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Multiplicity of positive solutions to second-order singular differential equations with a parameter

Shengjun Li12*, Fang-fang Liao3 and Hailong Zhu4

Author Affiliations

1 College of Science, Hohai University, Nanjing, 210098, China

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

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

4 School of Statistics and Applied Mathematics, Anhui University of Finance and Economics, Bengbu, 233030, China

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Boundary Value Problems 2014, 2014:115  doi:10.1186/1687-2770-2014-115

The electronic version of this article is the complete one and can be found online at: http://www.boundaryvalueproblems.com/content/2014/1/115


Received:15 January 2014
Accepted:28 April 2014
Published:14 May 2014

© 2014 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 credited.

Abstract

We study the existence and multiplicity of positive periodic solutions for second-order nonlinear damped differential equations by combing the analysis of positiveness of the Green function for a linear damped equation. Our nonlinearity may be singular in its dependent variable. The proof of the main result relies on the Guo-Krasnosel’skii fixed point theorem on compression and expansion of cones.

MSC: 34B16, 34C25.

Keywords:
positive solutions; singular; Guo-Krasnosel’skii fixed point theorem

1 Introduction

In this paper, we study the existence and multiplicity of positive T-periodic solutions for the following second-order singular differential equation:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/115/mathml/M1','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/115/mathml/M1">View MathML</a>

(1.1)

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/115/mathml/M2','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/115/mathml/M2">View MathML</a> is a positive parameter, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/115/mathml/M3','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/115/mathml/M3">View MathML</a> and the nonlinearity <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/115/mathml/M4','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/115/mathml/M4">View MathML</a>. In particular, the nonlinearity may change sign and have a repulsive singularity at <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/115/mathml/M5','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/115/mathml/M5">View MathML</a>, which means that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/115/mathml/M6','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/115/mathml/M6">View MathML</a>

Equation (1.1) is a particular case of a more general class of Sturm equations of the type

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/115/mathml/M7','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/115/mathml/M7">View MathML</a>

where p is a strictly positive absolutely continuous function. Such equations, even in the case <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/115/mathml/M8','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/115/mathml/M8">View MathML</a>, where they are referred to as being of Schrödinger or Klein-Gordon type, appear in many scientific areas including quantum field theory, gas dynamics, fluid mechanics and chemistry.

Electrostatic or gravitational forces are the most important examples of singular interactions. During the last few decades, the study of the existence of positive solutions for singular differential equations has deserved the attention of many researchers [1-7]. Some strong force conditions introduced by Gordon [8] are standard in the related earlier works [6,7,9]. Compared with the case of strong singularity, the study of the existence of periodic solutions under the presence of a weak singularity is more recent, but has also attracted many researchers [2,3,10,11]. In particular, the degree theory [6,7], the method of upper and lower solutions [11,12], Schauder’s fixed point theorem [2,10], some fixed point theorems in cones for completely continuous operators [13-16] and a nonlinear Leray-Schauder alternative principle [17-19] are the most relevant tools.

However, singular differential equation (1.1), in which the nonlinearity is dependent on the derivative and does not require f to be nonnegative, has not attracted much attention in the literature. There are not so many existence results for (1.1) even when the nonlinearity is independent of the derivative. In this paper, we try to fill this gap and establish the existence of positive T-periodic solutions of (1.1) using the Guo-Krasnosel’skii fixed point theorem on compression and expansion of cones, which has been used to study positive solutions for systems of ordinary, functional differential equations [14-16]. We remark that it is sufficient to prove that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/115/mathml/M9','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/115/mathml/M9">View MathML</a> is continuous and completely continuous in Lemma 3.2 (Section 3). This point is essential and advantageous.

As mentioned above, this paper is mainly motivated by the recent paper [14,16]. The aim of this paper is to study the multiplicity of positive solutions to (1.1). It is proved that such a problem has at least two positive solutions under reasonable conditions (see Theorem 3.5). And the remaining part of this paper is organized as follows. In Section 2, we find the Green function of the linear damped equation

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/115/mathml/M10','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/115/mathml/M10">View MathML</a>

(1.2)

subject to periodic boundary conditions

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/115/mathml/M11','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/115/mathml/M11">View MathML</a>

(1.3)

and prove its positiveness. The fact is very crucial to our arguments. Moreover, the one-signed property of the Green function implies that a maximum principle and an anti-maximum principle hold for the corresponding linear differential equations subject to various boundary conditions, which is an important topic in differential equations (see [20,21]). In Section 3, by employing the Guo-Krasnosel’skii fixed point theorem, we prove the existence of twin positive solutions for (1.1) under the positiveness of the Green function associated with (1.2)-(1.3). To illustrate the new results, some applications are also given.

