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

Open Access Research

Periodic solutions of radially symmetric systems with a singularity

Shengjun Li12*, Fang-fang Liao3 and Jianqiang Sun1

Author Affiliations

1 Department of Mathematics, Hainan University, Haikou, 570228, China

2 College of Science, Hohai University, Nanjing, 210046, China

3 Nanjing College of Information Technology, Nanjing, 210046, China

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Boundary Value Problems 2013, 2013:110  doi:10.1186/1687-2770-2013-110

The electronic version of this article is the complete one and can be found online at: http://www.boundaryvalueproblems.com/content/2013/1/110


Received:30 November 2012
Accepted:13 April 2013
Published:29 April 2013

© 2013 Li et al.; 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 infinitely many periodic solutions to planar radially symmetric systems with certain strong repulsive singularities near the origin and with some semilinear growth near infinity. The proof of the main result relies on topological degree theory. Recent results in the literature are generalized and complemented.

MSC: 34C25.

Keywords:
periodic solution; singular systems; topological degree

1 Introduction

In this work, we are concerned with the existence of positive periodic solutions for the following radically symmetric system:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/110/mathml/M1','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/110/mathml/M1">View MathML</a>

(1.1)

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/110/mathml/M2','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/110/mathml/M2">View MathML</a> is T-periodic in the time variable t for some <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/110/mathml/M3','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/110/mathml/M3">View MathML</a> and satisfies the <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/110/mathml/M4','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/110/mathml/M4">View MathML</a>-Carathéodory condition. Setting <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/110/mathml/M5','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/110/mathml/M5">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/110/mathml/M6','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/110/mathml/M6">View MathML</a> may be singular at <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/110/mathml/M7','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/110/mathml/M7">View MathML</a>, we therefore look for non-collision solutions, i.e., solutions which never attain the singularity.

Roughly speaking, system (1.1) is singular at 0 means that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/110/mathml/M6','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/110/mathml/M6">View MathML</a> becomes unbounded when <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/110/mathml/M9','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/110/mathml/M9">View MathML</a>. We say that (1.1) is of repulsive type (attractive type) if <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/110/mathml/M10','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/110/mathml/M10">View MathML</a> (respectively <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/110/mathml/M11','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/110/mathml/M11">View MathML</a>) when <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/110/mathml/M9','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/110/mathml/M9">View MathML</a>.

Such a type of singular systems appears in many problems of applications. Such as, if we take <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/110/mathml/M13','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/110/mathml/M13">View MathML</a> (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/110/mathml/M14','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/110/mathml/M14">View MathML</a>), it is the famous Newtonian equation

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/110/mathml/M15','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/110/mathml/M15">View MathML</a>

which describes the motion of a particle subjected to the gravitational attraction of a sun that lies at the origin. If we take <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/110/mathml/M13','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/110/mathml/M13">View MathML</a> (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/110/mathml/M17','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/110/mathml/M17">View MathML</a>), (1.1) may be used to model Rutherford’s scattering of α particles by heavy atomic nuclei.

The question about the existence of non-collision periodic orbits for scalar equations and dynamical systems with singularities has attracted much attention of many researchers over many years [1-10]. There are two main lines of research in this area. The first one is the variational approach [11-13]. Usually, the proof requires some strong force condition, which was first introduced with this name by Gordon in [14], although the idea goes back at least to Poincaré [15]. Gordon’s result, later improved by Capozzi, Greco and Salvatore [16], is stated as follows.

Theorem 1.1Let<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/110/mathml/M18','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/110/mathml/M18">View MathML</a>and the following assumptions hold.

(<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/110/mathml/M19','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/110/mathml/M19">View MathML</a>) The functionVisT-periodic int, differentiable in<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/110/mathml/M20','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/110/mathml/M20">View MathML</a>with continuous gradient, and such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/110/mathml/M21','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/110/mathml/M21">View MathML</a>

(<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/110/mathml/M22','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/110/mathml/M22">View MathML</a>) There exist<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/110/mathml/M23','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/110/mathml/M23">View MathML</a>and positive constants<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/110/mathml/M24','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/110/mathml/M24">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/110/mathml/M25','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/110/mathml/M25">View MathML</a>such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/110/mathml/M26','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/110/mathml/M26">View MathML</a>

for everytand<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/110/mathml/M20','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/110/mathml/M20">View MathML</a>.

