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Positive periodic solutions for a second-order functional differential equation

Yongxiang Li* and Qiang Li

Author Affiliations

Department of Mathematics, Northwest Normal University, Lanzhou, 730070, People’s Republic of China

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Boundary Value Problems 2012, 2012:140  doi:10.1186/1687-2770-2012-140


The electronic version of this article is the complete one and can be found online at: http://www.boundaryvalueproblems.com/content/2012/1/140


Received:12 June 2012
Accepted:12 November 2012
Published:27 November 2012

© 2012 Li and Li; 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, the existence results of positive ω-periodic solutions are obtained for the second-order functional differential equation

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/140/mathml/M1','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/140/mathml/M1">View MathML</a>

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/140/mathml/M2','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/140/mathml/M2">View MathML</a> is a continuous function which is ω-periodic in t, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/140/mathml/M3','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/140/mathml/M3">View MathML</a> is a ω-periodic function, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/140/mathml/M4','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/140/mathml/M4">View MathML</a>. Our discussion is based on the fixed point index theory in cones.

MSC: 34C25, 47H10.

Keywords:
functional differential equation; positive periodic solution; cone; fixed point index

1 Introduction

In this paper, we discuss the existence of positive ω-periodic solutions of the second-order functional differential equation with the delay terms of first-order derivative in nonlinearity,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/140/mathml/M5','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/140/mathml/M5">View MathML</a>

(1)

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/140/mathml/M6','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/140/mathml/M6">View MathML</a> is a continuous function which is ω-periodic in t and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/140/mathml/M7','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/140/mathml/M7">View MathML</a> is a ω-periodic delay function, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/140/mathml/M8','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/140/mathml/M8">View MathML</a>.

For the second-order differential equation without delay and the first-order derivative term in nonlinearity,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/140/mathml/M9','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/140/mathml/M9">View MathML</a>

(2)

the existence problems of periodic solutions have attracted many authors’ attention and concern. Many theorems and methods of nonlinear functional analysis have been applied to research the periodic problems of Equation (2), such as the upper and lower solutions method and monotone iterative technique [1-4], the continuation method of topological degree [5-7], variational method and critical point theory [8-10], the theory of the fixed point index in cones [11-16], etc.

In recent years, the existence of periodic solutions for the second-order delayed differential equations have also been researched by many authors; see [17-24] and the references therein. In some practice models, only positive periodic solutions are significant. In [20,21,23], the authors obtained the existence of positive periodic solutions for some delayed second-order differential equations as a special form of the following equation:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/140/mathml/M10','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/140/mathml/M10">View MathML</a>

(3)

by using Krasnoselskii’s fixed point theorem of cone mapping or the theory of the fixed point index in cones. In these works, the positivity of Green’s function of the corresponding linear second-order periodic problems plays an important role. The positivity guarantees that the integral operators of the second-order periodic problems are cone-preserving in the cone

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/140/mathml/M11','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/140/mathml/M11">View MathML</a>

(4)

in the Banach space <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/140/mathml/M12','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/140/mathml/M12">View MathML</a>, where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/140/mathml/M13','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/140/mathml/M13">View MathML</a> is a constant. Hence, the fixed point theorems of cone mapping can be applied to periodic problems of the second-order delay equation (3) as well as Equation (2) (for Equation (2), see [11-16]). However, few people consider the existence of positive periodic solutions of Equation (1). Since the nonlinearity of Equation (1) explicitly contains the delayed first-order derivative term, the corresponding integral operator has no definition on the cone P. Thus, the argument methods used in [20,21,23] are not applicable to Equation (1).

The purpose of this paper is to discuss the existence of positive periodic solutions of Equation (1). We will use a different method to treat Equation (1). Our main results will be given in Section 3. Some preliminaries to discuss Equation (1) are presented in Section 2.