2 The Green function and its positiveness

In this section, we consider the nonhomogeneous equation

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/115/mathml/M12','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/115/mathml/M12">View MathML</a>

(2.1)

We say that (1.2)-(1.3) is nonresonant if its unique T-periodic solution is the trivial one. When (1.2)-(1.3) is nonresonant, as a consequence of Fredholm’s alternative, equation (2.1) admits a unique T-periodic solution which can be written as

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/115/mathml/M13','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/115/mathml/M13">View MathML</a>

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/115/mathml/M14','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/115/mathml/M14">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/115/mathml/M15','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/115/mathml/M15">View MathML</a> is the Green function of (2.1), associated with (1.3), and we will prove its positiveness. Throughout this paper, we assume that the following condition is satisfied:

(H) <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/115/mathml/M16','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/115/mathml/M16">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/115/mathml/M17','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/115/mathml/M17">View MathML</a> are continuous functions and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/115/mathml/M18','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/115/mathml/M18">View MathML</a> for all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/115/mathml/M19','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/115/mathml/M19">View MathML</a>.

Lemma 2.1Let<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/115/mathml/M20','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/115/mathml/M20">View MathML</a>be a continuous function. Then, for any nonnegative continuous function<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/115/mathml/M21','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/115/mathml/M21">View MathML</a>defined on<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/115/mathml/M22','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/115/mathml/M22">View MathML</a>, the integral equation

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/115/mathml/M23','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/115/mathml/M23">View MathML</a>

(2.2)

has a unique solution<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/115/mathml/M24','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/115/mathml/M24">View MathML</a>, which is continuous on<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/115/mathml/M22','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/115/mathml/M22">View MathML</a>and satisfies the following inequality:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/115/mathml/M26','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/115/mathml/M26">View MathML</a>

(2.3)

Proof We solve equation (2.2) by the method of successive approximations. Let

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/115/mathml/M27','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/115/mathml/M27">View MathML</a>

Take

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/115/mathml/M28','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/115/mathml/M28">View MathML</a>

One can easily verify that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/115/mathml/M29','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/115/mathml/M29">View MathML</a>

which implies that the series <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/115/mathml/M30','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/115/mathml/M30">View MathML</a> converges uniformly with respect to <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/115/mathml/M31','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/115/mathml/M31">View MathML</a>. Obviously, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/115/mathml/M32','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/115/mathml/M32">View MathML</a> is a continuous solution of (2.2). Moreover, inequality (2.3) holds because <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/115/mathml/M21','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/115/mathml/M21">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/115/mathml/M34','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/115/mathml/M34">View MathML</a> are nonnegative functions.

Next we prove the uniqueness. To do so, we first show that the solution of (2.2) is unique on <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/115/mathml/M35','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/115/mathml/M35">View MathML</a> with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/115/mathml/M36','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/115/mathml/M36">View MathML</a>. Then the uniqueness of the solution on <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/115/mathml/M22','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/115/mathml/M22">View MathML</a> is direct using the continuation property.

On the contrary, suppose that (2.2) has two solutions <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/115/mathml/M38','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/115/mathml/M38">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/115/mathml/M39','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/115/mathml/M39">View MathML</a> on <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/115/mathml/M40','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/115/mathml/M40">View MathML</a>. Then, for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/115/mathml/M41','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/115/mathml/M41">View MathML</a>, we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/115/mathml/M42','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/115/mathml/M42">View MathML</a>

Hence it follows that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/115/mathml/M43','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/115/mathml/M43">View MathML</a> for all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/115/mathml/M41','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/115/mathml/M41">View MathML</a>. □

Let us denote by <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/115/mathml/M45','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/115/mathml/M45">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/115/mathml/M46','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/115/mathml/M46">View MathML</a> the solutions of (1.2) satisfying the initial conditions