(<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/110/mathml/M28','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/110/mathml/M28">View MathML</a>) There are a<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/110/mathml/M29','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/110/mathml/M29">View MathML</a>-function<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/110/mathml/M30','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/110/mathml/M30">View MathML</a>, a neighborhood<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/110/mathml/M31','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/110/mathml/M31">View MathML</a>of 0 and a positive constant<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/110/mathml/M32','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/110/mathml/M32">View MathML</a>such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/110/mathml/M33','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/110/mathml/M33">View MathML</a>

for every<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/110/mathml/M34','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/110/mathml/M34">View MathML</a>, then, for every integer<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/110/mathml/M35','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/110/mathml/M35">View MathML</a>, the system

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/110/mathml/M36','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/110/mathml/M36">View MathML</a>

has a periodic solution with a minimal periodkT.

The strong force conditions (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/110/mathml/M22','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/110/mathml/M22">View MathML</a>), (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/110/mathml/M28','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/110/mathml/M28">View MathML</a>) guarantee that the minimization procedure does not lead to a collision orbit. This similar condition has been widely used for a voiding collisions in the singularity case. For example, if we consider the system

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/110/mathml/M39','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/110/mathml/M39">View MathML</a>

the strong force condition corresponds to the case <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/110/mathml/M40','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/110/mathml/M40">View MathML</a>.

Besides the variational approach, topological methods have been widely applied, starting with the pioneering paper of Lazer and Solimini [17]. In particular, some classical tools have been used to study singular differential equations and dynamical systems in the literature, including the degree theory [18-23], the method of upper and lower solutions [24,25], Schauder’s fixed point theorem [26-28], some fixed point theorems in cones for completely continuous operators [29-32] and a nonlinear Leray-Schauder alternative principle [33-36]. Contrasting with the variational setting, the strong force condition plays here a different role linked to repulsive singularities. A counterexample in the paper of Lazer and Solimini [17] shows that a strong force assumption (unboundedness of the potential near the singularity) is necessary in some sense for the existence of positive periodic solutions in the scalar case.

However, compared with the case of strong singularities, the study of the existence of periodic solutions under the presence of weak singularities by topological methods is more recent and the number of references is much smaller. Several existence results can be found in [7,26,28].

As mentioned above, this paper is mainly motivated by the recent papers [19,20]. The aim of this paper is to show that the topological degree theorem can be applied to the periodic problem. We prove the existence of large-amplitude periodic solutions whose minimal period is an integer multiple of T.

The rest of this paper is organized as follows. In Section 2, some preliminary results will be given. In Section 3, by the use of topological degree theory, we will state and prove the main results.

2 Preliminaries

In this section, we present some results which will be applied in Section 3. We may write the solutions of (1.1) in polar coordinates as follows:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/110/mathml/M41','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/110/mathml/M41">View MathML</a>

(2.1)

Eq. (1.1) is then equivalent to the system

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/110/mathml/M42','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/110/mathml/M42">View MathML</a>

(2.2)

where μ is the (scalar) angular momentum of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/110/mathml/M43','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/110/mathml/M43">View MathML</a>. Recall that μ is constant in time along any solution. In the following, when considering a solution of (2.2), we will always implicitly assume that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/110/mathml/M44','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/110/mathml/M44">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/110/mathml/M45','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/110/mathml/M45">View MathML</a>.

If x is a T-radially periodic, then r must be T-periodic. We will prove the existence of a T-periodic solution r of the first equation in (2.2). We thus consider the boundary value problem

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/110/mathml/M46','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/110/mathml/M46">View MathML</a>

(2.3)

Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/110/mathml/M47','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/110/mathml/M47">View MathML</a>, (2.3) can be written as the T-periodic problem

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/110/mathml/M48','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/110/mathml/M48">View MathML</a>

(2.4)

Let X be a Banach space of functions such that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/110/mathml/M49','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/110/mathml/M49">View MathML</a> with continuous immersions, and set <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/110/mathml/M50','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/110/mathml/M50">View MathML</a>.