2 Preliminaries

Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/140/mathml/M14','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/140/mathml/M14">View MathML</a> denote the Banach space of all continuous ω-periodic function <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/140/mathml/M15','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/140/mathml/M15">View MathML</a> with the norm <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/140/mathml/M16','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/140/mathml/M16">View MathML</a>. Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/140/mathml/M17','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/140/mathml/M17">View MathML</a> be the Banach space of all continuous differentiable ω-periodic function <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/140/mathml/M15','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/140/mathml/M15">View MathML</a> with the norm

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/140/mathml/M19','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/140/mathml/M19">View MathML</a>

Generally, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/140/mathml/M20','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/140/mathml/M20">View MathML</a> denotes the nth-order continuous differentiable ω-periodic function space for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/140/mathml/M21','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/140/mathml/M21">View MathML</a>. Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/140/mathml/M22','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/140/mathml/M22">View MathML</a> be the cone of all nonnegative functions in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/140/mathml/M23','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/140/mathml/M23">View MathML</a>.

Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/140/mathml/M24','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/140/mathml/M24">View MathML</a> be a constant. For <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/140/mathml/M25','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/140/mathml/M25">View MathML</a>, we consider the linear second-order differential equation

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/140/mathml/M26','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/140/mathml/M26">View MathML</a>

(5)

The ω-periodic solutions of Equation (5) are closely related to the linear second-order boundary value problem

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/140/mathml/M27','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/140/mathml/M27">View MathML</a>

(6)

see [14]. It is easy to see that problem (6) has a unique solution which is explicitly given by

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/140/mathml/M28','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/140/mathml/M28">View MathML</a>

(7)

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/140/mathml/M29','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/140/mathml/M29">View MathML</a>. By [[14], Lemma 1], we have

Lemma 2.1Let<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/140/mathml/M24','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/140/mathml/M24">View MathML</a>. Then, for every<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/140/mathml/M31','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/140/mathml/M31">View MathML</a>, the linear equation (5) has a uniqueω-periodic solution<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/140/mathml/M15','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/140/mathml/M15">View MathML</a>which is given by

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/140/mathml/M33','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/140/mathml/M33">View MathML</a>

(8)

Moreover, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/140/mathml/M34','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/140/mathml/M34">View MathML</a>is a completely continuous linear operator.

Since <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/140/mathml/M35','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/140/mathml/M35">View MathML</a>, for every <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/140/mathml/M36','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/140/mathml/M36">View MathML</a>, by (8), if <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/140/mathml/M37','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/140/mathml/M37">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/140/mathml/M38','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/140/mathml/M38">View MathML</a>, then the ω-periodic solution of Equation (5) <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/140/mathml/M39','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/140/mathml/M39">View MathML</a> for every <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/140/mathml/M40','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/140/mathml/M40">View MathML</a>, and we term it the positive ω-periodic solution. Let

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/140/mathml/M41','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/140/mathml/M41">View MathML</a>

(9)

Define a set K in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/140/mathml/M17','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/140/mathml/M17">View MathML</a> by

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/140/mathml/M43','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/140/mathml/M43">View MathML</a>

(10)

It is easy to verify that K is a closed convex cone in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/140/mathml/M17','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/140/mathml/M17">View MathML</a>.

Lemma 2.2Let<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/140/mathml/M24','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/140/mathml/M24">View MathML</a>. Then, for every<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/140/mathml/M46','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/140/mathml/M46">View MathML</a>, the positiveω-periodic solution of Equation (5) <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/140/mathml/M47','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/140/mathml/M47">View MathML</a>. Namely, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/140/mathml/M48','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/140/mathml/M48">View MathML</a>.

Proof Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/140/mathml/M46','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/140/mathml/M46">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/140/mathml/M50','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/140/mathml/M50">View MathML</a>. For every <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/140/mathml/M40','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/140/mathml/M40">View MathML</a>, from (8) it follows that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/140/mathml/M52','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/140/mathml/M52">View MathML</a>

and therefore,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/140/mathml/M53','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/140/mathml/M53">View MathML</a>

Using (8), we obtain that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/140/mathml/M54','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/140/mathml/M54">View MathML</a>

For every <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/140/mathml/M55','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/140/mathml/M55">View MathML</a>, since

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/140/mathml/M56','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/140/mathml/M56">View MathML</a>

we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/140/mathml/M57','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/140/mathml/M57">View MathML</a>

Hence, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/140/mathml/M58','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/140/mathml/M58">View MathML</a>. □

Now we consider the nonlinear delay equation (1). Hereafter, we assume that the nonlinearity f satisfies the condition

