SpringerOpen Newsletter

Receive periodic news and updates relating to SpringerOpen.

Open Access Research

Existence of anti-periodic solutions with symmetry for some high-order ordinary differential equations

Hai Pu12* and Jinyun Yang3

Author affiliations

1 State Key Laboratory for Geomechanics and Deep Underground Engineering, China University of Mining and Technology, Xuzhou, Jiangsu, 221008, China

2 School of Mechanics and Civil Engineering, China University of Mining and Technology, Xuzhou, Jiangsu, 221116, China

3 School of Mathematics and Physical Sciences, Xuzhou Institute of Technology, Xuzhou, Jiangsu, 221008, China

For all author emails, please log on.

Citation and License

Boundary Value Problems 2012, 2012:108  doi:10.1186/1687-2770-2012-108


The electronic version of this article is the complete one and can be found online at: http://www.boundaryvalueproblems.com/content/2012/1/108


Received:25 May 2012
Accepted:24 September 2012
Published:9 October 2012

© 2012 Pu and Yang; 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

The existence of anti-periodic solutions with symmetry for high-order Duffing equations and a high-order Duffing type p-Laplacian equation has been studied by using degree theory. The results obtained enrich some known works to some extent.

MSC: 34B15, 34C25.

Keywords:
anti-periodic solution with symmetry; high-order ordinary differential equation; p-Laplacian operator; Leray-Schauder degree theory

1 Introduction

Anti-periodic problems arise naturally from the mathematical models of various physical processes (see [1,2]) and also appear in the study of partial differential equations and abstract differential equations (see [3-5]). For instance, electron beam focusing system in traveling-wave tube theories is an anti-periodic problem (see [6]).

In mechanics, the simplest model of oscillation equation is a single pendulum equation

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/108/mathml/M1','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/108/mathml/M1">View MathML</a>

whose anti-periodic solutions satisfy

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/108/mathml/M2','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/108/mathml/M2">View MathML</a>

During the past twenty years, anti-periodic problems have been studied extensively by numerous scholars. For example, for first-order ordinary differential equations, a Massera’s type criterion was presented in [7] and the validity of the monotone iterative technique was shown in [8]. Moreover, for higher-order ordinary differential equations, the existence of anti-periodic solutions was considered in [9-12]. Recently, existence results were extended to anti-periodic boundary value problems for impulsive differential equations (see [13]), and anti-periodic wavelets were discussed in [14].

It is well known that higher-order p-Laplacian equations are derived from many fields such as fluid mechanics and nonlinear elastic mechanics. In the past few decades, many important results on higher-order p-Laplacian equations with certain boundary conditions have been obtained. We refer the readers to [15-19] and the references cited therein.

In [10], the authors considered the existence of anti-periodic solutions for the high-order Duffing equation as follows:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/108/mathml/M3','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/108/mathml/M3">View MathML</a>

(1.1)

Moreover, in [15] the authors discussed the existence of anti-periodic solutions for the following higher-order Liénard type p-Laplacian equation:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/108/mathml/M4','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/108/mathml/M4">View MathML</a>

(1.2)

However, to the best of our knowledge, there exist relatively few results on the existence of anti-periodic solutions with symmetry for (1.1) and (1.2). Thus, it is worthwhile to continue to investigate the existence of anti-periodic solutions with symmetry for (1.1) and (1.2).

Motivated by the works mentioned previously, in this paper, we study the existence of anti-periodic solutions with symmetry for high-order Duffing equations of the forms:

(1.3)

(1.4)

and high-order Duffing type p-Laplacian equation of the form:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/108/mathml/M7','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/108/mathml/M7">View MathML</a>

(1.5)

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/108/mathml/M8','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/108/mathml/M8">View MathML</a> is a constant, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/108/mathml/M9','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/108/mathml/M9">View MathML</a> is an integer, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/108/mathml/M10','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/108/mathml/M10">View MathML</a>; <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/108/mathml/M11','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/108/mathml/M11">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/108/mathml/M12','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/108/mathml/M12">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/108/mathml/M13','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/108/mathml/M13">View MathML</a> with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/108/mathml/M14','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/108/mathml/M14">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/108/mathml/M15','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/108/mathml/M15">View MathML</a>. Obviously, the inverse operator of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/108/mathml/M16','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/108/mathml/M16">View MathML</a> is <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/108/mathml/M17','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/108/mathml/M17">View MathML</a>, where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/108/mathml/M18','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/108/mathml/M18">View MathML</a> is a constant such that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/108/mathml/M19','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/108/mathml/M19">View MathML</a>.