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/115/mathml/M47','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/115/mathml/M47">View MathML</a>

Lemma 2.2<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/115/mathml/M45','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/115/mathml/M45">View MathML</a>and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/115/mathml/M46','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/115/mathml/M46">View MathML</a>satisfy the following integral equations:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/115/mathml/M50','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/115/mathml/M50">View MathML</a>

(2.4)

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/115/mathml/M51','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/115/mathml/M51">View MathML</a>

(2.5)

Proof Since <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/115/mathml/M45','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/115/mathml/M45">View MathML</a> is a solution of (1.2), we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/115/mathml/M53','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/115/mathml/M53">View MathML</a>

(2.6)

Integrating (2.6) from 0 to t and noticing <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/115/mathml/M54','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/115/mathml/M54">View MathML</a>, we obtain

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/115/mathml/M55','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/115/mathml/M55">View MathML</a>

To obtain (2.4), we only need to integrate the above equality from 0 to t and notice <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/115/mathml/M56','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/115/mathml/M56">View MathML</a>. In a similar way, we can prove (2.5). □

Lemma 2.3For the solution<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/115/mathml/M24','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/115/mathml/M24">View MathML</a>of boundary value problem (2.1)-(1.3), the formula

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/115/mathml/M58','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/115/mathml/M58">View MathML</a>

(2.7)

holds, where

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/115/mathml/M59','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/115/mathml/M59">View MathML</a>

(2.8)

is the Green function, the numberDis defined by<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/115/mathml/M60','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/115/mathml/M60">View MathML</a>.

Proof It is easy to see that the general solution of equation (2.1) has the form

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/115/mathml/M61','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/115/mathml/M61">View MathML</a>

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/115/mathml/M62','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/115/mathml/M62">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/115/mathml/M63','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/115/mathml/M63">View MathML</a> are arbitrary constants. Substituting this expression for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/115/mathml/M24','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/115/mathml/M24">View MathML</a> in boundary condition (1.3), we can obtain

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/115/mathml/M65','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/115/mathml/M65">View MathML</a>

After not very complicated calculations, we can get (2.7) and (2.8). □

Remark 2.4 As a direct application of Lemma 2.3, if <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/115/mathml/M66','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/115/mathml/M66">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/115/mathml/M67','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/115/mathml/M67">View MathML</a>, then the Green function <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/115/mathml/M15','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/115/mathml/M15">View MathML</a> of boundary value problem (2.1)-(1.3) has the form

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/115/mathml/M69','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/115/mathml/M69">View MathML</a>

Lemma 2.5Assume that (H) holds. Then the Green function<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/115/mathml/M15','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/115/mathml/M15">View MathML</a>associated with (2.1)-(1.3) is positive for all<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/115/mathml/M71','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/115/mathml/M71">View MathML</a>.

Proof Since <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/115/mathml/M72','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/115/mathml/M72">View MathML</a>, it is enough to prove that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/115/mathml/M73','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/115/mathml/M73">View MathML</a> for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/115/mathml/M74','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/115/mathml/M74">View MathML</a>. Recall that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/115/mathml/M45','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/115/mathml/M45">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/115/mathml/M46','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/115/mathml/M46">View MathML</a> satisfy integral equations (2.4) and (2.5). By condition (H) and Lemma 2.1, it follows that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/115/mathml/M77','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/115/mathml/M77">View MathML</a>

(2.9)

Now from (2.9) we get <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/115/mathml/M78','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/115/mathml/M78">View MathML</a>. Setting

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/115/mathml/M79','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/115/mathml/M79">View MathML</a>

for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/115/mathml/M80','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/115/mathml/M80">View MathML</a>, we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/115/mathml/M81','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/115/mathml/M81">View MathML</a>

Evidently, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/115/mathml/M82','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/115/mathml/M82">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/115/mathml/M83','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/115/mathml/M83">View MathML</a> for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/115/mathml/M84','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/115/mathml/M84">View MathML</a> holds. Let us now show that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/115/mathml/M85','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/115/mathml/M85">View MathML</a>

(2.10)

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/115/mathml/M86','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/115/mathml/M86">View MathML</a>

