150','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/84/mathml/M150">View MathML</a>. Then <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/84/mathml/M151','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/84/mathml/M151">View MathML</a>. Under this transformation, the system (2.15) is changed into the form

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/84/mathml/M152','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/84/mathml/M152">View MathML</a>

(2.16)

which is also Hamiltonian system with the new Hamiltonian function

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/84/mathml/M153','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/84/mathml/M153">View MathML</a>

Obviously, if <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/84/mathml/M154','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/84/mathml/M154">View MathML</a>, the solution <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/84/mathml/M155','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/84/mathml/M155">View MathML</a> of (2.16) with the initial date <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/84/mathml/M156','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/84/mathml/M156">View MathML</a> is defined in the interval <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/84/mathml/M157','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/84/mathml/M157">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/84/mathml/M158','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/84/mathml/M158">View MathML</a>. So the Poincaré map of (2.16) is well defined in the domain <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/84/mathml/M159','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/84/mathml/M159">View MathML</a>.

Lemma 7 [[6], Lemma 5.1]

The Poincaré map of (2.16) has intersection property.

The proof is similar to the corresponding one in [6].

For convenience, we introduce the notation <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/84/mathml/M160','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/84/mathml/M160">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/84/mathml/M161','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/84/mathml/M161">View MathML</a>. We say a function <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/84/mathml/M162','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/84/mathml/M162">View MathML</a> if f is smooth in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/84/mathml/M163','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/84/mathml/M163">View MathML</a> and for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/84/mathml/M164','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/84/mathml/M164">View MathML</a>,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/84/mathml/M165','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/84/mathml/M165">View MathML</a>

for some constant <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/84/mathml/M166','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/84/mathml/M166">View MathML</a> which is independent of the arguments t, ρ, θ, ϵ.

Similarly, we say <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/84/mathml/M167','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/84/mathml/M167">View MathML</a> if f is smooth in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/84/mathml/M163','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/84/mathml/M163">View MathML</a> and for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/84/mathml/M164','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/84/mathml/M164">View MathML</a>,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/84/mathml/M170','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/84/mathml/M170">View MathML</a>

uniformly in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/84/mathml/M171','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/84/mathml/M171">View MathML</a>.

2.3 Poincaré map and twist theorems

We will use Ortega’s small twist theorem to prove that the Poincaré map P has an invariant closed curve, if ϵ is sufficiently small. Let us first recall the theorem in [7].

Lemma 8 (Ortega’s theorem)

Let<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/84/mathml/M172','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/84/mathml/M172">View MathML</a>be a finite cylinder with universal cover<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/84/mathml/M173','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/84/mathml/M173">View MathML</a>. The coordinate in<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/84/mathml/M174','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/84/mathml/M174">View MathML</a>is denoted by<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/84/mathml/M175','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/84/mathml/M175">View MathML</a>. Consider a map

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/84/mathml/M176','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/84/mathml/M176">View MathML</a>

We assume that the map has the intersection property. Suppose that<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/84/mathml/M177','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/84/mathml/M177">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/84/mathml/M178','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/84/mathml/M178">View MathML</a>is a lift of<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/84/mathml/M179','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/84/mathml/M179">View MathML</a>and it has the form

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/84/mathml/M180','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/84/mathml/M180">View MathML</a>

(2.17)

whereNis an integer, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/84/mathml/M181','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/84/mathml/M181">View MathML</a>is a parameter. The functions<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/84/mathml/M182','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/84/mathml/M182">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/84/mathml/M183','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/84/mathml/M183">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/84/mathml/M184','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/84/mathml/M184">View MathML</a>and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/84/mathml/M185','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/84/mathml/M185">View MathML</a>satisfy

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/84/mathml/M186','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/84/mathml/M186">View MathML</a>

(2.18)

In addition, we assume that there is a function<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/84/mathml/M187','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/84/mathml/M187">View MathML</a>satisfying

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/84/mathml/M188','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/84/mathml/M188">View MathML</a>