Notice that, when <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/108/mathml/M20','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/108/mathml/M20">View MathML</a>, the nonlinear operator <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/108/mathml/M21','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/108/mathml/M21">View MathML</a> reduces to the linear operator <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/108/mathml/M22','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/108/mathml/M22">View MathML</a>. On the other hand, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/108/mathml/M23','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/108/mathml/M23">View MathML</a> is also a 2π-periodic solution if <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/108/mathml/M23','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/108/mathml/M23">View MathML</a> is a π-anti-periodic solution. Hence, from the arguments in this paper, we can also obtain the existence results on periodic solutions for the above equations.

The rest of this paper is organized as follows. Section 2 contains some necessary preliminaries. In Section 3 and Section 4, basing on the Leray-Schauder principle, we establish some existence theorems on anti-periodic solutions with symmetry of (1.3), (1.4) and (1.5). Our results are different from those of bibliographies listed in the previous texts.

2 Preliminaries

For the sake of convenience, we set

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/108/mathml/M25','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/108/mathml/M25">View MathML</a>

with the norm

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/108/mathml/M26','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/108/mathml/M26">View MathML</a>

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/108/mathml/M27','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/108/mathml/M27">View MathML</a>, and

with the norm <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/108/mathml/M29','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/108/mathml/M29">View MathML</a>.

Notice that, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/108/mathml/M30','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/108/mathml/M30">View MathML</a> may be written as Fourier series as follows:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/108/mathml/M31','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/108/mathml/M31">View MathML</a>

and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/108/mathml/M32','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/108/mathml/M32">View MathML</a> may be written as the following Fourier series:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/108/mathml/M33','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/108/mathml/M33">View MathML</a>

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/108/mathml/M34','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/108/mathml/M34">View MathML</a>. We define the mapping <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/108/mathml/M35','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/108/mathml/M35">View MathML</a> by

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/108/mathml/M36','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/108/mathml/M36">View MathML</a>

and the mapping <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/108/mathml/M37','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/108/mathml/M37">View MathML</a> by

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/108/mathml/M38','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/108/mathml/M38">View MathML</a>

It is easy to prove that the mappings <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/108/mathml/M39','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/108/mathml/M39">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/108/mathml/M40','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/108/mathml/M40">View MathML</a> are completely continuous by using the Arzelà-Ascoli theorem.

Next, we introduce a continuation theorem (see [20]) as follows.

Lemma 2.1 (Continuation theorem)

Let Ω be open bounded in a linear normal spaceX. Suppose thatfis a completely continuous field on<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/108/mathml/M41','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/108/mathml/M41">View MathML</a>. Moreover, assume that the Leray-Schauder degree

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/108/mathml/M42','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/108/mathml/M42">View MathML</a>

Then the equation<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/108/mathml/M43','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/108/mathml/M43">View MathML</a>has at least one solution in Ω.

3 Anti-periodic solutions with symmetry of (1.3) and (1.4)

In this section, some existence results on anti-periodic solutions with symmetry of (1.3) and (1.4) will be given.

Theorem 3.1Assume that

(H1) the functions<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/108/mathml/M44','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/108/mathml/M44">View MathML</a>and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/108/mathml/M45','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/108/mathml/M45">View MathML</a>are odd int, i.e.,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/108/mathml/M46','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/108/mathml/M46">View MathML</a>

(H2) there exist non-negative functions<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/108/mathml/M47','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/108/mathml/M47">View MathML</a>such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/108/mathml/M48','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/108/mathml/M48">View MathML</a>

(H3) <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/108/mathml/M49','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/108/mathml/M49">View MathML</a>.