(2.11)

To prove (2.10), we note that for fixed <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/115/mathml/M87','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/115/mathml/M87">View MathML</a>,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/115/mathml/M88','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/115/mathml/M88">View MathML</a>

and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/115/mathml/M89','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/115/mathml/M89">View MathML</a>

Hence it follows that for all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/115/mathml/M90','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/115/mathml/M90">View MathML</a>, we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/115/mathml/M91','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/115/mathml/M91">View MathML</a>

(2.12)

Using Lemma 2.1, we get from (2.12) that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/115/mathml/M92','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/115/mathml/M92">View MathML</a> for all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/115/mathml/M93','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/115/mathml/M93">View MathML</a>.

Next, we prove (2.11), note that for fixed <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/115/mathml/M94','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/115/mathml/M94">View MathML</a>,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/115/mathml/M95','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/115/mathml/M95">View MathML</a>

and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/115/mathml/M96','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/115/mathml/M96">View MathML</a>

Hence it follows that, for all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/115/mathml/M97','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/115/mathml/M97">View MathML</a>, we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/115/mathml/M98','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/115/mathml/M98">View MathML</a>

(2.13)

Again using Lemma 2.1, we get from (2.13) that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/115/mathml/M99','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/115/mathml/M99">View MathML</a> if <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/115/mathml/M100','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/115/mathml/M100">View MathML</a>, and the proof is completed. □

Under hypothesis (H), we always define

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/115/mathml/M101','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/115/mathml/M101">View MathML</a>

(2.14)

Thus <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/115/mathml/M102','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/115/mathml/M102">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/115/mathml/M103','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/115/mathml/M103">View MathML</a>. When <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/115/mathml/M66','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/115/mathml/M66">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/115/mathml/M67','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/115/mathml/M67">View MathML</a>, and a direct calculation shows that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/115/mathml/M106','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/115/mathml/M106">View MathML</a>

3 Main results

In this section, we state and prove the new existence results for (1.1). The proof is based on the following well-known fixed point theorem on compression and expansion of cones, which we state here for the convenience of the reader, after introducing the definition of a cone.

Definition 3.1 Let X be a Banach space and let K be a closed, nonempty subset of X. K is a cone if

(i) <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/115/mathml/M107','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/115/mathml/M107">View MathML</a> for all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/115/mathml/M108','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/115/mathml/M108">View MathML</a> and all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/115/mathml/M109','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/115/mathml/M109">View MathML</a>,

(ii) <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/115/mathml/M110','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/115/mathml/M110">View MathML</a> implies <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/115/mathml/M111','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/115/mathml/M111">View MathML</a>.

We also recall that a compact operator means an operator which transforms every bounded set into a relatively compact set. Let us define the function <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/115/mathml/M112','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/115/mathml/M112">View MathML</a> and use <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/115/mathml/M113','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/115/mathml/M113">View MathML</a> to denote the usual <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/115/mathml/M114','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/115/mathml/M114">View MathML</a>-norm over <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/115/mathml/M115','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/115/mathml/M115">View MathML</a>, by <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/115/mathml/M116','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/115/mathml/M116">View MathML</a> we denote the supremum norm of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/115/mathml/M117','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/115/mathml/M117">View MathML</a>.

Lemma 3.2[22]

LetXbe a Banach space andK (⊂X) be a cone. Assume that<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/115/mathml/M118','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/115/mathml/M118">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/115/mathml/M119','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/115/mathml/M119">View MathML</a>are open subsets ofXwith<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/115/mathml/M120','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/115/mathml/M120">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/115/mathml/M121','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/115/mathml/M121">View MathML</a>, and let

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/115/mathml/M122','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/115/mathml/M122">View MathML</a>

be a completely continuous operator such that either

(i) <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/115/mathml/M123','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/115/mathml/M123">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/115/mathml/M124','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/115/mathml/M124">View MathML</a>and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/115/mathml/M125','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/115/mathml/M125">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/115/mathml/M126','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/115/mathml/M126">View MathML</a>; or

(ii) <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/115/mathml/M125','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/115/mathml/M125">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/115/mathml/M124','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/115/mathml/M124">View MathML</a>and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/115/mathml/M123','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/115/mathml/M123">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/115/mathml/M130','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/115/mathml/M130">View MathML</a>.