Define the following two operators:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/110/mathml/M51','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/110/mathml/M51">View MathML</a>

and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/110/mathml/M52','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/110/mathml/M52">View MathML</a>

Taking <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/110/mathml/M53','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/110/mathml/M53">View MathML</a> not belonging to the spectrum of L, (2.4) can be translated to the fixed problem

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/110/mathml/M54','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/110/mathml/M54">View MathML</a>

We will say that a set <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/110/mathml/M55','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/110/mathml/M55">View MathML</a> is uniformly positively bounded below if there is a constant <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/110/mathml/M56','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/110/mathml/M56">View MathML</a> such that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/110/mathml/M57','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/110/mathml/M57">View MathML</a> for every <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/110/mathml/M58','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/110/mathml/M58">View MathML</a>. In order to prove the main result of this paper, we need the following theorem, which has been proved in [18].

Theorem 2.1Let Ω be an open bounded subset ofX, uniformly positively bounded below. Assume that there is no solution of (2.4) on the boundaryΩ, and that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/110/mathml/M59','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/110/mathml/M59">View MathML</a>

Then, there exists a<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/110/mathml/M60','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/110/mathml/M60">View MathML</a>such that, for every integer<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/110/mathml/M61','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/110/mathml/M61">View MathML</a>, system (1.1) has a periodic solution<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/110/mathml/M62','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/110/mathml/M62">View MathML</a>with a minimal periodkT, which makes exactly one revolution around the origin in the period timekT. The function<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/110/mathml/M63','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/110/mathml/M63">View MathML</a>isT-periodic and, when restricted to<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/110/mathml/M64','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/110/mathml/M64">View MathML</a>, it belongs to Ω. Moreover, if<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/110/mathml/M65','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/110/mathml/M65">View MathML</a>denotes the angular momentum associated to<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/110/mathml/M62','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/110/mathml/M62">View MathML</a>, then

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/110/mathml/M67','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/110/mathml/M67">View MathML</a>

3 Main results

First we introduce some known results on eigenvalues. Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/110/mathml/M68','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/110/mathml/M68">View MathML</a> be a T-periodic potential such that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/110/mathml/M69','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/110/mathml/M69">View MathML</a>. Consider the eigenvalue problems of

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/110/mathml/M70','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/110/mathml/M70">View MathML</a>

(3.1)

with the periodic boundary condition (PC): <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/110/mathml/M71','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/110/mathml/M71">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/110/mathml/M72','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/110/mathml/M72">View MathML</a>, or with the antiperiodic boundary condition (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/110/mathml/M73','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/110/mathml/M73">View MathML</a>): <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/110/mathml/M74','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/110/mathml/M74">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/110/mathml/M75','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/110/mathml/M75">View MathML</a>. We use <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/110/mathml/M76','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/110/mathml/M76">View MathML</a> to denote all the eigenvalues of (3.1) with the Dirichlet boundary condition (DC): <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/110/mathml/M77','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/110/mathml/M77">View MathML</a>.

The following are the standard results for eigenvalues. See, e.g., reference [37].

(<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/110/mathml/M78','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/110/mathml/M78">View MathML</a>) With respect to the periodic and anti-periodic eigenvalues, there exist sequences

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/110/mathml/M79','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/110/mathml/M79">View MathML</a>

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/110/mathml/M80','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/110/mathml/M80">View MathML</a> (as <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/110/mathml/M81','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/110/mathml/M81">View MathML</a>), such that λ is an eigenvalue of (3.1)-(PC) if and only if <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/110/mathml/M82','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/110/mathml/M82">View MathML</a> or <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/110/mathml/M83','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/110/mathml/M83">View MathML</a> with n is even; and λ is an eigenvalue of (3.1)-(<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/110/mathml/M73','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/110/mathml/M73">View MathML</a>) if and only if <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/110/mathml/M85','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/110/mathml/M85">View MathML</a> or <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/110/mathml/M83','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/110/mathml/M83">View MathML</a> with n is odd.

