Open Access Research

Existence of anti-periodic solutions for second-order ordinary differential equations involving the Fučík spectrum

Xin Zhao1 and Xiaojun Chang2*

Author Affiliations

1 College of Information Technology, Jilin Agricultural University, Changchun, 130118, P.R. China

2 College of Mathematics, Jilin University, Changchun, Jilin, 130012, China

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Boundary Value Problems 2012, 2012:149  doi:10.1186/1687-2770-2012-149


The electronic version of this article is the complete one and can be found online at: http://www.boundaryvalueproblems.com/content/2012/1/149


Received:20 July 2012
Accepted:20 September 2012
Published:21 December 2012

© 2012 Zhao and Chang; licensee Springer

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

Abstract

In this paper, we study the existence of anti-periodic solutions for a second-order ordinary differential equation. Using the interaction of the nonlinearity with the Fučík spectrum related to the anti-periodic boundary conditions, we apply the Leray-Schauder degree theory and the Borsuk theorem to establish new results on the existence of anti-periodic solutions of second-order ordinary differential equations. Our nonlinearity may cross multiple consecutive branches of the Fučík spectrum curves, and recent results in the literature are complemented and generalized.

Keywords:
anti-periodic solutions; Fučík spectrum; Leray-Schauder degree theory; Borsuk theorem

1 Introduction and main results

In this paper, we study the existence of anti-periodic solutions for the following second-order ordinary differential equation:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/149/mathml/M1','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/149/mathml/M1">View MathML</a>

(1.1)

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/149/mathml/M2','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/149/mathml/M2">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/149/mathml/M3','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/149/mathml/M3">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/149/mathml/M4','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/149/mathml/M4">View MathML</a> and T is a positive constant. A function <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/149/mathml/M5','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/149/mathml/M5">View MathML</a> is called an anti-periodic solution of (1.1) if <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/149/mathml/M5','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/149/mathml/M5">View MathML</a> satisfies (1.1) and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/149/mathml/M7','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/149/mathml/M7">View MathML</a> for all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/149/mathml/M8','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/149/mathml/M8">View MathML</a>. Note that to obtain anti-periodic solutions of (1.1), it suffices to find solutions of the following anti-periodic boundary value problem:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/149/mathml/M9','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/149/mathml/M9">View MathML</a>

(1.2)

In what follows, we will consider problem (1.2) directly.

The problem of the existence of solutions of (1.1) under various boundary conditions has been widely investigated in the literature and many results have been obtained (see [1-13]). Usually, the asymptotic interaction of the ratio <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/149/mathml/M10','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/149/mathml/M10">View MathML</a> with the Fučík spectrum of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/149/mathml/M11','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/149/mathml/M11">View MathML</a> under various boundary conditions was required as a nonresonance condition to obtain the solvability of equation (1.1). Recall that the Fučík spectrum of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/149/mathml/M11','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/149/mathml/M11">View MathML</a> with an anti-periodic boundary condition is the set of real number pairs <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/149/mathml/M13','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/149/mathml/M13">View MathML</a> such that the problem

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/149/mathml/M14','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/149/mathml/M14">View MathML</a>

(1.3)

has nontrivial solutions, where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/149/mathml/M15','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/149/mathml/M15">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/149/mathml/M16','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/149/mathml/M16">View MathML</a>; while the concept of Fučík spectrum was firstly introduced in the 1970s by Fučík [14] and Dancer [15] independently under the periodic boundary condition. Since the work of Fonda [6], some investigation has been devoted to the nonresonance condition of (1.1) by studying the asymptotic interaction of the ratio <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/149/mathml/M17','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/149/mathml/M17">View MathML</a>, where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/149/mathml/M18','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/149/mathml/M18">View MathML</a>, with the spectrum of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/149/mathml/M11','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/149/mathml/M11">View MathML</a> under different boundary conditions; for instance, see [10] for the periodic boundary condition, [16] for the two-point boundary condition. Note that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/149/mathml/M20','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/149/mathml/M20">View MathML</a>

we can see that the conditions on the ratio <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/149/mathml/M17','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/149/mathml/M17">View MathML</a> are more general than those on the ratio <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/149/mathml/M22','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/149/mathml/M22">View MathML</a>. In fact, by using the asymptotic interaction of the ratio <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/149/mathml/M17','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/149/mathml/M17">View MathML</a> with the spectrum of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/149/mathml/M11','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/149/mathml/M11">View MathML</a>, the ratio <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/149/mathml/M22','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/149/mathml/M22">View MathML</a> can cross multiple spectrum curves of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/149/mathml/M11','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/149/mathml/M11">View MathML</a>. In this paper, we are interested in the nonresonance condition on the ratio <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/149/mathml/M17','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/149/mathml/M17">View MathML</a> for the solvability of (1.1) involving the Fučík spectrum of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/149/mathml/M11','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/149/mathml/M11">View MathML</a> under the anti-periodic boundary condition.