Then (1.3) has at least one even anti-periodic solution<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/108/mathml/M23','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/108/mathml/M23">View MathML</a>, i.e., <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/108/mathml/M23','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/108/mathml/M23">View MathML</a>satisfies

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/108/mathml/M52','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/108/mathml/M52">View MathML</a>

Proof For making use of the Leray-Schauder degree theory to prove the existence of even anti-periodic solutions for (1.3), we consider the following homotopic equation of (1.3):

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/108/mathml/M53','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/108/mathml/M53">View MathML</a>

(3.1)

Define the operator <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/108/mathml/M54','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/108/mathml/M54">View MathML</a> by

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/108/mathml/M55','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/108/mathml/M55">View MathML</a>

Obviously, the operator <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/108/mathml/M56','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/108/mathml/M56">View MathML</a> is invertible. Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/108/mathml/M57','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/108/mathml/M57">View MathML</a> be the Nemytskii operator

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/108/mathml/M58','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/108/mathml/M58">View MathML</a>

By hypothesis (H1), it is easy to see that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/108/mathml/M59','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/108/mathml/M59">View MathML</a>

Thus, the operator <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/108/mathml/M60','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/108/mathml/M60">View MathML</a> sends <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/108/mathml/M61','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/108/mathml/M61">View MathML</a> into <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/108/mathml/M62','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/108/mathml/M62">View MathML</a>. Hence, the problem of even anti-periodic solutions for (3.1) is equivalent to the operator equation

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/108/mathml/M63','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/108/mathml/M63">View MathML</a>

From hypotheses (H2), (H3) and (5) in [10], for the possible even anti-periodic solution <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/108/mathml/M23','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/108/mathml/M23">View MathML</a> of (3.1), there exists a prior bounds in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/108/mathml/M65','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/108/mathml/M65">View MathML</a>, i.e., <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/108/mathml/M23','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/108/mathml/M23">View MathML</a> satisfies

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/108/mathml/M67','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/108/mathml/M67">View MathML</a>

(3.2)

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/108/mathml/M68','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/108/mathml/M68">View MathML</a> is a positive constant independent of λ. So, our problem is reduced to construct one completely continuous operator <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/108/mathml/M69','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/108/mathml/M69">View MathML</a>, which sends <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/108/mathml/M65','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/108/mathml/M65">View MathML</a> into <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/108/mathml/M65','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/108/mathml/M65">View MathML</a>, such that the fixed points of operator <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/108/mathml/M72','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/108/mathml/M72">View MathML</a> in some open bounded set are the even anti-periodic solutions of (1.3).

With this in mind, let us define the set as follows:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/108/mathml/M73','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/108/mathml/M73">View MathML</a>

Obviously, the set <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/108/mathml/M74','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/108/mathml/M74">View MathML</a> is a open bounded set in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/108/mathml/M75','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/108/mathml/M75">View MathML</a> and zero element <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/108/mathml/M76','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/108/mathml/M76">View MathML</a>. Define the completely continuous operator <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/108/mathml/M77','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/108/mathml/M77">View MathML</a> by

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/108/mathml/M78','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/108/mathml/M78">View MathML</a>

Let us define the completely continuous field <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/108/mathml/M79','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/108/mathml/M79">View MathML</a> by

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/108/mathml/M80','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/108/mathml/M80">View MathML</a>

By (3.2), we get that zero element <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/108/mathml/M81','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/108/mathml/M81">View MathML</a> for all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/108/mathml/M82','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/108/mathml/M82">View MathML</a>. So, the following Leray-Schauder degrees are well defined and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/108/mathml/M83','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/108/mathml/M83">View MathML</a>

Consequently, the operator <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/108/mathml/M72','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/108/mathml/M72">View MathML</a> has at least one fixed point in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/108/mathml/M74','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/108/mathml/M74">View MathML</a> by using Lemma 2.1. Namely, (1.3) has at least one even anti-periodic solution. The proof is complete. □

Theorem 3.2Assume that

(H4) the function<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/108/mathml/M44','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/108/mathml/M44">View MathML</a>is even int, xand<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/108/mathml/M45','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/108/mathml/M45">View MathML</a>is even int, i.e.,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/108/mathml/M88','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/108/mathml/M88">View MathML</a>

and the assumptions (H2), (H3) are true.

