This article is part of the series Jean Mawhin’s Achievements in Nonlinear Analysis.

Open Access Research

Existence and uniqueness of anti-periodic solutions for prescribed mean curvature Rayleigh equations

Jin Li12* and Zaihong Wang1

Author Affiliations

1 School of Mathematical Sciences, Capital Normal University, Beijing, 100048, China

2 College of Science, Jiujiang University, Jiujiang, 332005, China

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Boundary Value Problems 2012, 2012:109  doi:10.1186/1687-2770-2012-109


The electronic version of this article is the complete one and can be found online at: http://www.boundaryvalueproblems.com/content/2012/1/109


Received:12 May 2012
Accepted:22 September 2012
Published:9 October 2012

© 2012 Li and Wang; 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

By means of the Leray-Schauder degree theory, we establish some sufficient conditions for the existence and uniqueness of anti-periodic solutions for prescribed mean curvature Rayleigh equations.

MSC: 34C25, 34D40.

Keywords:
prescribed mean curvature Rayleigh equation; anti-periodic solutions; Leray-Schauder degree

1 Introduction

We are concerned with the existence and uniqueness of anti-periodic solutions of the following prescribed mean curvature Rayleigh equation:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/109/mathml/M1','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/109/mathml/M1">View MathML</a>

(1.1)

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/109/mathml/M2','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/109/mathml/M2">View MathML</a> is T-periodic, and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/109/mathml/M3','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/109/mathml/M3">View MathML</a> are T-periodic in the first argument, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/109/mathml/M4','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/109/mathml/M4">View MathML</a> is a constant.

In recent years, the existence of periodic solutions and anti-periodic solutions for some types of second-order differential equations, especially for the Rayleigh ones, were widely studied (see [1-7]) and the references cited therein). For example, Liu [7] discussed the Rayleigh equation

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/109/mathml/M5','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/109/mathml/M5">View MathML</a>

and established the existence and uniqueness of anti-periodic solutions. At the same time, a kind of prescribed mean curvature equations attracted many people’s attention (see [8-11] and the references cited therein). Feng [8] investigated the prescribed mean curvature Liénard equation

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/109/mathml/M6','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/109/mathml/M6">View MathML</a>

and obtained some existence results on periodic solutions. However, to the best of our knowledge, the existence and uniqueness of anti-periodic solution for Eq. (1.1) have not been investigated till now. Motivated by [7,8], we establish some sufficient conditions for the existence and uniqueness of anti-periodic solutions via the Leray-Schauder degree theory.

The rest of the paper is organized as follows. In Section 2, we shall state and prove some basic lemmas. In Section 3, we shall prove the main result. An example will be given to show the applications of our main result in the final section.

2 Preliminaries

We first give the definition of an anti-periodic function. Assume that N is a positive integer. Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/109/mathml/M7','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/109/mathml/M7">View MathML</a> be a continuous function. We call <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/109/mathml/M8','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/109/mathml/M8">View MathML</a> an anti-periodic function on ℝ if u satisfies the following condition:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/109/mathml/M9','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/109/mathml/M9">View MathML</a>

Obviously, a <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/109/mathml/M10','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/109/mathml/M10">View MathML</a>-anti-periodic function u is a T-periodic function.

Throughout this paper, we will adopt the following notations:

which is a linear normal space endowed with the norm <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/109/mathml/M12','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/109/mathml/M12">View MathML</a> defined by

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/109/mathml/M13','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/109/mathml/M13">View MathML</a>

The following lemmas will be useful to prove our main results.

Lemma 2.1[12]

If<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/109/mathml/M14','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/109/mathml/M14">View MathML</a>and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/109/mathml/M15','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/109/mathml/M15">View MathML</a>, then

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/109/mathml/M16','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/109/mathml/M16">View MathML</a>

(Wirtinger inequality) and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/109/mathml/M17','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/109/mathml/M17">View MathML</a>

(Sobolev inequality).

Lemma 2.2Suppose that the following condition holds:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/109/mathml/M18','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/109/mathml/M18">View MathML</a><a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/109/mathml/M19','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/109/mathml/M19">View MathML</a>, for all<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/109/mathml/M20','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/109/mathml/M20">View MathML</a>and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/109/mathml/M21','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/109/mathml/M21">View MathML</a>.

Then Eq. (1.1) has at most oneT-periodic solution.