Note that the study of anti-periodic solutions for nonlinear differential equations is closely related to the study of periodic solutions. In fact, since <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/149/mathml/M29','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/149/mathml/M29">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/149/mathml/M5','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/149/mathml/M5">View MathML</a> is a T-periodic solution of (1.1) if <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/149/mathml/M5','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/149/mathml/M5">View MathML</a> is a <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/149/mathml/M32','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/149/mathml/M32">View MathML</a>-anti-periodic solution of (1.1). Many results on the periodic solutions of (1.1) have been worked out. For some recent work, one can see [2-5,8-10,17]. As special periodic solutions, the existence of anti-periodic solutions plays a key role in characterizing the behavior of nonlinear differential equations coming from some models in applied sciences. During the last thirty years, anti-periodic problems of nonlinear differential equations have been extensively studied since the pioneering work by Okochi [18]. For example, in [19], anti-periodic trigonometric polynomials are used to investigate the interpolation problems, and anti-periodic wavelets are studied in [20]. Also, some existence results of ordinary differential equations are presented in [17,21-24]. Anti-periodic boundary conditions for partial differential equations and abstract differential equations are considered in [25-32]. For recent developments involving the existence of anti-periodic solutions, one can also see [33-35] and the references therein.

Denote by Σ the Fuc̆ík spectrum of the operator <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/149/mathml/M11','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/149/mathml/M11">View MathML</a> under the anti-periodic boundary condition. Simple computation implies that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/149/mathml/M34','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/149/mathml/M34">View MathML</a>, where

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/149/mathml/M35','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/149/mathml/M35">View MathML</a>

It is easily seen that the set Σ can be seen as a subset of the Fuc̆ík spectrum of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/149/mathml/M11','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/149/mathml/M11">View MathML</a> under the corresponding Dirichlet boundary condition; one can see the definition of the set <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/149/mathml/M37','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/149/mathml/M37">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/149/mathml/M38','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/149/mathml/M38">View MathML</a>, or Figure 1 in [12]. Without loss of generality, we assume that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/149/mathml/M39','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/149/mathml/M39">View MathML</a> is an eigenfunction of (1.3) corresponding to <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/149/mathml/M40','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/149/mathml/M40">View MathML</a> such that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/149/mathml/M41','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/149/mathml/M41">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/149/mathml/M42','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/149/mathml/M42">View MathML</a>. Denote <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/149/mathml/M43','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/149/mathml/M43">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/149/mathml/M44','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/149/mathml/M44">View MathML</a>. Then if <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/149/mathml/M45','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/149/mathml/M45">View MathML</a>, we obtain only a one-dimensional function <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/149/mathml/M39','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/149/mathml/M39">View MathML</a>, denoted by <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/149/mathml/M47','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/149/mathml/M47">View MathML</a>, corresponding to the point <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/149/mathml/M48','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/149/mathml/M48">View MathML</a>, and if <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/149/mathml/M49','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/149/mathml/M49">View MathML</a>, we obtain only a one-dimensional function <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/149/mathml/M39','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/149/mathml/M39">View MathML</a>, denoted by <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/149/mathml/M51','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/149/mathml/M51">View MathML</a>, corresponding to the point <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/149/mathml/M52','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/149/mathml/M52">View MathML</a>.

In this paper, together with the Leray-Schauder degree theory and the Borsuk theorem, we obtain new existence results of anti-periodic solutions of (1.1) when the nonlinearity <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/149/mathml/M53','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/149/mathml/M53">View MathML</a> is asymptotically linear in s at infinity and the ratio <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/149/mathml/M54','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/149/mathml/M54">View MathML</a> stays asymptotically at infinity in some rectangular domain between Fučík spectrum curves <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/149/mathml/M55','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/149/mathml/M55">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/149/mathml/M56','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/149/mathml/M56">View MathML</a>.