Then (1.3) has at least one odd anti-periodic solution<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/108/mathml/M23','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/108/mathml/M23">View MathML</a>, i.e., <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/108/mathml/M23','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/108/mathml/M23">View MathML</a>satisfies

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/108/mathml/M91','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/108/mathml/M91">View MathML</a>

Proof We consider the homotopic equation (3.1) of (1.3). Define the operator <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/108/mathml/M92','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/108/mathml/M92">View MathML</a> by

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/108/mathml/M93','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/108/mathml/M93">View MathML</a>

Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/108/mathml/M94','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/108/mathml/M94">View MathML</a> be the Nemytskii operator

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/108/mathml/M95','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/108/mathml/M95">View MathML</a>

By hypothesis (H4), it is easy to see that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/108/mathml/M96','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/108/mathml/M96">View MathML</a>

Thus, the operator <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/108/mathml/M97','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/108/mathml/M97">View MathML</a> sends <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/108/mathml/M98','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/108/mathml/M98">View MathML</a> into <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/108/mathml/M99','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/108/mathml/M99">View MathML</a>. Hence, the problem of odd anti-periodic solutions for (3.1) is equivalent to the operator equation

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/108/mathml/M100','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/108/mathml/M100">View MathML</a>

Our problem is reduced to construct one completely continuous operator <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/108/mathml/M101','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/108/mathml/M101">View MathML</a>, which sends <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/108/mathml/M102','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/108/mathml/M102">View MathML</a> into <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/108/mathml/M102','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/108/mathml/M102">View MathML</a>, such that the fixed points of operator <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/108/mathml/M104','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/108/mathml/M104">View MathML</a> in some open bounded set are the odd anti-periodic solutions of (1.3). With this in mind, let us define the following set:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/108/mathml/M105','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/108/mathml/M105">View MathML</a>

Define the completely continuous operator <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/108/mathml/M106','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/108/mathml/M106">View MathML</a> by

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/108/mathml/M107','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/108/mathml/M107">View MathML</a>

The remainder of the proof work is quite similar to the proof of Theorem 3.1, so we omit the details. The proof is complete. □

Theorem 3.3Assume that

(H5) the functions<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/108/mathml/M44','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/108/mathml/M44">View MathML</a>and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/108/mathml/M45','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/108/mathml/M45">View MathML</a>are even int, i.e.,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/108/mathml/M110','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/108/mathml/M110">View MathML</a>

and the assumptions (H2), (H3) are true.

Then (1.4) has at least one even anti-periodic solution.

Proof We consider the homotopic equation of (1.4) as follows:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/108/mathml/M111','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/108/mathml/M111">View MathML</a>

(3.3)

Define the operator <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/108/mathml/M112','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/108/mathml/M112">View MathML</a> by

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/108/mathml/M113','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/108/mathml/M113">View MathML</a>

Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/108/mathml/M114','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/108/mathml/M114">View MathML</a> be the Nemytskii operator

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/108/mathml/M115','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/108/mathml/M115">View MathML</a>

By hypothesis (H5), it is easy to see that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/108/mathml/M116','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/108/mathml/M116">View MathML</a>

Thus, the operator <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/108/mathml/M117','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/108/mathml/M117">View MathML</a> sends <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/108/mathml/M118','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/108/mathml/M118">View MathML</a> into <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/108/mathml/M99','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/108/mathml/M99">View MathML</a>. Hence, the problem of even anti-periodic solutions for (3.3) is equivalent to the operator equation

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/108/mathml/M120','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/108/mathml/M120">View MathML</a>

Our problem is reduced to construct one completely continuous operator <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/108/mathml/M121','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/108/mathml/M121">View MathML</a>, which sends <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/108/mathml/M122','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/108/mathml/M122">View MathML</a> into <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/108/mathml/M122','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/108/mathml/M122">View MathML</a>, such that the fixed points of operator <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/108/mathml/M124','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/108/mathml/M124">View MathML</a> in some open bounded set are the even anti-periodic solutions of (1.4). With this in mind, let us define the following set:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/108/mathml/M125','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/108/mathml/M125">View MathML</a>