Proof Assume that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/109/mathml/M22','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/109/mathml/M22">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/109/mathml/M23','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/109/mathml/M23">View MathML</a> are two T-periodic solutions of Eq. (1.1). Then we obtain

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/109/mathml/M24','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/109/mathml/M24">View MathML</a>

(2.1)

It is easy to see that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/109/mathml/M25','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/109/mathml/M25">View MathML</a> (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/109/mathml/M26','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/109/mathml/M26">View MathML</a>). From (2.1), we know

(2.2)

Set <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/109/mathml/M28','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/109/mathml/M28">View MathML</a>. Now, we prove

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/109/mathml/M29','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/109/mathml/M29">View MathML</a>

Otherwise, we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/109/mathml/M30','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/109/mathml/M30">View MathML</a>

Then there exists a <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/109/mathml/M31','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/109/mathml/M31">View MathML</a> such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/109/mathml/M32','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/109/mathml/M32">View MathML</a>

(2.3)

which implies that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/109/mathml/M33','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/109/mathml/M33">View MathML</a>

(2.4)

and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/109/mathml/M34','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/109/mathml/M34">View MathML</a>

(2.5)

It follows from (2.2), (2.4) and (2.5) that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/109/mathml/M35','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/109/mathml/M35">View MathML</a>

From <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/109/mathml/M18','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/109/mathml/M18">View MathML</a>, we get

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/109/mathml/M37','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/109/mathml/M37">View MathML</a>

which contradicts (2.3). Thus,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/109/mathml/M38','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/109/mathml/M38">View MathML</a>

By using a similar argument, we can also show

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/109/mathml/M39','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/109/mathml/M39">View MathML</a>

Hence,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/109/mathml/M40','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/109/mathml/M40">View MathML</a>

Therefore, Eq. (1.1) has at most one T-periodic solution. The proof is completed. □

To prove the main result of this paper, we shall use a continuation theorem [13,14] as follows.

Lemma 2.3Let Ω be open bounded in a linear normal spaceX. Suppose that<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/109/mathml/M41','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/109/mathml/M41">View MathML</a>is a complete continuous field on<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/109/mathml/M42','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/109/mathml/M42">View MathML</a>. Moreover, assume that the Leray-Schauder degree

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/109/mathml/M43','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/109/mathml/M43">View MathML</a>

Then the equation<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/109/mathml/M44','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/109/mathml/M44">View MathML</a>has at least one solution in Ω.

3 Main result

In this section, we present and prove our main result concerning the existence and uniqueness of anti-periodic solutions of Eq. (1.1).

Theorem 3.1Let<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/109/mathml/M18','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/109/mathml/M18">View MathML</a>hold. Moreover, assume that the following conditions hold:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/109/mathml/M46','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/109/mathml/M46">View MathML</a>there exists<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/109/mathml/M47','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/109/mathml/M47">View MathML</a>such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/109/mathml/M48','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/109/mathml/M48">View MathML</a>

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/109/mathml/M49','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/109/mathml/M49">View MathML</a>there exists<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/109/mathml/M50','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/109/mathml/M50">View MathML</a>such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/109/mathml/M51','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/109/mathml/M51">View MathML</a>

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/109/mathml/M52','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/109/mathml/M52">View MathML</a>for all<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/109/mathml/M53','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/109/mathml/M53">View MathML</a>,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/109/mathml/M54','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/109/mathml/M54">View MathML</a>

Then Eq. (1.1) has a unique anti-periodic solution for<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/109/mathml/M55','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/109/mathml/M55">View MathML</a>.

Proof Rewrite Eq. (1.1) in the equivalent form:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/109/mathml/M56','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/109/mathml/M56">View MathML</a>

(3.1)

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/109/mathml/M57','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/109/mathml/M57">View MathML</a>. Now, we consider the auxiliary equation of (3.1),

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/109/mathml/M58','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/109/mathml/M58">View MathML</a>

(3.2)

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/109/mathml/M59','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/109/mathml/M59">View MathML</a> is a parameter. Set

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/109/mathml/M60','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/109/mathml/M60">View MathML</a>

Then Eq. (3.2) can be reduced to the equation as follows:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/109/mathml/M61','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/109/mathml/M61">View MathML</a>

By Lemma 2.2 and condition <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/109/mathml/M18','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/109/mathml/M18">View MathML</a>, it is easy to see that Eq. (1.1) has at most one anti-periodic solution. Thus, to prove Theorem 3.1, it suffices to show that Eq. (1.1) has at least one anti-periodic solution. To do this, we shall apply Lemma 2.3. Firstly, we will prove that the set of all possible anti-periodic solutions of Eq. (3.2) is bounded.

Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/109/mathml/M63','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/109/mathml/M63">View MathML</a> be an arbitrary possible anti-periodic solution of Eq. (3.2). Then <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/109/mathml/M64','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/109/mathml/M64">View MathML</a>. Thus, we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/109/mathml/M65','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/109/mathml/M65">View MathML</a>

It follows from Lemma 2.1 that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/109/mathml/M66','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/109/mathml/M66">View MathML</a>

Obviously, Eq. (3.2) is equivalent to the following equation:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/109/mathml/M67','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/109/mathml/M67">View MathML</a>

(3.3)

Multiplying (3.3) by <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/109/mathml/M68','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/109/mathml/M68">View MathML</a> and integrating from 0 to T, we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/109/mathml/M69','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/109/mathml/M69">View MathML</a>

(3.4)

Since <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/109/mathml/M70','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/109/mathml/M70">View MathML</a>, there exists a constant <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/109/mathml/M71','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/109/mathml/M71">View MathML</a> such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/109/mathml/M72','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/109/mathml/M72">View MathML</a>

(3.5)

For such a <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/109/mathml/M71','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/109/mathml/M71">View MathML</a>, in view of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/109/mathml/M49','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/109/mathml/M49">View MathML</a>, there exists <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/109/mathml/M75','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/109/mathml/M75">View MathML</a> such that for all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/109/mathml/M53','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/109/mathml/M53">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/109/mathml/M77','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/109/mathml/M77">View MathML</a>. Hence,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/109/mathml/M78','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/109/mathml/M78">View MathML</a>

(3.6)

It follows from (3.4) and (3.6) that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/109/mathml/M79','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/109/mathml/M79">View MathML</a>

For <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/109/mathml/M80','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/109/mathml/M80">View MathML</a>, we have the Schwarz inequality

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/109/mathml/M81','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/109/mathml/M81">View MathML</a>

Hence,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/109/mathml/M82','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/109/mathml/M82">View MathML</a>

(3.7)

From (3.5) and (3.7), we know that there exists a constant <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/109/mathml/M83','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/109/mathml/M83">View MathML</a> such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/109/mathml/M84','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/109/mathml/M84">View MathML</a>

(3.8)

By the first equation of (3.2), we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/109/mathml/M85','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/109/mathml/M85">View MathML</a>

Then there exists <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/109/mathml/M86','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/109/mathml/M86">View MathML</a> such that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/109/mathml/M87','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/109/mathml/M87">View MathML</a>. It follows that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/109/mathml/M88','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/109/mathml/M88">View MathML</a>, and so

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/109/mathml/M89','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/109/mathml/M89">View MathML</a>

According to <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/109/mathml/M49','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/109/mathml/M49">View MathML</a>, we know there exists <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/109/mathml/M91','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/109/mathml/M91">View MathML</a> such that for all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/109/mathml/M53','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/109/mathml/M53">View MathML</a>,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/109/mathml/M93','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/109/mathml/M93">View MathML</a>

From the second equation of (3.2), we get

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/109/mathml/M94','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/109/mathml/M94">View MathML</a>

From (3.8), we know that there exists a constant <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/109/mathml/M95','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/109/mathml/M95">View MathML</a> such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/109/mathml/M96','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/109/mathml/M96">View MathML</a>

Thus,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/109/mathml/M97','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/109/mathml/M97">View MathML</a>

which implies that there exists a constant <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/109/mathml/M98','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/109/mathml/M98">View MathML</a> such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/109/mathml/M99','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/109/mathml/M99">View MathML</a>

Let

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/109/mathml/M100','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/109/mathml/M100">View MathML</a>

(3.9)

Set

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/109/mathml/M101','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/109/mathml/M101">View MathML</a>

Then Eq. (3.2) has no anti-periodic solution on Ω for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/109/mathml/M59','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/109/mathml/M59">View MathML</a>.