Our main result is as follows.

Theorem 1.1Assume that<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/149/mathml/M2','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/149/mathml/M2">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/149/mathml/M3','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/149/mathml/M3">View MathML</a>. If the following conditions:

(i) There exist positive constantsρ, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/149/mathml/M59','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/149/mathml/M59">View MathML</a>, Msuch that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/149/mathml/M60','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/149/mathml/M60">View MathML</a>

(1.4)

(ii) There exist connect subset<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/149/mathml/M61','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/149/mathml/M61">View MathML</a>, constants<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/149/mathml/M62','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/149/mathml/M62">View MathML</a>and a point of the type<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/149/mathml/M63','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/149/mathml/M63">View MathML</a>such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/149/mathml/M64','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/149/mathml/M64">View MathML</a>

(1.5)

and

hold uniformly for all<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/149/mathml/M8','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/149/mathml/M8">View MathML</a>,

then (1.1) admits a<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/149/mathml/M32','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/149/mathml/M32">View MathML</a>-anti-periodic solution.

In particular, if <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/149/mathml/M68','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/149/mathml/M68">View MathML</a>, then problem (1.3) becomes the following linear eigenvalue problem:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/149/mathml/M69','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/149/mathml/M69">View MathML</a>

(1.6)

Simple computation implies that the operator <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/149/mathml/M11','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/149/mathml/M11">View MathML</a> with the anti-periodic boundary condition has a sequence of eigenvalues <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/149/mathml/M71','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/149/mathml/M71">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/149/mathml/M72','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/149/mathml/M72">View MathML</a>, and the corresponding eigenspace is two-dimensional.

Corollary 1.2Assume that<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/149/mathml/M2','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/149/mathml/M2">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/149/mathml/M74','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/149/mathml/M74">View MathML</a>. If (1.4) holds and there exist constantsp, qand<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/149/mathml/M72','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/149/mathml/M72">View MathML</a>such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/149/mathml/M76','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/149/mathml/M76">View MathML</a>

holds uniformly for all<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/149/mathml/M8','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/149/mathml/M8">View MathML</a>, then (1.1) admits a<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/149/mathml/M32','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/149/mathml/M32">View MathML</a>-anti-periodic solution.

Remark It is well known that (1.1) has a <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/149/mathml/M32','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/149/mathml/M32">View MathML</a>-anti-periodic solution if

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/149/mathml/M80','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/149/mathml/M80">View MathML</a>

for some <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/149/mathml/M81','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/149/mathml/M81">View MathML</a> (see Theorem 3.1 in [22]), which implies that the ratio <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/149/mathml/M10','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/149/mathml/M10">View MathML</a> stays at infinity asymptotically below the first eigenvalue <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/149/mathml/M83','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/149/mathml/M83">View MathML</a> of (1.6). In this paper, this requirement on the ratio <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/149/mathml/M84','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/149/mathml/M84">View MathML</a> can be relaxed to (1.4), with some additional restrictions imposed on the ratio <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/149/mathml/M85','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/149/mathml/M85">View MathML</a>. In fact, the conditions relative to the ratios <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/149/mathml/M10','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/149/mathml/M10">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/149/mathml/M85','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/149/mathml/M85">View MathML</a> as in Theorem 1.1 and Corollary 1.2 may lead to that the ratio <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/149/mathml/M10','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/149/mathml/M10">View MathML</a> oscillates and crosses multiple consecutive eigenvalues or branches of the Fučík spectrum curves of the operator <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/149/mathml/M11','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/149/mathml/M11">View MathML</a>. In what follows, we give an example to show this.

Denote <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/149/mathml/M71','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/149/mathml/M71">View MathML</a> for some positive integer <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/149/mathml/M91','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/149/mathml/M91">View MathML</a>. Define

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/149/mathml/M92','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/149/mathml/M92">View MathML</a>

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/149/mathml/M93','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/149/mathml/M93">View MathML</a>. Clearly,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/149/mathml/M94','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/149/mathml/M94">View MathML</a>

In addition,

for all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/149/mathml/M8','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/149/mathml/M8">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/149/mathml/M97','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/149/mathml/M97">View MathML</a>, which imply that

(1.7)