(<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/110/mathml/M87','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/110/mathml/M87">View MathML</a>) The comparison results hold for all of these eigenvalues. If <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/110/mathml/M88','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/110/mathml/M88">View MathML</a>, then

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/110/mathml/M89','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/110/mathml/M89">View MathML</a>

for any <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/110/mathml/M90','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/110/mathml/M90">View MathML</a>.

(<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/110/mathml/M91','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/110/mathml/M91">View MathML</a>) The eigenvalues <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/110/mathml/M92','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/110/mathml/M92">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/110/mathml/M83','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/110/mathml/M83">View MathML</a> can be recovered from the Dirichlet eigenvalues in the following way. For any <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/110/mathml/M90','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/110/mathml/M90">View MathML</a>,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/110/mathml/M95','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/110/mathml/M95">View MathML</a>

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/110/mathml/M96','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/110/mathml/M96">View MathML</a> denotes the translation of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/110/mathml/M68','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/110/mathml/M68">View MathML</a>: <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/110/mathml/M98','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/110/mathml/M98">View MathML</a>.

Now we present our main result.

Theorem 3.1Let the following assumptions hold.

(<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/110/mathml/M99','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/110/mathml/M99">View MathML</a>) There exist a constant<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/110/mathml/M100','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/110/mathml/M100">View MathML</a>and a function<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/110/mathml/M101','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/110/mathml/M101">View MathML</a>such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/110/mathml/M102','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/110/mathml/M102">View MathML</a>

for alltand all<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/110/mathml/M103','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/110/mathml/M103">View MathML</a>, where<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/110/mathml/M104','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/110/mathml/M104">View MathML</a>satisfies

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/110/mathml/M105','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/110/mathml/M105">View MathML</a>

and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/110/mathml/M106','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/110/mathml/M106">View MathML</a>

(<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/110/mathml/M107','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/110/mathml/M107">View MathML</a>) There exist positiveT-periodic continuous functionsϕ, Φ such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/110/mathml/M108','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/110/mathml/M108">View MathML</a>

(3.2)

uniformly int. Moreover,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/110/mathml/M109','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/110/mathml/M109">View MathML</a>

(3.3)

Then Eq. (2.4) has aT-periodic solution, and there exists a<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/110/mathml/M60','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/110/mathml/M60">View MathML</a>such that, for every integer<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/110/mathml/M61','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/110/mathml/M61">View MathML</a>, Eq. (1.1) has a periodic solution with a minimal periodkT, which makes exactly one revolution around the origin in the period timekT. Moreover, there exists a constant<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/110/mathml/M112','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/110/mathml/M112">View MathML</a> (independent ofμandk) such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/110/mathml/M113','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/110/mathml/M113">View MathML</a>

and if<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/110/mathml/M65','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/110/mathml/M65">View MathML</a>denotes the angular momentum associated to<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/110/mathml/M62','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/110/mathml/M62">View MathML</a>, then

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/110/mathml/M116','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/110/mathml/M116">View MathML</a>

In order to apply Theorem 2.1, we consider the T-periodic problem (2.4).

Lemma 3.2Suppose that<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/110/mathml/M6','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/110/mathml/M6">View MathML</a>satisfies (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/110/mathml/M118','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/110/mathml/M118">View MathML</a>) andϕ, Φ satisfy (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/110/mathml/M107','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/110/mathml/M107">View MathML</a>). Then Eq. (2.4) has at least one positiveT-periodic solution.

Now we begin by showing that Lemma 3.2 holds, and use topological degree theory. To this end, we deform (2.4) to a simpler singular autonomous equation

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/110/mathml/M120','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/110/mathml/M120">View MathML</a>

where a for some positive constant satisfies <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/110/mathml/M121','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/110/mathml/M121">View MathML</a> for all t. Consider the following homotopy equation:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/110/mathml/M122','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/110/mathml/M122">View MathML</a>

(3.4)

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/110/mathml/M123','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/110/mathml/M123">View MathML</a>. We need to find a priori estimates for the possible positive T-periodic solutions of (3.4).