(F0) There exists <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/140/mathml/M24','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/140/mathml/M24">View MathML</a> such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/140/mathml/M60','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/140/mathml/M60">View MathML</a>

Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/140/mathml/M61','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/140/mathml/M61">View MathML</a>, then <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/140/mathml/M62','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/140/mathml/M62">View MathML</a> for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/140/mathml/M63','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/140/mathml/M63">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/140/mathml/M40','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/140/mathml/M40">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/140/mathml/M65','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/140/mathml/M65">View MathML</a>, and Equation (1) is rewritten to

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/140/mathml/M66','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/140/mathml/M66">View MathML</a>

(11)

For every <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/140/mathml/M58','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/140/mathml/M58">View MathML</a>, set

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/140/mathml/M68','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/140/mathml/M68">View MathML</a>

(12)

Then <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/140/mathml/M69','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/140/mathml/M69">View MathML</a> is continuous. We define an integral operator <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/140/mathml/M70','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/140/mathml/M70">View MathML</a> by

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/140/mathml/M71','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/140/mathml/M71">View MathML</a>

(13)

By the definition of the operator S, the positive ω-periodic solution of Equation (1) is equivalent to the nontrivial fixed point of A. From assumption (F0), Lemma 2.1 and Lemma 2.2, we easily see that

Lemma 2.3<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/140/mathml/M72','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/140/mathml/M72">View MathML</a>and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/140/mathml/M73','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/140/mathml/M73">View MathML</a>is completely continuous.

We will find the non-zero fixed point of A by using the fixed point index theory in cones. We recall some concepts and conclusions on the fixed point index in [25,26]. Let E be a Banach space and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/140/mathml/M74','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/140/mathml/M74">View MathML</a> be a closed convex cone in E. Assume Ω is a bounded open subset of E with the boundary Ω, and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/140/mathml/M75','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/140/mathml/M75">View MathML</a>. Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/140/mathml/M76','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/140/mathml/M76">View MathML</a> be a completely continuous mapping. If <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/140/mathml/M77','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/140/mathml/M77">View MathML</a> for any <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/140/mathml/M78','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/140/mathml/M78">View MathML</a>, then the fixed point index <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/140/mathml/M79','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/140/mathml/M79">View MathML</a> has a definition. One important fact is that if <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/140/mathml/M80','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/140/mathml/M80">View MathML</a>, then A has a fixed point in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/140/mathml/M81','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/140/mathml/M81">View MathML</a>. The following two lemmas are needed in our argument.

Lemma 2.4 ([26])

Let Ω be a bounded open subset ofEwith<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/140/mathml/M82','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/140/mathml/M82">View MathML</a>and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/140/mathml/M83','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/140/mathml/M83">View MathML</a>be a completely continuous mapping. If<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/140/mathml/M84','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/140/mathml/M84">View MathML</a>for every<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/140/mathml/M78','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/140/mathml/M78">View MathML</a>and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/140/mathml/M86','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/140/mathml/M86">View MathML</a>, then<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/140/mathml/M87','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/140/mathml/M87">View MathML</a>.

Lemma 2.5 ([26])

Let Ω be a bounded open subset ofEand<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/140/mathml/M88','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/140/mathml/M88">View MathML</a>be a completely continuous mapping. If there exists an<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/140/mathml/M89','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/140/mathml/M89">View MathML</a>such that<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/140/mathml/M90','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/140/mathml/M90">View MathML</a>for every<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/140/mathml/M91','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/140/mathml/M91">View MathML</a>and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/140/mathml/M92','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/140/mathml/M92">View MathML</a>, then<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/140/mathml/M93','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/140/mathml/M93">View MathML</a>.

In the next section, we will use Lemma 2.4 and Lemma 2.5 to discuss the existence of positive ω-periodic solutions of Equation (1).

3 Main results

We consider the existence of positive ω-periodic solutions of the functional differential equation (1). Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/140/mathml/M94','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/140/mathml/M94">View MathML</a> satisfy assumption (F0) and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/140/mathml/M95','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/140/mathml/M95">View MathML</a> be ω-periodic in t. Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/140/mathml/M96','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/140/mathml/M96">View MathML</a> be the constant defined by (9) and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/140/mathml/M97','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/140/mathml/M97">View MathML</a>. For convenience, we introduce the notations

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/140/mathml/M98','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/140/mathml/M98">View MathML</a>

Our main results are as follows.