(1.8)

for all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/149/mathml/M8','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/149/mathml/M8">View MathML</a>. It is obvious that (1.7) implies that the assumption (i) of Theorem 1.1 holds. Take <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/149/mathml/M101','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/149/mathml/M101">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/149/mathml/M102','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/149/mathml/M102">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/149/mathml/M103','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/149/mathml/M103">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/149/mathml/M104','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/149/mathml/M104">View MathML</a> such that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/149/mathml/M105','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/149/mathml/M105">View MathML</a>. Then (1.8) implies that the assumption (ii) of Theorem 1.1 holds. Thus, by Theorem 1.1 we can obtain a <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/149/mathml/M32','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/149/mathml/M32">View MathML</a>-anti-periodic solution of equation (1.1). Here the ratio <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/149/mathml/M17','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/149/mathml/M17">View MathML</a> stays at infinity in the rectangular domain <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/149/mathml/M108','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/149/mathml/M108">View MathML</a> between Fučík spectrum curves <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/149/mathml/M55','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/149/mathml/M55">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/149/mathml/M56','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/149/mathml/M56">View MathML</a>, while the ratio <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/149/mathml/M22','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/149/mathml/M22">View MathML</a> can cross at infinity multiple Fučík spectrum curves <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/149/mathml/M112','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/149/mathml/M112">View MathML</a>.

This paper is organized as follows. In Section 2, some necessary preliminaries are presented. In Section 3, we give the proof of Theorem 1.1.

2 Preliminaries

Assume that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/149/mathml/M113','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/149/mathml/M113">View MathML</a>. Define

For <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/149/mathml/M115','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/149/mathml/M115">View MathML</a>, we can write the Fourier series expansion as follows:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/149/mathml/M116','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/149/mathml/M116">View MathML</a>

Define an operator <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/149/mathml/M117','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/149/mathml/M117">View MathML</a> by

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/149/mathml/M118','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/149/mathml/M118">View MathML</a>

Clearly,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/149/mathml/M119','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/149/mathml/M119">View MathML</a>

which implies that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/149/mathml/M120','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/149/mathml/M120">View MathML</a>

(2.1)

Furthermore, we obtain

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/149/mathml/M121','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/149/mathml/M121">View MathML</a>

Note that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/149/mathml/M122','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/149/mathml/M122">View MathML</a>

using the Parseval equality <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/149/mathml/M123','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/149/mathml/M123">View MathML</a>, we get

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/149/mathml/M124','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/149/mathml/M124">View MathML</a>

which implies that the operator J is continuous. In view of the Arzela-Ascoli theorem, it is easy to see that J is completely continuous.

Denote by deg the Leray-Schauder degree. We need the following results.

Lemma 2.1 ([[36], p.58])

Let Ω be a bounded open region in a real Banach spaceX. Assume that<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/149/mathml/M125','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/149/mathml/M125">View MathML</a>is completely continuous and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/149/mathml/M126','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/149/mathml/M126">View MathML</a>. Then the equation<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/149/mathml/M127','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/149/mathml/M127">View MathML</a>has a solution in Ω if<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/149/mathml/M128','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/149/mathml/M128">View MathML</a>.

Lemma 2.2 ([[36], Borsuk theorem, p.58])

Assume thatXis a real Banach space. Let Ω be a symmetric bounded open region with<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/149/mathml/M129','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/149/mathml/M129">View MathML</a>. Assume that<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/149/mathml/M130','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/149/mathml/M130">View MathML</a>is completely continuous and odd with<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/149/mathml/M131','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/149/mathml/M131">View MathML</a>. Then<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/149/mathml/M132','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/149/mathml/M132">View MathML</a>is odd.

3 Proof of Theorem 1.1

Proof of Theorem 1.1 Consider the following homotopy problem:

(3.1)

(3.2)

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/149/mathml/M135','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/149/mathml/M135">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/149/mathml/M136','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/149/mathml/M136">View MathML</a>.