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/108/mathml/M126','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/108/mathml/M126">View MathML</a> is a positive constant independent of λ. Define the completely continuous operator <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/108/mathml/M127','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/108/mathml/M127">View MathML</a> by

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/108/mathml/M128','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/108/mathml/M128">View MathML</a>

The remainder of the proof work is quite similar to the proof of Theorem 3.1, so we omit the details. The proof is complete. □

Theorem 3.4Assume that

(H6) the function<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/108/mathml/M44','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/108/mathml/M44">View MathML</a>is odd int, xand<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/108/mathml/M45','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/108/mathml/M45">View MathML</a>is odd int, i.e.,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/108/mathml/M131','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/108/mathml/M131">View MathML</a>

and the assumptions (H2), (H3) are true.

Then (1.4) has at least one odd anti-periodic solution.

Proof We consider the homotopic equation (3.3) of (1.4). Define the operator <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/108/mathml/M132','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/108/mathml/M132">View MathML</a> by

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/108/mathml/M133','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/108/mathml/M133">View MathML</a>

Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/108/mathml/M134','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/108/mathml/M134">View MathML</a> be the Nemytskii operator

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/108/mathml/M135','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/108/mathml/M135">View MathML</a>

By hypothesis (H6), it is easy to see that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/108/mathml/M136','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/108/mathml/M136">View MathML</a>

Thus, the operator <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/108/mathml/M137','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/108/mathml/M137">View MathML</a> sends <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/108/mathml/M138','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/108/mathml/M138">View MathML</a> into <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/108/mathml/M62','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/108/mathml/M62">View MathML</a>. Hence, the problem of odd anti-periodic solutions for (3.3) is equivalent to the operator equation

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/108/mathml/M140','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/108/mathml/M140">View MathML</a>

Our problem is reduced to construct one completely continuous operator <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/108/mathml/M141','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/108/mathml/M141">View MathML</a> which sends <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/108/mathml/M142','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/108/mathml/M142">View MathML</a> into <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/108/mathml/M142','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/108/mathml/M142">View MathML</a>, such that the fixed points of operator <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/108/mathml/M144','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/108/mathml/M144">View MathML</a> in some open bounded set are the odd anti-periodic solutions of (1.4). With this in mind, let us define the set as follows:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/108/mathml/M145','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/108/mathml/M145">View MathML</a>

Define the completely continuous operator <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/108/mathml/M146','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/108/mathml/M146">View MathML</a> by

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/108/mathml/M147','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/108/mathml/M147">View MathML</a>

The remainder of the proof work is quite similar to the proof of Theorem 3.1, so we omit the details. The proof is complete. □

When <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/108/mathml/M148','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/108/mathml/M148">View MathML</a>, we can remove the assumption (H2) in Theorem 3.1, Theorem 3.2 and obtain the following results.

Theorem 3.5Assume that

(H7) <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/108/mathml/M149','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/108/mathml/M149">View MathML</a>and the assumption (H1) is true.

Then (1.3) (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/108/mathml/M148','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/108/mathml/M148">View MathML</a>) has at least one even anti-periodic solution.

Theorem 3.6Suppose that the assumptions (H4), (H7) are true. Then (1.3) (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/108/mathml/M148','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/108/mathml/M148">View MathML</a>) has at least one odd anti-periodic solution.

Basing on the proof of Theorem 2 in [10], for the possible anti-periodic solution <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/108/mathml/M23','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/108/mathml/M23">View MathML</a> of (3.1) (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/108/mathml/M148','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/108/mathml/M148">View MathML</a>), the hypothesis (H7) yields that there exists a prior bounds in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/108/mathml/M154','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/108/mathml/M154">View MathML</a>, i.e., <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/108/mathml/M23','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/108/mathml/M23">View MathML</a> satisfies

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/108/mathml/M156','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/108/mathml/M156">View MathML</a>

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/108/mathml/M157','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/108/mathml/M157">View MathML</a> is a positive constant independent of λ. The remainder of the proof work of Theorem 3.5 and Theorem 3.6 is quite similar to the proof of Theorem 3.1 and Theorem 3.2, so we omit the details.