Thenhas a fixed point in<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/115/mathml/M132','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/115/mathml/M132">View MathML</a>.

Let<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/115/mathml/M133','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/115/mathml/M133">View MathML</a>and define

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/115/mathml/M134','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/115/mathml/M134">View MathML</a>

whereσis as in (2.14).

One may readily verify thatKis a cone inX. Now, suppose that<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/115/mathml/M135','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/115/mathml/M135">View MathML</a>is a continuous function. Define an operator

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/115/mathml/M136','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/115/mathml/M136">View MathML</a>

for<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/115/mathml/M137','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/115/mathml/M137">View MathML</a>and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/115/mathml/M19','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/115/mathml/M19">View MathML</a>.

Lemma 3.3<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/115/mathml/M139','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/115/mathml/M139">View MathML</a>is well defined.

Proof Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/115/mathml/M137','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/115/mathml/M137">View MathML</a>, then we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/115/mathml/M141','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/115/mathml/M141">View MathML</a>

This implies that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/115/mathml/M142','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/115/mathml/M142">View MathML</a> and the proof is completed. □

It is easy to prove.

Lemma 3.4is continuous and completely continuous.

Now we present our main result.

Theorem 3.5Suppose that (1.1) satisfies (H). Furthermore, assume that

(H1) <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/115/mathml/M144','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/115/mathml/M144">View MathML</a>is continuous and there exists a constant<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/115/mathml/M145','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/115/mathml/M145">View MathML</a>such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/115/mathml/M146','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/115/mathml/M146">View MathML</a>

(H2) <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/115/mathml/M147','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/115/mathml/M147">View MathML</a>and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/115/mathml/M148','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/115/mathml/M148">View MathML</a>uniformly<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/115/mathml/M149','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/115/mathml/M149">View MathML</a>.

Then (1.1) has at least two positiveT-periodic solutions for sufficiently smallλ.

Proof To show that (1.1) has a positive solution, we should only show that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/115/mathml/M150','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/115/mathml/M150">View MathML</a>

(3.1)

has a positive solution x satisfying (1.3) and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/115/mathml/M151','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/115/mathml/M151">View MathML</a> for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/115/mathml/M19','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/115/mathml/M19">View MathML</a>. If it is right, then <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/115/mathml/M153','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/115/mathml/M153">View MathML</a> is a solution of (1.1) since

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/115/mathml/M154','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/115/mathml/M154">View MathML</a>

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/115/mathml/M155','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/115/mathml/M155">View MathML</a> is used.

Problem (3.1)-(1.3) is equivalent to the following fixed point of the operator equation:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/115/mathml/M156','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/115/mathml/M156">View MathML</a>

where is a completely continuous operator defined by

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/115/mathml/M158','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/115/mathml/M158">View MathML</a>

Since <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/115/mathml/M159','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/115/mathml/M159">View MathML</a>, there exists <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/115/mathml/M160','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/115/mathml/M160">View MathML</a> such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/115/mathml/M161','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/115/mathml/M161">View MathML</a>

For <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/115/mathml/M162','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/115/mathml/M162">View MathML</a>, let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/115/mathml/M163','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/115/mathml/M163">View MathML</a> and note that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/115/mathml/M164','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/115/mathml/M164">View MathML</a>.

First we show

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/115/mathml/M165','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/115/mathml/M165">View MathML</a>

Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/115/mathml/M166','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/115/mathml/M166">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/115/mathml/M167','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/115/mathml/M167">View MathML</a>. For any <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/115/mathml/M168','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/115/mathml/M168">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/115/mathml/M169','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/115/mathml/M169">View MathML</a>, we can verify that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/115/mathml/M170','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/115/mathml/M170">View MathML</a>

Then we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/115/mathml/M171','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/115/mathml/M171">View MathML</a>

This implies <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/115/mathml/M172','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/115/mathml/M172">View MathML</a>.