Note that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/110/mathml/M124','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/110/mathml/M124">View MathML</a> satisfies the conditions (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/110/mathml/M99','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/110/mathml/M99">View MathML</a>) uniformly with respect to <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/110/mathml/M126','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/110/mathml/M126">View MathML</a>. Moreover, for each <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/110/mathml/M127','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/110/mathml/M127">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/110/mathml/M124','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/110/mathml/M124">View MathML</a> satisfies (3.2) with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/110/mathml/M129','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/110/mathml/M129">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/110/mathml/M130','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/110/mathml/M130">View MathML</a>. We will prove that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/110/mathml/M131','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/110/mathml/M131">View MathML</a> satisfy (3.3) uniformly in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/110/mathml/M126','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/110/mathml/M126">View MathML</a>. The usual <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/110/mathml/M133','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/110/mathml/M133">View MathML</a>-norm is denoted by <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/110/mathml/M134','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/110/mathml/M134">View MathML</a>, and the supremum norm of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/110/mathml/M135','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/110/mathml/M135">View MathML</a> is denoted by <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/110/mathml/M136','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/110/mathml/M136">View MathML</a>.

This follows from the convexity of the first eigenvalues with respect to potentials.

Lemma 3.3Given<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/110/mathml/M137','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/110/mathml/M137">View MathML</a>. Then, for all<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/110/mathml/M126','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/110/mathml/M126">View MathML</a>,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/110/mathml/M139','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/110/mathml/M139">View MathML</a>

(3.5)

Proof Put <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/110/mathml/M140','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/110/mathml/M140">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/110/mathml/M126','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/110/mathml/M126">View MathML</a>. Then

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/110/mathml/M142','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/110/mathml/M142">View MathML</a>

For (3.5), applying <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/110/mathml/M143','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/110/mathml/M143">View MathML</a> to <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/110/mathml/M144','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/110/mathml/M144">View MathML</a>, where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/110/mathml/M145','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/110/mathml/M145">View MathML</a>, we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/110/mathml/M146','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/110/mathml/M146">View MathML</a>

for all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/110/mathml/M147','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/110/mathml/M147">View MathML</a>. Thus

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/110/mathml/M148','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/110/mathml/M148">View MathML</a>

Hence (3.5) holds. □

Applying Lemma 3.3 to <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/110/mathml/M149','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/110/mathml/M149">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/110/mathml/M150','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/110/mathml/M150">View MathML</a>, we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/110/mathml/M151','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/110/mathml/M151">View MathML</a>

Thus <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/110/mathml/M131','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/110/mathml/M131">View MathML</a> defined above satisfy (3.3) uniformly in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/110/mathml/M127','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/110/mathml/M127">View MathML</a>.

In the obtention of a priori estimates for all possible positive solutions to (3.4)-(PC), we simply prove this for all possible positive solutions to (2.4)-(PC), because <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/110/mathml/M154','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/110/mathml/M154">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/110/mathml/M155','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/110/mathml/M155">View MathML</a> satisfy (3.3) and also (3.2) uniformly in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/110/mathml/M126','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/110/mathml/M126">View MathML</a>.

Lemma 3.4Assume that<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/110/mathml/M157','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/110/mathml/M157">View MathML</a>of the equation<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/110/mathml/M158','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/110/mathml/M158">View MathML</a>, then

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/110/mathml/M159','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/110/mathml/M159">View MathML</a>

Proof By the results for eigenvalues in (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/110/mathml/M91','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/110/mathml/M91">View MathML</a>), we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/110/mathml/M161','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/110/mathml/M161">View MathML</a>

for all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/110/mathml/M162','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/110/mathml/M162">View MathML</a>.