4 Anti-periodic solutions with symmetry of (1.5)

In this section, we will give some existence results on anti-periodic solutions with symmetry of (1.5).

Theorem 4.1Assume that

(H8) there exist non-negative functions<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/108/mathml/M158','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/108/mathml/M158">View MathML</a>such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/108/mathml/M159','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/108/mathml/M159">View MathML</a>

(H9) <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/108/mathml/M160','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/108/mathml/M160">View MathML</a>and the assumption (H5) is true.

Then (1.5) has at least one even anti-periodic solution.

Proof We consider the following homotopic equation of (1.5):

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/108/mathml/M161','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/108/mathml/M161">View MathML</a>

(4.1)

Define the operator <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/108/mathml/M162','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/108/mathml/M162">View MathML</a> by

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/108/mathml/M163','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/108/mathml/M163">View MathML</a>

where

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/108/mathml/M164','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/108/mathml/M164">View MathML</a>

Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/108/mathml/M165','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/108/mathml/M165">View MathML</a> be the Nemytskii operator

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/108/mathml/M166','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/108/mathml/M166">View MathML</a>

Obviously, the operator <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/108/mathml/M167','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/108/mathml/M167">View MathML</a> is invertible and the problem of even anti-periodic solutions for (4.1) is equivalent to the operator equation

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/108/mathml/M168','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/108/mathml/M168">View MathML</a>

From hypotheses (H8), (H9) and (3.8) in [15], for the possible even anti-periodic solution <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/108/mathml/M23','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/108/mathml/M23">View MathML</a> of (4.1), there exists a prior bounds in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/108/mathml/M99','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/108/mathml/M99">View MathML</a>, i.e., <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/108/mathml/M23','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/108/mathml/M23">View MathML</a> satisfies

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/108/mathml/M172','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/108/mathml/M172">View MathML</a>

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/108/mathml/M173','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/108/mathml/M173">View MathML</a> is a positive constant independent of λ. So, our problem is reduced to construct one completely continuous operator <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/108/mathml/M174','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/108/mathml/M174">View MathML</a>, which sends <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/108/mathml/M99','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/108/mathml/M99">View MathML</a> into <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/108/mathml/M99','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/108/mathml/M99">View MathML</a>, such that the fixed points of operator <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/108/mathml/M177','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/108/mathml/M177">View MathML</a> in some open bounded set are the even anti-periodic solutions of (1.5).

With this in mind, let us define the set as follows:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/108/mathml/M178','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/108/mathml/M178">View MathML</a>

By hypothesis (H5), it is easy to see that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/108/mathml/M179','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/108/mathml/M179">View MathML</a>

Hence, the operator <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/108/mathml/M180','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/108/mathml/M180">View MathML</a> sends <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/108/mathml/M99','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/108/mathml/M99">View MathML</a> into <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/108/mathml/M99','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/108/mathml/M99">View MathML</a>. Define the completely continuous operator <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/108/mathml/M183','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/108/mathml/M183">View MathML</a> by

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/108/mathml/M184','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/108/mathml/M184">View MathML</a>

or

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/108/mathml/M185','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/108/mathml/M185">View MathML</a>

The remainder of the proof work is quite similar to the proof of Theorem 3.1, so we omit the details. The proof is complete. □

Theorem 4.2Suppose that the assumptions (H6), (H8), (H9) are true. Then (1.5) has at least one odd anti-periodic solution.