We first prove that the set of all possible solutions of problem (3.1)-(3.2) is bounded. Assume by contradiction that there exist a sequence of number <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/149/mathml/M137','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/149/mathml/M137">View MathML</a> and corresponding solutions <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/149/mathml/M138','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/149/mathml/M138">View MathML</a> of (3.1)-(3.2) such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/149/mathml/M139','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/149/mathml/M139">View MathML</a>

(3.3)

Set <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/149/mathml/M140','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/149/mathml/M140">View MathML</a>. Obviously, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/149/mathml/M141','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/149/mathml/M141">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/149/mathml/M142','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/149/mathml/M142">View MathML</a> satisfies

(3.4)

(3.5)

By (1.4), (3.3) and the fact that f is continuous, there exist <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/149/mathml/M145','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/149/mathml/M145">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/149/mathml/M146','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/149/mathml/M146">View MathML</a> such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/149/mathml/M147','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/149/mathml/M147">View MathML</a>

In view of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/149/mathml/M148','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/149/mathml/M148">View MathML</a>, together with the choice of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/149/mathml/M149','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/149/mathml/M149">View MathML</a>, it follows that there exists <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/149/mathml/M150','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/149/mathml/M150">View MathML</a> such that, for all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/149/mathml/M151','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/149/mathml/M151">View MathML</a>,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/149/mathml/M152','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/149/mathml/M152">View MathML</a>

It is easily seen that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/149/mathml/M153','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/149/mathml/M153">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/149/mathml/M154','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/149/mathml/M154">View MathML</a> are uniformly bounded and equicontinuous on <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/149/mathml/M155','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/149/mathml/M155">View MathML</a>. Then, using the Arzela-Ascoli theorem, there exist uniformly convergent subsequences on <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/149/mathml/M155','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/149/mathml/M155">View MathML</a> for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/149/mathml/M153','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/149/mathml/M153">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/149/mathml/M154','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/149/mathml/M154">View MathML</a> respectively, which are still denoted as <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/149/mathml/M153','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/149/mathml/M153">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/149/mathml/M154','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/149/mathml/M154">View MathML</a>, such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/149/mathml/M161','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/149/mathml/M161">View MathML</a>

(3.6)

Clearly, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/149/mathml/M162','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/149/mathml/M162">View MathML</a>. Since <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/149/mathml/M163','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/149/mathml/M163">View MathML</a> is a solution of (3.1)-(3.2), for each n, we get

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/149/mathml/M164','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/149/mathml/M164">View MathML</a>

which implies that there exists <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/149/mathml/M165','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/149/mathml/M165">View MathML</a> such that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/149/mathml/M166','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/149/mathml/M166">View MathML</a>. Then

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/149/mathml/M167','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/149/mathml/M167">View MathML</a>

(3.7)

Owing to that the sequences <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/149/mathml/M168','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/149/mathml/M168">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/149/mathml/M169','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/149/mathml/M169">View MathML</a> are uniformly bounded, there exist <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/149/mathml/M170','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/149/mathml/M170">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/149/mathml/M171','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/149/mathml/M171">View MathML</a> such that, passing to subsequences if possible,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/149/mathml/M172','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/149/mathml/M172">View MathML</a>

(3.8)

Multiplying both sides of (3.4) by <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/149/mathml/M173','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/149/mathml/M173">View MathML</a> and integrating from <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/149/mathml/M174','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/149/mathml/M174">View MathML</a> to t, we get

Taking a superior limit as <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/149/mathml/M176','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/149/mathml/M176">View MathML</a>, by (3.3) and (3.6)-(3.8), we obtain

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/149/mathml/M177','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/149/mathml/M177">View MathML</a>

By the assumption (ii) and the choice of λ, if <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/149/mathml/M178','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/149/mathml/M178">View MathML</a>, we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/149/mathml/M179','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/149/mathml/M179">View MathML</a>

Similarly, we obtain

Note that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/149/mathml/M181','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/149/mathml/M181">View MathML</a>, the above inequalities can be rewritten as the following equivalent forms:

(3.9)

(3.10)

It is easy to see that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/149/mathml/M184','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/149/mathml/M184">View MathML</a>. In fact, if not, in view of (3.7)-(3.10), we get <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/149/mathml/M185','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/149/mathml/M185">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/149/mathml/M186','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/149/mathml/M186">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/149/mathml/M187','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/149/mathml/M187">View MathML</a>, which is contrary to <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/149/mathml/M162','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/149/mathml/M162">View MathML</a>.