Theorem 3.1Let<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/140/mathml/M99','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/140/mathml/M99">View MathML</a>and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/140/mathml/M100','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/140/mathml/M100">View MathML</a>beω-periodic int, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/140/mathml/M101','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/140/mathml/M101">View MathML</a>. Iffsatisfies assumption (F0) and the condition

(F1) <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/140/mathml/M102','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/140/mathml/M102">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/140/mathml/M103','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/140/mathml/M103">View MathML</a>,

then Equation (1) has at least one positiveω-periodic solution.

Theorem 3.2Let<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/140/mathml/M99','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/140/mathml/M99">View MathML</a>and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/140/mathml/M100','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/140/mathml/M100">View MathML</a>beω-periodic int, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/140/mathml/M101','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/140/mathml/M101">View MathML</a>. Iffsatisfies assumption (F0) and the conditions

(F2) <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/140/mathml/M107','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/140/mathml/M107">View MathML</a>,

then Equation (1) has at least one positiveω-periodic solution.

In Theorem 3.1, the condition (F1) allows <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/140/mathml/M108','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/140/mathml/M108">View MathML</a> to be superlinear growth on x and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/140/mathml/M109','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/140/mathml/M109">View MathML</a>. For example,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/140/mathml/M110','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/140/mathml/M110">View MathML</a>

satisfies (F0) with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/140/mathml/M111','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/140/mathml/M111">View MathML</a> and (F1) with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/140/mathml/M112','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/140/mathml/M112">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/140/mathml/M113','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/140/mathml/M113">View MathML</a>.

In Theorem 3.2, the condition (F2) allows <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/140/mathml/M108','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/140/mathml/M108">View MathML</a> to be sublinear growth on x and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/140/mathml/M109','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/140/mathml/M109">View MathML</a>. For example,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/140/mathml/M116','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/140/mathml/M116">View MathML</a>

satisfies (F0) with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/140/mathml/M111','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/140/mathml/M111">View MathML</a> and (F2) with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/140/mathml/M118','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/140/mathml/M118">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/140/mathml/M119','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/140/mathml/M119">View MathML</a>.

Proof of Theorem 3.1 Choose the working space <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/140/mathml/M120','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/140/mathml/M120">View MathML</a>. Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/140/mathml/M121','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/140/mathml/M121">View MathML</a> be the closed convex cone in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/140/mathml/M17','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/140/mathml/M17">View MathML</a> defined by (10) and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/140/mathml/M73','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/140/mathml/M73">View MathML</a> be the operator defined by (13). Then the positive ω-periodic solution of Equation (1) is equivalent to the nontrivial fixed point of A. Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/140/mathml/M124','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/140/mathml/M124">View MathML</a> and set

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/140/mathml/M125','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/140/mathml/M125">View MathML</a>

(14)

We show that the operator A has a fixed point in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/140/mathml/M126','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/140/mathml/M126">View MathML</a> when r is small enough and R is large enough.

By <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/140/mathml/M102','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/140/mathml/M102">View MathML</a> and the definition of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/140/mathml/M128','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/140/mathml/M128">View MathML</a>, there exist <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/140/mathml/M129','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/140/mathml/M129">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/140/mathml/M130','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/140/mathml/M130">View MathML</a> such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/140/mathml/M131','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/140/mathml/M131">View MathML</a>

(15)

Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/140/mathml/M132','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/140/mathml/M132">View MathML</a>. We now prove that A satisfies the condition of Lemma 2.4 in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/140/mathml/M133','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/140/mathml/M133">View MathML</a>, namely <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/140/mathml/M84','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/140/mathml/M84">View MathML</a> for every <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/140/mathml/M135','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/140/mathml/M135">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/140/mathml/M136','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/140/mathml/M136">View MathML</a>. In fact, if there exist <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/140/mathml/M137','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/140/mathml/M137">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/140/mathml/M138','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/140/mathml/M138">View MathML</a> such that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/140/mathml/M139','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/140/mathml/M139">View MathML</a>, then by the definition of A and Lemma 2.1, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/140/mathml/M140','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/140/mathml/M140">View MathML</a> satisfies the delay differential equation