Proof We consider the homotopic equation (4.1) of (1.5). Define the operator <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/108/mathml/M186','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/108/mathml/M186">View MathML</a> by

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/108/mathml/M187','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/108/mathml/M187">View MathML</a>

where

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/108/mathml/M188','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/108/mathml/M188">View MathML</a>

Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/108/mathml/M189','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/108/mathml/M189">View MathML</a> be the Nemytskii operator

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/108/mathml/M190','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/108/mathml/M190">View MathML</a>

Thus, the problem of odd anti-periodic solutions for (4.1) is equivalent to the operator equation

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/108/mathml/M191','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/108/mathml/M191">View MathML</a>

Our problem is reduced to construct one completely continuous operator <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/108/mathml/M192','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/108/mathml/M192">View MathML</a>, which sends <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/108/mathml/M62','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/108/mathml/M62">View MathML</a> into <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/108/mathml/M62','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/108/mathml/M62">View MathML</a>, such that the fixed points of operator <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/108/mathml/M195','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/108/mathml/M195">View MathML</a> in some open bounded set are the odd anti-periodic solutions of (1.5). With this in mind, let us define the following set:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/108/mathml/M196','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/108/mathml/M196">View MathML</a>

By hypothesis (H6), it is easy to see that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/108/mathml/M197','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/108/mathml/M197">View MathML</a>

Hence, the operator <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/108/mathml/M198','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/108/mathml/M198">View MathML</a> sends <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/108/mathml/M62','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/108/mathml/M62">View MathML</a> into <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/108/mathml/M62','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/108/mathml/M62">View MathML</a>. Define the completely continuous operator <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/108/mathml/M201','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/108/mathml/M201">View MathML</a> by

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/108/mathml/M202','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/108/mathml/M202">View MathML</a>

or

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/108/mathml/M203','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/108/mathml/M203">View MathML</a>

The remainder of the proof work is quite similar to the proof of Theorem 3.1, so we omit the details. The proof is complete. □

Theorem 4.3Assume that<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/108/mathml/M44','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/108/mathml/M44">View MathML</a>has the decomposition

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/108/mathml/M205','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/108/mathml/M205">View MathML</a>

such that

(H10) there exist non-negative constantsγ, rwith<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/108/mathml/M206','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/108/mathml/M206">View MathML</a>, such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/108/mathml/M207','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/108/mathml/M207">View MathML</a>

(H11) there are non-negative functions<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/108/mathml/M208','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/108/mathml/M208">View MathML</a>such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/108/mathml/M209','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/108/mathml/M209">View MathML</a>

(H12) <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/108/mathml/M210','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/108/mathml/M210">View MathML</a>and the assumption (H5) is true.

Then (1.5) has at least one even anti-periodic solution.

Theorem 4.4Suppose that the assumptions (H6), (H10), (H11), (H12) are true. Then (1.5) has at least one odd anti-periodic solution.

Basing on the proof of Theorem 3.2 in [15], for the possible anti-periodic solution <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/108/mathml/M23','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/108/mathml/M23">View MathML</a> of (4.1), the hypotheses (H10), (H11), (H12) yield that there exists a prior bounds in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/108/mathml/M154','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/108/mathml/M154">View MathML</a>, i.e., <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/108/mathml/M23','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/108/mathml/M23">View MathML</a> satisfies

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/108/mathml/M214','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/108/mathml/M214">View MathML</a>

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/108/mathml/M215','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/108/mathml/M215">View MathML</a> is a positive constant independent of λ. The remainder of the proof work of Theorem 4.3 and Theorem 4.4 is quite similar to the proof of Theorem 4.1 and Theorem 4.2, so we omit the details.

Remark Assumptions (H10), (H11), (H12) guarantee that the degree with respect to x of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/108/mathml/M44','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/108/mathml/M44">View MathML</a> is allowed to be greater than <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/108/mathml/M217','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/108/mathml/M217">View MathML</a>, which is different from the hypothesis (H8) of Theorem 4.1 and Theorem 4.2.

Competing interests

The authors declare that they have no competing interests.

Authors’ contributions

HP carried out the theoretical analysis. JY drafted the manuscript. All authors read and approved the final manuscript.

Acknowledgements

This work was supported by the National Natural Science Foundations of China (50904065) and the Program for New Century Excellent Talents in University (NCET-09-0728). As well, this work was sponsored by the Qing Lan Project and the Fundamental Research Funds for the Central Universities (China University of Mining and Technology).