Then, by the theory of linear second-order differential operators [38], the eigenvalues of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/110/mathml/M163','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/110/mathml/M163">View MathML</a> with Dirichlet boundary conditions form a sequence <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/110/mathml/M164','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/110/mathml/M164">View MathML</a> which tends to +∞, and the corresponding eigenfunctions <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/110/mathml/M165','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/110/mathml/M165">View MathML</a> are an orthonormal base of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/110/mathml/M166','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/110/mathml/M166">View MathML</a>. Hence, given <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/110/mathml/M167','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/110/mathml/M167">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/110/mathml/M168','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/110/mathml/M168">View MathML</a>, we can write

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/110/mathml/M169','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/110/mathml/M169">View MathML</a>

and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/110/mathml/M170','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/110/mathml/M170">View MathML</a>

This completes the proof. □

Lemma 3.5Under the assumptions as in Theorem 3.1, there exist<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/110/mathml/M171','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/110/mathml/M171">View MathML</a>such that any positiveT-periodic solution<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/110/mathml/M172','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/110/mathml/M172">View MathML</a>of (2.4)-(PC) satisfies

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/110/mathml/M173','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/110/mathml/M173">View MathML</a>

(3.6)

for some<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/110/mathml/M145','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/110/mathml/M145">View MathML</a>.

Proof Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/110/mathml/M172','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/110/mathml/M172">View MathML</a> be a positive T-periodic solution of (2.4)-(PC). By (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/110/mathml/M99','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/110/mathml/M99">View MathML</a>), there is <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/110/mathml/M177','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/110/mathml/M177">View MathML</a> such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/110/mathml/M178','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/110/mathml/M178">View MathML</a>

Integrating (2.4) from 0 to T, we get

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/110/mathml/M179','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/110/mathml/M179">View MathML</a>

Thus <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/110/mathml/M180','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/110/mathml/M180">View MathML</a>, there exist <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/110/mathml/M181','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/110/mathml/M181">View MathML</a> such that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/110/mathml/M182','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/110/mathml/M182">View MathML</a>.

Take some constant <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/110/mathml/M183','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/110/mathml/M183">View MathML</a>, where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/110/mathml/M184','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/110/mathml/M184">View MathML</a> is the average of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/110/mathml/M185','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/110/mathml/M185">View MathML</a>. From (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/110/mathml/M107','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/110/mathml/M107">View MathML</a>) there is <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/110/mathml/M187','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/110/mathml/M187">View MathML</a> (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/110/mathml/M188','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/110/mathml/M188">View MathML</a>) large enough such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/110/mathml/M189','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/110/mathml/M189">View MathML</a>

(3.7)

for all t and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/110/mathml/M190','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/110/mathml/M190">View MathML</a>. We assert that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/110/mathml/M191','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/110/mathml/M191">View MathML</a> for some <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/110/mathml/M192','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/110/mathml/M192">View MathML</a>. Otherwise, assume that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/110/mathml/M193','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/110/mathml/M193">View MathML</a> for all t.

Let

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/110/mathml/M194','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/110/mathml/M194">View MathML</a>

Moreover, write r as <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/110/mathml/M195','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/110/mathml/M195">View MathML</a>, then <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/110/mathml/M196','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/110/mathml/M196">View MathML</a> satisfies the following differential equation:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/110/mathml/M197','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/110/mathml/M197">View MathML</a>

(3.8)

Integrating (3.8) from 0 to T, we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/110/mathml/M198','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/110/mathml/M198">View MathML</a>

(3.9)

Multiplying (3.8) by <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/110/mathml/M196','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/110/mathml/M196">View MathML</a> and integrating, we get

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/110/mathml/M200','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/110/mathml/M200">View MathML</a>

(3.10)

where the fact <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/110/mathml/M201','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/110/mathml/M201">View MathML</a> is used.

Note that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/110/mathml/M202','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/110/mathml/M202">View MathML</a> for some <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/110/mathml/M147','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/110/mathml/M147">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/110/mathml/M204','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/110/mathml/M204">View MathML</a>, so <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/110/mathml/M205','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/110/mathml/M205">View MathML</a>. We assert that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/110/mathml/M206','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/110/mathml/M206">View MathML</a>. On the contrary, assume that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/110/mathml/M207','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/110/mathml/M207">View MathML</a>. Now, by (3.10), the first Dirichlet eigenvalue

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/110/mathml/M208','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/110/mathml/M208">View MathML</a>

So,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/110/mathml/M209','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/110/mathml/M209">View MathML</a>

On the other hand, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/110/mathml/M210','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/110/mathml/M210">View MathML</a>,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/110/mathml/M211','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/110/mathml/M211">View MathML</a>

This is a contradiction.