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/140/mathml/M141','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/140/mathml/M141">View MathML</a>

(16)

Since <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/140/mathml/M137','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/140/mathml/M137">View MathML</a>, by the definitions of K and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/140/mathml/M143','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/140/mathml/M143">View MathML</a>, we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/140/mathml/M144','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/140/mathml/M144">View MathML</a>

(17)

Hence, from (15) it follows that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/140/mathml/M145','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/140/mathml/M145">View MathML</a>

By this, (16) and the definition of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/140/mathml/M146','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/140/mathml/M146">View MathML</a>, we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/140/mathml/M147','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/140/mathml/M147">View MathML</a>

Integrating both sides of this inequality from 0 to ω and using the periodicity of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/140/mathml/M148','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/140/mathml/M148">View MathML</a>, we obtain that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/140/mathml/M149','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/140/mathml/M149">View MathML</a>

Since <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/140/mathml/M150','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/140/mathml/M150">View MathML</a>, it follows that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/140/mathml/M151','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/140/mathml/M151">View MathML</a>, which is a contradiction. Hence, A satisfies the condition of Lemma 2.4 in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/140/mathml/M152','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/140/mathml/M152">View MathML</a>. By Lemma 2.4, we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/140/mathml/M153','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/140/mathml/M153">View MathML</a>

(18)

On the other hand, since <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/140/mathml/M103','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/140/mathml/M103">View MathML</a>, by the definition of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/140/mathml/M155','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/140/mathml/M155">View MathML</a>, there exist <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/140/mathml/M156','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/140/mathml/M156">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/140/mathml/M157','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/140/mathml/M157">View MathML</a> such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/140/mathml/M158','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/140/mathml/M158">View MathML</a>

(19)

Choose <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/140/mathml/M159','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/140/mathml/M159">View MathML</a> and let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/140/mathml/M160','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/140/mathml/M160">View MathML</a>. Clearly, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/140/mathml/M161','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/140/mathml/M161">View MathML</a>. We show that A satisfies the condition of Lemma 2.5 in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/140/mathml/M162','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/140/mathml/M162">View MathML</a>, namely <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/140/mathml/M163','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/140/mathml/M163">View MathML</a> for every <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/140/mathml/M164','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/140/mathml/M164">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/140/mathml/M92','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/140/mathml/M92">View MathML</a>. In fact, if there exist <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/140/mathml/M166','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/140/mathml/M166">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/140/mathml/M167','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/140/mathml/M167">View MathML</a> such that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/140/mathml/M168','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/140/mathml/M168">View MathML</a>, since <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/140/mathml/M169','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/140/mathml/M169">View MathML</a>, by the definition of A and Lemma 2.1, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/140/mathml/M170','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/140/mathml/M170">View MathML</a> satisfies the differential equation

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/140/mathml/M171','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/140/mathml/M171">View MathML</a>

(20)

Since <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/140/mathml/M166','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/140/mathml/M166">View MathML</a>, by the definition of K, we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/140/mathml/M173','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/140/mathml/M173">View MathML</a>

(21)

By the latter inequality of (21), we have that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/140/mathml/M174','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/140/mathml/M174">View MathML</a>. This implies that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/140/mathml/M175','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/140/mathml/M175">View MathML</a>. Consequently,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/140/mathml/M176','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/140/mathml/M176">View MathML</a>

(22)

By (22) and the former inequality of (21), we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/140/mathml/M177','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/140/mathml/M177">View MathML</a>

From this, the latter inequality of (21) and (19), it follows that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/140/mathml/M178','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/140/mathml/M178">View MathML</a>

By this inequality, (20) and the definition of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/140/mathml/M146','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/140/mathml/M146">View MathML</a>, we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/140/mathml/M180','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/140/mathml/M180">View MathML</a>

Integrating this inequality on I and using the periodicity of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/140/mathml/M181','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/140/mathml/M181">View MathML</a>, we get that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/140/mathml/M182','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/140/mathml/M182">View MathML</a>