We claim that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/149/mathml/M189','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/149/mathml/M189">View MathML</a> has only finite zero points on <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/149/mathml/M190','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/149/mathml/M190">View MathML</a>. In fact, if not, we may assume that there are infinitely many zero points <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/149/mathml/M191','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/149/mathml/M191">View MathML</a> of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/149/mathml/M189','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/149/mathml/M189">View MathML</a>. Without loss of generality, we assume that there exists <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/149/mathml/M193','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/149/mathml/M193">View MathML</a> such that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/149/mathml/M194','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/149/mathml/M194">View MathML</a>. Letting <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/149/mathml/M195','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/149/mathml/M195">View MathML</a> in (3.9)-(3.10) and taking <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/149/mathml/M196','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/149/mathml/M196">View MathML</a>, we can obtain that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/149/mathml/M197','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/149/mathml/M197">View MathML</a>. Without loss of generality, we assume that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/149/mathml/M198','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/149/mathml/M198">View MathML</a>. Since <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/149/mathml/M199','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/149/mathml/M199">View MathML</a> is continuous, there exist <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/149/mathml/M200','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/149/mathml/M200">View MathML</a> such that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/149/mathml/M201','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/149/mathml/M201">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/149/mathml/M202','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/149/mathml/M202">View MathML</a>. Then there exists <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/149/mathml/M203','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/149/mathml/M203">View MathML</a> such that, if <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/149/mathml/M204','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/149/mathml/M204">View MathML</a>, we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/149/mathml/M205','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/149/mathml/M205">View MathML</a>

(3.11)

Clearly, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/149/mathml/M206','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/149/mathml/M206">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/149/mathml/M202','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/149/mathml/M202">View MathML</a>. Take <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/149/mathml/M208','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/149/mathml/M208">View MathML</a> with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/149/mathml/M209','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/149/mathml/M209">View MathML</a> such that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/149/mathml/M210','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/149/mathml/M210">View MathML</a>. Integrating (3.4) from <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/149/mathml/M211','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/149/mathml/M211">View MathML</a> to <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/149/mathml/M212','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/149/mathml/M212">View MathML</a>,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/149/mathml/M213','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/149/mathml/M213">View MathML</a>

(3.12)

By (3.3), (3.11), we obtain

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/149/mathml/M214','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/149/mathml/M214">View MathML</a>

holds uniformly for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/149/mathml/M215','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/149/mathml/M215">View MathML</a>. Thus, using (1.4), we get

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/149/mathml/M216','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/149/mathml/M216">View MathML</a>

which implies that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/149/mathml/M217','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/149/mathml/M217">View MathML</a>

Then, together with (3.6), (3.8) and (3.12), we obtain

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/149/mathml/M218','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/149/mathml/M218">View MathML</a>

a contradiction.

Now, we show that (3.9)-(3.10) has only a trivial anti-periodic solution. In fact, if not, we assume that (3.9)-(3.10) has a nontrivial anti-periodic solution <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/149/mathml/M219','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/149/mathml/M219">View MathML</a>. Without loss of generality, we assume <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/149/mathml/M220','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/149/mathml/M220">View MathML</a>. Firstly, we consider the case that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/149/mathml/M221','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/149/mathml/M221">View MathML</a>. Assume that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/149/mathml/M222','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/149/mathml/M222">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/149/mathml/M223','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/149/mathml/M223">View MathML</a> satisfy the following equations respectively:

(3.13)

(3.14)

with

(3.15)

(3.16)

Take <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/149/mathml/M228','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/149/mathml/M228">View MathML</a> as the first zero point of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/149/mathml/M219','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/149/mathml/M219">View MathML</a> on <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/149/mathml/M230','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/149/mathml/M230">View MathML</a>. Then by (3.13)-(3.16) it follows that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/149/mathml/M231','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/149/mathml/M231">View MathML</a>

(3.17)

In fact, by (3.15)-(3.16) and the fact that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/149/mathml/M232','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/149/mathml/M232">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/149/mathml/M222','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/149/mathml/M222">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/149/mathml/M223','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/149/mathml/M223">View MathML</a> are continuous differential, it is easy to see that there exists sufficiently small <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/149/mathml/M235','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/149/mathml/M235">View MathML</a> such that

If there is <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/149/mathml/M237','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/149/mathml/M237">View MathML</a> such that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/149/mathml/M238','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/149/mathml/M238">View MathML</a>, then comparing (3.9) with (3.13), we can obtain that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/149/mathml/M239','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/149/mathml/M239">View MathML</a>, which implies that if <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/149/mathml/M240','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/149/mathml/M240">View MathML</a>, we have <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/149/mathml/M241','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/149/mathml/M241">View MathML</a>. Then <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/149/mathml/M242','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/149/mathml/M242">View MathML</a> for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/149/mathml/M243','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/149/mathml/M243">View MathML</a>. Similarly, we have <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/149/mathml/M244','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/149/mathml/M244">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/149/mathml/M245','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/149/mathml/M245">View MathML</a>. Hence, (3.17) holds.