Now it follows from (3.9) that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/110/mathml/M212','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/110/mathml/M212">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/110/mathml/M213','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/110/mathml/M213">View MathML</a>, a contradiction to the positiveness of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/110/mathml/M172','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/110/mathml/M172">View MathML</a>. We have proved that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/110/mathml/M182','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/110/mathml/M182">View MathML</a> for some <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/110/mathml/M181','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/110/mathml/M181">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/110/mathml/M191','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/110/mathml/M191">View MathML</a> for some <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/110/mathml/M218','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/110/mathml/M218">View MathML</a>. Thus the intermediate value theorem implies that (3.6) holds. □

Lemma 3.6There exist<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/110/mathml/M219','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/110/mathml/M219">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/110/mathml/M220','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/110/mathml/M220">View MathML</a>such that any positiveT-periodic solution<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/110/mathml/M172','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/110/mathml/M172">View MathML</a>of (2.4)-(PC) satisfies

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/110/mathml/M222','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/110/mathml/M222">View MathML</a>

Proof From (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/110/mathml/M107','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/110/mathml/M107">View MathML</a>) and (3.7), we know that there is <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/110/mathml/M224','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/110/mathml/M224">View MathML</a> such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/110/mathml/M225','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/110/mathml/M225">View MathML</a>

for all t and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/110/mathml/M226','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/110/mathml/M226">View MathML</a>.

Multiplying (2.4) by r and then integrating over <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/110/mathml/M64','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/110/mathml/M64">View MathML</a>, we get

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/110/mathml/M228','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/110/mathml/M228">View MathML</a>

(3.11)

Note from Lemma 3.5 that there exists <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/110/mathml/M147','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/110/mathml/M147">View MathML</a> satisfying <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/110/mathml/M230','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/110/mathml/M230">View MathML</a>. Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/110/mathml/M231','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/110/mathml/M231">View MathML</a>, then <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/110/mathml/M232','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/110/mathml/M232">View MathML</a>. Thus

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/110/mathml/M233','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/110/mathml/M233">View MathML</a>

The other terms in (3.11) by the Hölder inequality can be estimated as follows:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/110/mathml/M234','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/110/mathml/M234">View MathML</a>

Thus (3.11) reads as

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/110/mathml/M235','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/110/mathml/M235">View MathML</a>

(3.12)

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/110/mathml/M236','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/110/mathml/M236">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/110/mathml/M237','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/110/mathml/M237">View MathML</a> are positive constants.

On the other hand, using Lemma 3.4,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/110/mathml/M238','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/110/mathml/M238">View MathML</a>

we get from (3.12) that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/110/mathml/M239','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/110/mathml/M239">View MathML</a>

Consequently, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/110/mathml/M240','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/110/mathml/M240">View MathML</a> for some <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/110/mathml/M241','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/110/mathml/M241">View MathML</a>. By (3.12), one has <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/110/mathml/M242','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/110/mathml/M242">View MathML</a> for some <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/110/mathml/M243','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/110/mathml/M243">View MathML</a>. From these, for any <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/110/mathml/M244','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/110/mathml/M244">View MathML</a>,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/110/mathml/M245','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/110/mathml/M245">View MathML</a>

Thus <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/110/mathml/M246','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/110/mathml/M246">View MathML</a> is obtained.

As <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/110/mathml/M180','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/110/mathml/M180">View MathML</a>, thus <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/110/mathml/M248','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/110/mathml/M248">View MathML</a>. Since <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/110/mathml/M249','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/110/mathml/M249">View MathML</a>, there exists <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/110/mathml/M250','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/110/mathml/M250">View MathML</a> such that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/110/mathml/M251','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/110/mathml/M251">View MathML</a>. Therefore

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/110/mathml/M252','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/110/mathml/M252">View MathML</a>

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/110/mathml/M253','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/110/mathml/M253">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/110/mathml/M254','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/110/mathml/M254">View MathML</a>. □

Next, the positive lower estimates for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/110/mathml/M255','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/110/mathml/M255">View MathML</a> are obtained from the condition (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/110/mathml/M99','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/110/mathml/M99">View MathML</a>).