Since <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/140/mathml/M183','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/140/mathml/M183">View MathML</a>, from this inequality it follows that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/140/mathml/M184','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/140/mathml/M184">View MathML</a>, which is a contradiction. This means that A satisfies the condition of Lemma 2.5 in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/140/mathml/M162','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/140/mathml/M162">View MathML</a>. By Lemma 2.5,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/140/mathml/M186','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/140/mathml/M186">View MathML</a>

(23)

Now, by the additivity of fixed point index, (18) and (23), we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/140/mathml/M187','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/140/mathml/M187">View MathML</a>

Hence, A has a fixed-point in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/140/mathml/M188','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/140/mathml/M188">View MathML</a>, which is a positive ω-periodic solution of Equation (1). □

Proof of Theorem 3.2 Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/140/mathml/M189','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/140/mathml/M189">View MathML</a> be defined by (14). We prove that the operator A defined by (13) has a fixed point in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/140/mathml/M126','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/140/mathml/M126">View MathML</a> if r is small enough and R is large enough.

By <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/140/mathml/M191','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/140/mathml/M191">View MathML</a> and the definition of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/140/mathml/M192','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/140/mathml/M192">View MathML</a>, there exist <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/140/mathml/M193','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/140/mathml/M193">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/140/mathml/M130','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/140/mathml/M130">View MathML</a> such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/140/mathml/M195','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/140/mathml/M195">View MathML</a>

(24)

Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/140/mathml/M132','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/140/mathml/M132">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/140/mathml/M160','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/140/mathml/M160">View MathML</a>. We prove that A satisfies the condition of Lemma 2.5 in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/140/mathml/M133','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/140/mathml/M133">View MathML</a>, namely <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/140/mathml/M199','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/140/mathml/M199">View MathML</a> for every <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/140/mathml/M135','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/140/mathml/M135">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/140/mathml/M92','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/140/mathml/M92">View MathML</a>. In fact, if there exist <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/140/mathml/M137','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/140/mathml/M137">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/140/mathml/M203','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/140/mathml/M203">View MathML</a> such that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/140/mathml/M204','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/140/mathml/M204">View MathML</a>, since <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/140/mathml/M205','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/140/mathml/M205">View MathML</a>, by the definition of A and Lemma 2.1, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/140/mathml/M140','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/140/mathml/M140">View MathML</a> satisfies the delay differential equation

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/140/mathml/M207','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/140/mathml/M207">View MathML</a>

(25)

Since <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/140/mathml/M137','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/140/mathml/M137">View MathML</a>, by the definitions of K and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/140/mathml/M143','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/140/mathml/M143">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/140/mathml/M148','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/140/mathml/M148">View MathML</a> satisfies (17). From (17) and (24) it follows that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/140/mathml/M211','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/140/mathml/M211">View MathML</a>

By this, (25) and the definition of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/140/mathml/M146','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/140/mathml/M146">View MathML</a>, we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/140/mathml/M213','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/140/mathml/M213">View MathML</a>

Integrating this inequality on <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/140/mathml/M214','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/140/mathml/M214">View MathML</a> and using the periodicity of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/140/mathml/M215','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/140/mathml/M215">View MathML</a>, we obtain that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/140/mathml/M216','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/140/mathml/M216">View MathML</a>

Since <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/140/mathml/M150','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/140/mathml/M150">View MathML</a>, from this inequality it follows that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/140/mathml/M218','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/140/mathml/M218">View MathML</a>, which is a contradiction. Hence, A satisfies the condition of Lemma 2.5 in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/140/mathml/M133','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/140/mathml/M133">View MathML</a>. By Lemma 2.5, we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/140/mathml/M220','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/140/mathml/M220">View MathML</a>

(26)

Since <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/140/mathml/M221','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/140/mathml/M221">View MathML</a>, by the definition of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/140/mathml/M222','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/140/mathml/M222">View MathML</a>, there exist <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/140/mathml/M223','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/140/mathml/M223">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/140/mathml/M157','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/140/mathml/M157">View MathML</a> such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/140/mathml/M225','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/140/mathml/M225">View MathML</a>

(27)