Next, we consider the Fourier series expansions of two functions <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/109/mathml/M103','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/109/mathml/M103">View MathML</a> (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/109/mathml/M104','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/109/mathml/M104">View MathML</a>). We have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/109/mathml/M105','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/109/mathml/M105">View MathML</a>

Define an operator <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/109/mathml/M106','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/109/mathml/M106">View MathML</a> by setting

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/109/mathml/M107','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/109/mathml/M107">View MathML</a>

Then

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/109/mathml/M108','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/109/mathml/M108">View MathML</a>

and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/109/mathml/M109','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/109/mathml/M109">View MathML</a>

Since

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/109/mathml/M110','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/109/mathml/M110">View MathML</a>

and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/109/mathml/M111','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/109/mathml/M111">View MathML</a>

we obtain

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/109/mathml/M112','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/109/mathml/M112">View MathML</a>

Define <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/109/mathml/M113','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/109/mathml/M113">View MathML</a> by setting

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/109/mathml/M114','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/109/mathml/M114">View MathML</a>

Then <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/109/mathml/M115','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/109/mathml/M115">View MathML</a>, and thus L is continuous.

For any <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/109/mathml/M116','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/109/mathml/M116">View MathML</a>, we know from <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/109/mathml/M52','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/109/mathml/M52">View MathML</a> that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/109/mathml/M118','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/109/mathml/M118">View MathML</a>

Therefore, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/109/mathml/M119','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/109/mathml/M119">View MathML</a>. Define an operator <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/109/mathml/M120','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/109/mathml/M120">View MathML</a> by setting

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/109/mathml/M121','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/109/mathml/M121">View MathML</a>

It is easy to see that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/109/mathml/M122','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/109/mathml/M122">View MathML</a> is a compact homotopy, and the fixed point of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/109/mathml/M123','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/109/mathml/M123">View MathML</a> on <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/109/mathml/M42','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/109/mathml/M42">View MathML</a> is the anti-periodic of Eq. (3.1).

Define a homotopic field as follows:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/109/mathml/M125','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/109/mathml/M125">View MathML</a>

From (3.9), we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/109/mathml/M126','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/109/mathml/M126">View MathML</a>

Using the homotopy invariance property of degree, we obtain

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/109/mathml/M127','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/109/mathml/M127">View MathML</a>

Till now, we have proved that Ω satisfies all the requirements in Lemma 2.3. Consequently, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/109/mathml/M128','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/109/mathml/M128">View MathML</a> has at least one solution in Ω, i.e., <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/109/mathml/M123','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/109/mathml/M123">View MathML</a> has a fixed point <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/109/mathml/M130','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/109/mathml/M130">View MathML</a> on <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/109/mathml/M42','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/109/mathml/M42">View MathML</a>. Therefore, Eq. (1.1) has at least one anti-periodic solution <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/109/mathml/M22','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/109/mathml/M22">View MathML</a>. This completes the proof. □

4 An example

In this section, we shall construct an example to show the applications of Theorem 3.1.

Example 4.1 Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/109/mathml/M133','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/109/mathml/M133">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/109/mathml/M134','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/109/mathml/M134">View MathML</a>. Then the prescribed mean curvature Rayleigh equation

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/109/mathml/M135','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/109/mathml/M135">View MathML</a>

(4.1)

has a unique anti-periodic solution with period 2π.

Proof Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/109/mathml/M136','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/109/mathml/M136">View MathML</a>. From the definitions of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/109/mathml/M137','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/109/mathml/M137">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/109/mathml/M138','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/109/mathml/M138">View MathML</a>, we can easily check that conditions <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/109/mathml/M18','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/109/mathml/M18">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/109/mathml/M52','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/109/mathml/M52">View MathML</a> hold. Moreover, it is easy to see that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/109/mathml/M46','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/109/mathml/M46">View MathML</a> holds for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/109/mathml/M142','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/109/mathml/M142">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/109/mathml/M49','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/109/mathml/M49">View MathML</a> holds for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/109/mathml/M144','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/109/mathml/M144">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/109/mathml/M145','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/109/mathml/M145">View MathML</a>. Since <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/109/mathml/M146','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/109/mathml/M146">View MathML</a>, we know from Theorem 3.1 that Eq. (4.1) has a unique anti-periodic solution with period 2π. □

Competing interests

The authors declare that they have no competing interests.

Authors’ contributions

Both authors, AA and MHA, contributed to each part of this work equally and read and approved the final version of the manuscript.

Acknowledgements

The authors would like to express their thanks to the Editor of the journal and the anonymous referees for their carefully reading of the first draft of the manuscript and making many helpful comments and suggestions which improved the presentation of the paper. Research supported by National Science foundation of China, No. 10771145 and Beijing Natural Science Foundation (Existence and multiplicity of periodic solutions in nonlinear oscillations), No. 1112006.

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