In view of the assumption

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/115/mathml/M173','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/115/mathml/M173">View MathML</a>

then there is <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/115/mathml/M174','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/115/mathml/M174">View MathML</a> such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/115/mathml/M175','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/115/mathml/M175">View MathML</a>

Hence, we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/115/mathml/M176','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/115/mathml/M176">View MathML</a>

Next, we show that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/115/mathml/M177','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/115/mathml/M177">View MathML</a>

To see this, let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/115/mathml/M178','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/115/mathml/M178">View MathML</a>, then

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/115/mathml/M179','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/115/mathml/M179">View MathML</a>

It follows from Lemma 3.2 that has a fixed point <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/115/mathml/M181','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/115/mathml/M181">View MathML</a> such that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/115/mathml/M182','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/115/mathml/M182">View MathML</a>, which is a positive periodic solution of (3.1) for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/115/mathml/M183','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/115/mathml/M183">View MathML</a> satisfying

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/115/mathml/M184','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/115/mathml/M184">View MathML</a>

So, equation (1.1) has a positive solution <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/115/mathml/M185','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/115/mathml/M185">View MathML</a>.

On the other hand, since

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/115/mathml/M186','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/115/mathml/M186">View MathML</a>

hence, there exists a positive number <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/115/mathml/M187','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/115/mathml/M187">View MathML</a> such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/115/mathml/M188','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/115/mathml/M188">View MathML</a>

problem (1.1)-(1.3) is equivalent to the following fixed point of the operator equation:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/115/mathml/M189','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/115/mathml/M189">View MathML</a>

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/115/mathml/M190','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/115/mathml/M190">View MathML</a> is a continuous and completely continuous operator defined by

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/115/mathml/M191','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/115/mathml/M191">View MathML</a>

And for any <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/115/mathml/M192','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/115/mathml/M192">View MathML</a>, define

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/115/mathml/M193','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/115/mathml/M193">View MathML</a>

Furthermore, for any <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/115/mathml/M194','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/115/mathml/M194">View MathML</a>, we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/115/mathml/M195','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/115/mathml/M195">View MathML</a>

Thus, from the above inequality, there exists <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/115/mathml/M196','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/115/mathml/M196">View MathML</a> such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/115/mathml/M197','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/115/mathml/M197">View MathML</a>

Since <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/115/mathml/M198','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/115/mathml/M198">View MathML</a>, then there is a positive number <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/115/mathml/M199','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/115/mathml/M199">View MathML</a> such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/115/mathml/M200','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/115/mathml/M200">View MathML</a>

where γ satisfies <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/115/mathml/M201','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/115/mathml/M201">View MathML</a>.

If <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/115/mathml/M202','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/115/mathml/M202">View MathML</a>, then

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/115/mathml/M203','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/115/mathml/M203">View MathML</a>

It follows from Lemma 3.2 that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/115/mathml/M190','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/115/mathml/M190">View MathML</a> has a fixed point <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/115/mathml/M205','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/115/mathml/M205">View MathML</a> such that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/115/mathml/M206','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/115/mathml/M206">View MathML</a>, which is a positive periodic solution of (1.1) for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/115/mathml/M207','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/115/mathml/M207">View MathML</a> satisfying

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/115/mathml/M208','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/115/mathml/M208">View MathML</a>

Noting that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/115/mathml/M209','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/115/mathml/M209">View MathML</a>

we can conclude that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/115/mathml/M38','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/115/mathml/M38">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/115/mathml/M39','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/115/mathml/M39">View MathML</a> are the desired distinct positive periodic solutions of (1.1) for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/115/mathml/M212','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/115/mathml/M212">View MathML</a>. □

Example Let the nonlinearity in (1.1) be

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/115/mathml/M213','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/115/mathml/M213">View MathML</a>

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/115/mathml/M214','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/115/mathml/M214">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/115/mathml/M215','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/115/mathml/M215">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/115/mathml/M216','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/115/mathml/M216">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/115/mathml/M217','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/115/mathml/M217">View MathML</a>. It is clear that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/115/mathml/M218','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/115/mathml/M218">View MathML</a> satisfies conditions (H1), (H2). Then (1.1) has at least two positive T-periodic solutions for sufficiently small λ.

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, No. 11301001, No. 11301139), Hainan Natural Science Foundation (Grant No.113001), Excellent Youth Scholars Foundation and the Natural Science Foundation of Anhui Province of PR China (No. 2013SQRL030ZD).

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