Choosing <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/140/mathml/M159','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/140/mathml/M159">View MathML</a>, we show that A satisfies the condition of Lemma 2.4 in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/140/mathml/M162','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/140/mathml/M162">View MathML</a>, namely <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/140/mathml/M228','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/140/mathml/M228">View MathML</a> for every <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/140/mathml/M164','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/140/mathml/M164">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/140/mathml/M230','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/140/mathml/M230">View MathML</a>. In fact, if there exist <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/140/mathml/M166','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/140/mathml/M166">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/140/mathml/M232','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/140/mathml/M232">View MathML</a> such that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/140/mathml/M233','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/140/mathml/M233">View MathML</a>, then by the definition of A and Lemma 2.1, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/140/mathml/M234','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/140/mathml/M234">View MathML</a> satisfies the differential equation

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/140/mathml/M235','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/140/mathml/M235">View MathML</a>

(28)

Since <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/140/mathml/M166','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/140/mathml/M166">View MathML</a>, by the definition of K, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/140/mathml/M181','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/140/mathml/M181">View MathML</a> satisfies (21). From the second inequality of (21), it follows that (22) holds. By (22) and the first inequality of (21), we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/140/mathml/M238','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/140/mathml/M238">View MathML</a>

From this, the second inequality of (21) and (27), it follows that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/140/mathml/M239','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/140/mathml/M239">View MathML</a>

By this and (28), we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/140/mathml/M240','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/140/mathml/M240">View MathML</a>

Integrating this inequality on <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/140/mathml/M214','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/140/mathml/M214">View MathML</a> and using the periodicity of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/140/mathml/M242','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/140/mathml/M242">View MathML</a>, we obtain that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/140/mathml/M243','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/140/mathml/M243">View MathML</a>

Since <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/140/mathml/M244','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/140/mathml/M244">View MathML</a>, from this inequality it follows that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/140/mathml/M245','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/140/mathml/M245">View MathML</a>, which is a contradiction. This means that A satisfies the condition of Lemma 2.4 in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/140/mathml/M162','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/140/mathml/M162">View MathML</a>. By Lemma 2.4,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/140/mathml/M247','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/140/mathml/M247">View MathML</a>

(29)

Now, from (26) and (29), it follows that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/140/mathml/M248','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/140/mathml/M248">View MathML</a>

Hence, A has a fixed-point in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/140/mathml/M188','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/140/mathml/M188">View MathML</a>, which is a positive ω-periodic solution of Equation (1). □

Example 1 Consider the following second-order differential equation with delay:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/140/mathml/M250','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/140/mathml/M250">View MathML</a>

(30)

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/140/mathml/M251','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/140/mathml/M251">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/140/mathml/M252','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/140/mathml/M252">View MathML</a>. If <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/140/mathml/M253','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/140/mathml/M253">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/140/mathml/M254','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/140/mathml/M254">View MathML</a> for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/140/mathml/M36','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/140/mathml/M36">View MathML</a>, we can verify that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/140/mathml/M256','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/140/mathml/M256">View MathML</a>

satisfies the conditions (F0) and (F1) for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/140/mathml/M257','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/140/mathml/M257">View MathML</a>. By Theorem 3.1, the delay equation (30) has at least one positive ω-periodic solution.

Example 2 Consider the functional differential equation

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/140/mathml/M258','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/140/mathml/M258">View MathML</a>

(31)

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/140/mathml/M259','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/140/mathml/M259">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/140/mathml/M252','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/140/mathml/M252">View MathML</a>, and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/140/mathml/M261','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/140/mathml/M261">View MathML</a>. If <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/140/mathml/M262','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/140/mathml/M262">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/140/mathml/M263','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/140/mathml/M263">View MathML</a> for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/140/mathml/M36','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/140/mathml/M36">View MathML</a>. We easily see that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/140/mathml/M265','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/140/mathml/M265">View MathML</a>

satisfies the conditions (F0) and (F2) for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/140/mathml/M257','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/140/mathml/M257">View MathML</a>. By Theorem 3.2, the functional differential equation (31) has a positive ω-periodic solution.

Competing interests

The authors declare that they have no competing interests.

Authors’ contributions

YL carried out the main part of this article. All authors read and approved the final manuscript.

Acknowledgements

The research was supported by the NNSFs of China (11261053, 11061031).

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