Similarly, if <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/149/mathml/M222','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/149/mathml/M222">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/149/mathml/M223','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/149/mathml/M223">View MathML</a> satisfy

(3.18)

(3.19)

and

then we obtain

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/149/mathml/M251','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/149/mathml/M251">View MathML</a>

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/149/mathml/M252','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/149/mathml/M252">View MathML</a> is the first zero point on <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/149/mathml/M253','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/149/mathml/M253">View MathML</a>.

Since <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/149/mathml/M189','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/149/mathml/M189">View MathML</a> has finite zero points, (3.13), (3.14), (3.18), (3.19) can be transformed into the following equations respectively:

(3.20)

(3.21)

Then there exist <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/149/mathml/M257','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/149/mathml/M257">View MathML</a> such that

It is easy to get

Since <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/149/mathml/M232','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/149/mathml/M232">View MathML</a> is anti-periodic and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/149/mathml/M221','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/149/mathml/M221">View MathML</a>, there exists <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/149/mathml/M72','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/149/mathml/M72">View MathML</a> such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/149/mathml/M263','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/149/mathml/M263">View MathML</a>

which implies that there exists a real number pair <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/149/mathml/M264','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/149/mathml/M264">View MathML</a> such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/149/mathml/M265','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/149/mathml/M265">View MathML</a>

(3.22)

On the other hand, in view of the assumption (ii), by the definition of Σ and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/149/mathml/M264','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/149/mathml/M264">View MathML</a>, it follows that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/149/mathml/M267','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/149/mathml/M267">View MathML</a>

which is contrary to (3.22).

If <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/149/mathml/M268','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/149/mathml/M268">View MathML</a>, then by the assumption (ii), we can obtain a contradiction using similar arguments.

In a word, we can see that there exists <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/149/mathml/M269','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/149/mathml/M269">View MathML</a> independent of μ such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/149/mathml/M270','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/149/mathml/M270">View MathML</a>

(3.23)

Set

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/149/mathml/M271','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/149/mathml/M271">View MathML</a>

Clearly, Ω is a bounded open set in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/149/mathml/M272','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/149/mathml/M272">View MathML</a>. Note that, for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/149/mathml/M273','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/149/mathml/M273">View MathML</a>, using the assumption on f, we obtain

which implies that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/149/mathml/M275','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/149/mathml/M275">View MathML</a>.

Define <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/149/mathml/M276','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/149/mathml/M276">View MathML</a> by

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/149/mathml/M277','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/149/mathml/M277">View MathML</a>

Clearly, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/149/mathml/M278','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/149/mathml/M278">View MathML</a> is completely continuous, and by (2.1) and (3.1) it follows that the fixed point of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/149/mathml/M279','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/149/mathml/M279">View MathML</a> in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/149/mathml/M280','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/149/mathml/M280">View MathML</a> is the anti-periodic solution of problem (1.1). Define the homotopy <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/149/mathml/M281','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/149/mathml/M281">View MathML</a> as follows:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/149/mathml/M282','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/149/mathml/M282">View MathML</a>

In view of (3.23), it follows that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/149/mathml/M283','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/149/mathml/M283">View MathML</a>

Hence,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/149/mathml/M284','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/149/mathml/M284">View MathML</a>

Note that the operator <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/149/mathml/M285','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/149/mathml/M285">View MathML</a> is odd. By Lemma 2.2 it follows that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/149/mathml/M286','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/149/mathml/M286">View MathML</a>. Thus,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/149/mathml/M287','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/149/mathml/M287">View MathML</a>

Now, using Lemma 2.1, we can see that (1.2) has a solution and hence (1.1) has a <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/149/mathml/M32','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/149/mathml/M32">View MathML</a>-anti-periodic solution. The proof is complete. □

Competing interests

The authors declare that they have no competing interests.

Authors’ contributions

All authors read and approved the final manuscript.

Acknowledgements

The authors sincerely thank Prof. Yong Li for his instructions and many invaluable suggestions. This work was supported financially by NSFC Grant (11101178), NSFJP Grant (201215184), and the 985 Program of Jilin University.

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