Lemma 3.7There exists a constant<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/110/mathml/M257','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/110/mathml/M257">View MathML</a>such that any positive solution<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/110/mathml/M172','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/110/mathml/M172">View MathML</a>of (2.4)-(PC) satisfies

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/110/mathml/M259','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/110/mathml/M259">View MathML</a>

Proof From (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/110/mathml/M99','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/110/mathml/M99">View MathML</a>), we fix some <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/110/mathml/M261','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/110/mathml/M261">View MathML</a> such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/110/mathml/M262','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/110/mathml/M262">View MathML</a>

for all t and all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/110/mathml/M263','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/110/mathml/M263">View MathML</a>. Assume now that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/110/mathml/M264','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/110/mathml/M264">View MathML</a>

By Lemma 3.5, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/110/mathml/M265','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/110/mathml/M265">View MathML</a>. Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/110/mathml/M266','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/110/mathml/M266">View MathML</a> be the first time instant such that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/110/mathml/M267','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/110/mathml/M267">View MathML</a>. Then, for any <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/110/mathml/M268','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/110/mathml/M268">View MathML</a>, we have <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/110/mathml/M269','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/110/mathml/M269">View MathML</a>. Hence, for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/110/mathml/M268','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/110/mathml/M268">View MathML</a>,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/110/mathml/M271','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/110/mathml/M271">View MathML</a>

As <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/110/mathml/M272','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/110/mathml/M272">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/110/mathml/M273','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/110/mathml/M273">View MathML</a> for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/110/mathml/M274','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/110/mathml/M274">View MathML</a>. Therefore, the function <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/110/mathml/M275','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/110/mathml/M275">View MathML</a> has an inverse denoted by ξ.

Now multiplying (2.4) by <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/110/mathml/M276','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/110/mathml/M276">View MathML</a> and integrating over <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/110/mathml/M277','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/110/mathml/M277">View MathML</a>, we get

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/110/mathml/M278','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/110/mathml/M278">View MathML</a>

for some <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/110/mathml/M279','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/110/mathml/M279">View MathML</a>, where the results from Lemma 3.6 are used. By (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/110/mathml/M99','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/110/mathml/M99">View MathML</a>),

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/110/mathml/M281','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/110/mathml/M281">View MathML</a>

(3.13)

if <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/110/mathml/M282','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/110/mathml/M282">View MathML</a>. Thus we know from (3.13) that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/110/mathml/M283','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/110/mathml/M283">View MathML</a> for some constant <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/110/mathml/M284','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/110/mathml/M284">View MathML</a>. □

Now we give the proof of Lemma 3.2. Consider the homotopy equation (3.4), we can get a priori estimates as in Lemmas 3.5, 3.6, 3.7. That is, any positive T-periodic solution of (3.4) satisfies

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/110/mathml/M285','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/110/mathml/M285">View MathML</a>

for some positive constants <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/110/mathml/M286','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/110/mathml/M286">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/110/mathml/M287','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/110/mathml/M287">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/110/mathml/M288','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/110/mathml/M288">View MathML</a>. Define <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/110/mathml/M289','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/110/mathml/M289">View MathML</a> and let the open bounded in X be

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/110/mathml/M290','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/110/mathml/M290">View MathML</a>

By the homotopy invariance of degree and the result of Capietto, Mawhin and Zanolin [39],

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/110/mathml/M291','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/110/mathml/M291">View MathML</a>

Thus (3.4), with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/110/mathml/M292','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/110/mathml/M292">View MathML</a>, has at least one solution in Ω, which is a positive T-periodic solution of (2.4). By Theorem 2.1, the proof of Theorem 3.1 is thus completed.

Competing interests

The authors declare that they have no competing interests.

Authors’ contributions

All authors read and approved the final manuscript.

Acknowledgements

The authors express their thanks to the referees for their valuable comments and suggestions. This work is supported by the National Natural Science Foundation of China (Grant No. 11161017), Hainan Natural Science Foundation (Grant No. 113001).

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