(2.19)

and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/84/mathml/M189','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/84/mathml/M189">View MathML</a>

(2.20)

Moreover, suppose that there are two numbers<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/84/mathml/M190','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/84/mathml/M190">View MathML</a>, and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/84/mathml/M191','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/84/mathml/M191">View MathML</a>such that<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/84/mathml/M192','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/84/mathml/M192">View MathML</a>and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/84/mathml/M193','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/84/mathml/M193">View MathML</a>

(2.21)

where

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/84/mathml/M194','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/84/mathml/M194">View MathML</a>

Then there exist<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/84/mathml/M149','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/84/mathml/M149">View MathML</a>and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/84/mathml/M196','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/84/mathml/M196">View MathML</a>such that, if<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/84/mathml/M197','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/84/mathml/M197">View MathML</a>and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/84/mathml/M198','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/84/mathml/M198">View MathML</a>

the mapping<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/84/mathml/M179','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/84/mathml/M179">View MathML</a>has an invariant curve in<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/84/mathml/M200','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/84/mathml/M200">View MathML</a>. The constantϵis independent ofδ.

We make the ansatz that the solution of (2.16) with the initial condition <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/84/mathml/M201','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/84/mathml/M201">View MathML</a> is of the form

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/84/mathml/M202','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/84/mathml/M202">View MathML</a>

Then the Poincaré map of (2.16) is

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/84/mathml/M203','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/84/mathml/M203">View MathML</a>

(2.22)

The functions <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/84/mathml/M204','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/84/mathml/M204">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/84/mathml/M205','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/84/mathml/M205">View MathML</a> satisfy

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/84/mathml/M206','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/84/mathml/M206">View MathML</a>

(2.23)

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/84/mathml/M207','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/84/mathml/M207">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/84/mathml/M208','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/84/mathml/M208">View MathML</a>. By Lemmas 4, 6 and 7, we know that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/84/mathml/M209','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/84/mathml/M209">View MathML</a>

(2.24)

Hence, for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/84/mathml/M210','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/84/mathml/M210">View MathML</a>, we may choose ϵ sufficiently small such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/84/mathml/M211','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/84/mathml/M211">View MathML</a>

(2.25)

Moreover, we can prove that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/84/mathml/M212','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/84/mathml/M212">View MathML</a>

(2.26)

Lemma 9The following estimates hold:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/84/mathml/M213','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/84/mathml/M213">View MathML</a>

Proof

Let

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/84/mathml/M214','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/84/mathml/M214">View MathML</a>

By Lemma 2 and (2.25), we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/84/mathml/M215','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/84/mathml/M215">View MathML</a>

Take the derivative with respect to <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/84/mathml/M216','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/84/mathml/M216">View MathML</a> in the both sides of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/84/mathml/M217','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/84/mathml/M217">View MathML</a>, we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/84/mathml/M218','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/84/mathml/M218">View MathML</a>

Using Lemma 2 and noticing <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/84/mathml/M219','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/84/mathml/M219">View MathML</a>, we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/84/mathml/M220','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/84/mathml/M220">View MathML</a>

Analogously, one may obtain, by a direct but cumbersome commutation that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/84/mathml/M221','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/84/mathml/M221">View MathML</a>

which means that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/84/mathml/M222','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/84/mathml/M222">View MathML</a>

The estimates for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/84/mathml/M223','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/84/mathml/M223">View MathML</a> follow from a similar argument, we omit it here. Thus, Lemma 9 is proved. □

Now we turn to give an asymptotic expression of Poincaré map of (2.15), that is, we study the behavior of the functions <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/84/mathml/M204','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/84/mathml/M204">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/84/mathml/M205','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/84/mathml/M205">View MathML</a> at <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/84/mathml/M226','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/84/mathml/M226">View MathML</a> as <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/84/mathml/M227','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/84/mathml/M227">View MathML</a>. In order to estimate <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/84/mathml/M204','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/84/mathml/M204">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/84/mathml/M205','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/84/mathml/M205">View MathML</a>, we need introduce the following definition and lemma. Let

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/84/mathml/M230','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/84/mathml/M230">View MathML</a>

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/84/mathml/M231','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/84/mathml/M231">View MathML</a>.