References

  1. Ahn, C, Rim, C: Boundary flows in general coset theories. J. Phys. A. 32(13), 2509–2525 (1999). Publisher Full Text OpenURL

  2. Kleinert, H, Chervyakov, A: Functional determinants from Wronski Green function. J. Math. Phys.. 40(11), 6044–6051 (1999). Publisher Full Text OpenURL

  3. Aizicovici, S, McKibben, M, Reich, S: Anti-periodic solutions to nonmonotone evolution equations with discontinuous nonlinearities. Nonlinear Anal.. 43(2), 233–251 (2001). Publisher Full Text OpenURL

  4. Nakao, M: Existence of an anti-periodic solution for the quasilinear wave equation with viscosity. J. Math. Anal. Appl.. 204(3), 754–764 (1996). Publisher Full Text OpenURL

  5. Souplet, P: Optimal uniqueness condition for the antiperiodic solutions of some nonlinear parabolic equations. Nonlinear Anal.. 32(2), 279–286 (1998). Publisher Full Text OpenURL

  6. Lu, Z: Travelling Tube, Shanghai Sci. Technol., Shanghai (1962)

  7. Chen, Y: On Massera’s theorem for anti-periodic solution. Adv. Math. Sci. Appl.. 9(1), 125–128 (1999)

  8. Yin, Y: Monotone iterative technique and quasilinearization for some anti-periodic problems. Nonlinear World. 3(2), 253–266 (1996)

  9. Aftabizadeh, AR, Pavel, NH, Huang, Y: Anti-periodic oscillations of some second-order differential equations and optimal control problems. J. Comput. Appl. Math.. 52(1-3), 3–21 Oscillations in nonlinear systems: Applications and numerical aspects (1994)

    Oscillations in nonlinear systems: Applications and numerical aspects

    Publisher Full Text OpenURL

  10. Chen, T, Liu, W, Zhang, J: The existence of anti-periodic solutions for high order Duffing equation. J. Appl. Math. Comput.. 27(1-2), 271–280 (2008). Publisher Full Text OpenURL

  11. Liu, B: Anti-periodic solutions for forced Rayleigh-type equations. Nonlinear Anal., Real World Appl.. 10(5), 2850–2856 (2009). Publisher Full Text OpenURL

  12. Liu, W, Zhang, J, Chen, T: Anti-symmetric periodic solutions for the third order differential systems. Appl. Math. Lett.. 22(5), 668–673 (2009). Publisher Full Text OpenURL

  13. Luo, Z, Shen, J, Nieto, JJ: Antiperiodic boundary value problem for first-order impulsive ordinary differential equation. Comput. Math. Appl.. 49(2-3), 253–261 (2005). Publisher Full Text OpenURL

  14. Chen, H: Antiperiodic wavelets. J. Comput. Math.. 14(1), 32–39 (1996)

  15. Chen, T, Liu, W: Anti-periodic solutions for higher-order Liénard type differential equation with p-Laplacian operator. Bull. Korean Math. Soc.. 49(3), 455–463 (2012). Publisher Full Text OpenURL

  16. Li, X: Existence and uniqueness of periodic solutions for a kind of high-order p-Laplacian Duffing differential equation with sign-changing coefficient ahead of linear term. Nonlinear Anal.. 71(7-8), 2764–2770 (2009). Publisher Full Text OpenURL

  17. Pang, H, Ge, W, Tian, M: Solvability of nonlocal boundary value problems for ordinary differential equation of higher order with a p-Laplacian. Comput. Math. Appl.. 56(1), 127–142 (2008). Publisher Full Text OpenURL

  18. Su, H, Wang, B, Wei, Z, Zhang, X: Positive solutions of four-point boundary value problems for higher-order p-Laplacian operator. J. Math. Anal. Appl.. 330(2), 836–851 (2007). Publisher Full Text OpenURL

  19. Xu, F, Liu, L, Wu, Y: Multiple positive solutions of four-point nonlinear boundary value problems for a higher-order p-Laplacian operator with all derivatives. Nonlinear Anal.. 71(9), 4309–4319 (2009). Publisher Full Text OpenURL

  20. Deimling, K: Nonlinear Functional Analysis, Springer, Berlin (1985)