Lemma 10

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/84/mathml/M232','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/84/mathml/M232">View MathML</a>

Proof This lemma was proved in [5], so we omit the details. □

For estimate <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/84/mathml/M204','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/84/mathml/M204">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/84/mathml/M205','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/84/mathml/M205">View MathML</a>, we need the estimates of x and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/84/mathml/M235','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/84/mathml/M235">View MathML</a>.

We recall that when <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/84/mathml/M133','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/84/mathml/M133">View MathML</a>, we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/84/mathml/M237','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/84/mathml/M237">View MathML</a>

When <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/84/mathml/M238','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/84/mathml/M238">View MathML</a>, by the definition of θ, we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/84/mathml/M239','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/84/mathml/M239">View MathML</a>

which yields that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/84/mathml/M240','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/84/mathml/M240">View MathML</a>

Now we can give the estimates of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/84/mathml/M204','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/84/mathml/M204">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/84/mathml/M205','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/84/mathml/M205">View MathML</a>.

Lemma 11The following estimates hold true:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/84/mathml/M243','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/84/mathml/M243">View MathML</a>

for<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/84/mathml/M227','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/84/mathml/M227">View MathML</a>.

Proof Firstly, we consider <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/84/mathml/M204','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/84/mathml/M204">View MathML</a>. By Lemmas 2, 6 and (2.23), we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/84/mathml/M246','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/84/mathml/M246">View MathML</a>

Now we consider <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/84/mathml/M205','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/84/mathml/M205">View MathML</a>.

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/84/mathml/M248','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/84/mathml/M248">View MathML</a>

Thus, Lemma 11 is proved. □

2.4 Proof of Theorem 1

Let

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/84/mathml/M249','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/84/mathml/M249">View MathML</a>

Then there are two functions <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/84/mathml/M250','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/84/mathml/M250">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/84/mathml/M251','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/84/mathml/M251">View MathML</a> such that the Poincaré map of (2.16), given by (2.22), is of the form

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/84/mathml/M252','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/84/mathml/M252">View MathML</a>

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/84/mathml/M253','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/84/mathml/M253">View MathML</a>.

Since <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/84/mathml/M254','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/84/mathml/M254">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/84/mathml/M255','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/84/mathml/M255">View MathML</a>, we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/84/mathml/M256','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/84/mathml/M256">View MathML</a>

Let

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/84/mathml/M257','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/84/mathml/M257">View MathML</a>

Then

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/84/mathml/M258','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/84/mathml/M258">View MathML</a>

The other assumptions of Ortega’s theorem are easily verified. Hence, there is an invariant curve of P in the annulus <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/84/mathml/M259','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/84/mathml/M259">View MathML</a> which imply that the boundedness of our original equation (1.4). Then Theorem 1 is proved.

2.5 Proof of Theorem 2

We apply Aubry-Mather theory. By Theorem B in [8] and the monotone twist property of the Poincaré map P guaranteed by <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/84/mathml/M260','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/84/mathml/M260">View MathML</a>. It is straightforward to check that Theorem 2 is correct.

Remark 4 In [9], the authors study the multiplicity of positive periodic solutions of singular Duffing equations

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/84/mathml/M261','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/84/mathml/M261">View MathML</a>

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/84/mathml/M262','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/84/mathml/M262">View MathML</a> satisfies the semilinear condition at infinity and the time map satisfies an oscillation condition, and prove that the given equation possesses infinitely many positive 2π-periodic solutions by using the Poincaré-Birkhoff theorem. By the methods and techniques in [9], we can also prove the existence of 2π-periodic solutions of (1.4) where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/84/mathml/M2','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/84/mathml/M2">View MathML</a> satisfies the sublinear condition.

Competing interests

The author declares that they have no competing interests.

Acknowledgements

Thanks are given to referees whose comments and suggestions were very helpful for revising our paper.

References

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