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Positive solutions for a fourth-order p-Laplacian boundary value problem with impulsive effects

Keyu Zhang12*, Jiafa Xu1 and Wei Dong3

Author Affiliations

1 School of Mathematics, Shandong University, Jinan, Shandong, 250100, China

2 Department of Mathematics, Qilu Normal University, Jinan, Shandong, 250013, China

3 Department of Mathematics, Hebei University of Engineering, Handan, Hebei, 056038, China

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Boundary Value Problems 2013, 2013:120  doi:10.1186/1687-2770-2013-120


The electronic version of this article is the complete one and can be found online at: http://www.boundaryvalueproblems.com/content/2013/1/120


Received:31 January 2013
Accepted:15 April 2013
Published:10 May 2013

© 2013 Zhang et al.; licensee Springer

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

Abstract

This paper is devoted to study the existence and multiplicity of positive solutions for the fourth-order p-Laplacian boundary value problem involving impulsive effects

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/120/mathml/M1','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/120/mathml/M1">View MathML</a>

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/120/mathml/M2','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/120/mathml/M2">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/120/mathml/M3','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/120/mathml/M3">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/120/mathml/M4','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/120/mathml/M4">View MathML</a> (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/120/mathml/M5','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/120/mathml/M5">View MathML</a>). Based on a priori estimates achieved by utilizing the properties of concave functions and Jensen’s inequality, we adopt fixed point index theory to establish our main results.

MSC: 34B18, 47H07, 47H11, 45M20, 26D15.

Keywords:
p-Laplacian boundary value problem with impulsive effects; positive solution; fixed point index; concave function; Jensen inequality

1 Introduction

In this paper, we mainly investigate the existence and multiplicity of positive solutions for the fourth-order p-Laplacian boundary value problem with impulsive effects

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/120/mathml/M6','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/120/mathml/M6">View MathML</a>

(1.1)

Here <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/120/mathml/M2','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/120/mathml/M2">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/120/mathml/M3','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/120/mathml/M3">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/120/mathml/M4','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/120/mathml/M4">View MathML</a>. Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/120/mathml/M10','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/120/mathml/M10">View MathML</a> be fixed, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/120/mathml/M11','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/120/mathml/M11">View MathML</a>, where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/120/mathml/M12','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/120/mathml/M12">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/120/mathml/M13','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/120/mathml/M13">View MathML</a> denote the right and left limit of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/120/mathml/M14','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/120/mathml/M14">View MathML</a> at <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/120/mathml/M15','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/120/mathml/M15">View MathML</a>, respectively.

Fourth-order boundary value problems, including those with the p-Laplacian operator, have their origin in beam theory [1,2], ice formation [3,4], fluids on lungs [5], brain warping [6,7], designing special curves on surfaces [6,8], etc. In beam theory, more specifically, a beam with a small deformation, a beam of a material which satisfies a nonlinear power-like stress and strain law, and a beam with two-sided links which satisfies a nonlinear power-like elasticity law can be described by fourth-order differential equations along with their boundary value conditions. For the case of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/120/mathml/M16','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/120/mathml/M16">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/120/mathml/M17','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/120/mathml/M17">View MathML</a>, and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/120/mathml/M18','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/120/mathml/M18">View MathML</a>, problem (1.1) reduces to the differential equation <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/120/mathml/M19','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/120/mathml/M19">View MathML</a> subject to boundary value conditions <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/120/mathml/M20','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/120/mathml/M20">View MathML</a>, which can be used to model the deflection of elastic beams simply supported at the endpoints [9-11]. This explains the reason that the last two decades have witnessed an overgrowing interest in the research of such problems, with many papers in this direction published. We refer the interested reader to [12-26] and references therein devoted to the existence of solutions for the equations with p-Laplacian operator.

In [17], Zhang et al. studied the existence and nonexistence of symmetric positive solutions of the following fourth-order boundary value problem with integral boundary conditions:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/120/mathml/M21','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/120/mathml/M21">View MathML</a>

(1.2)

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/120/mathml/M22','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/120/mathml/M22">View MathML</a> is nonnegative, symmetric on the interval <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/120/mathml/M23','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/120/mathml/M23">View MathML</a> (i.e., <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/120/mathml/M24','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/120/mathml/M24">View MathML</a> for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/120/mathml/M25','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/120/mathml/M25">View MathML</a>), <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/120/mathml/M26','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/120/mathml/M26">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/120/mathml/M27','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/120/mathml/M27">View MathML</a> for all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/120/mathml/M28','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/120/mathml/M28">View MathML</a>, and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/120/mathml/M29','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/120/mathml/M29">View MathML</a> are nonnegative, symmetric on <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/120/mathml/M30','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/120/mathml/M30">View MathML</a>. The arguments are based upon a specially constructed cone and the fixed point theory for cones. Moreover, they also studied the nonexistence of a positive solution.

In [16], Luo and Luo considered the existence, multiplicity, and nonexistence of symmetric positive solutions for (1.2) with a ϕ-Laplacian operator and the term f involving the first derivative.

Except that, many researchers considered and studied the existence of positive solutions for a lot of impulsive boundary value problems; see, for example, [21-29] and the references therein.

In [21], Feng considered the problem (1.2) with impulsive effects and he obtained the existence and multiplicity of positive solutions. The fundamental tool in this paper is Guo-Krasnosel’skii fixed point theorem on a cone. Moreover, the nonlinearity f can be allowed to grow both sublinear and superlinear. Therefore, he improved and generalized the results of [17] to some degree. However, we can easily find that these papers do only simple promotion based on their original papers, and no substantial changes.

Motivated by the works mentioned above, in this paper, we study the existence and multiplicity of positive solutions for (1.1). Nevertheless, our methodology and results in this paper are different from those in the papers cited above. The main features of this paper are as follows. Firstly, we convert the boundary value problem (1.1) into an equivalent integral equation. Next, we consider impulsive effect as a perturbation to the corresponding problem without the impulsive terms, so that we can construct an integral operator for an appropriate linear Dirichlet boundary value problem and obtain its first eigenvalue and eigenfunction. Our main results are formulated in terms of spectral radii of the linear integral operator, and our a priori estimates for positive solutions are derived by developing some properties of positive concave functions and using Jensen’s inequality. It is of interest to note that our nonlinearity f may grow superlinearly and sublinearly. The main tool used in the proofs is fixed point index theory, combined with the a priori estimates of positive solutions. Although our problem (1.1) merely involves Dirichlet boundary conditions, both our methodology and the results in this work improve and extend the corresponding ones from [21-29].

2 Preliminaries

Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/120/mathml/M31','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/120/mathml/M31">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/120/mathml/M32','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/120/mathml/M32">View MathML</a>. Then <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/120/mathml/M33','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/120/mathml/M33">View MathML</a> is a real Banach space. Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/120/mathml/M34','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/120/mathml/M34">View MathML</a> and introduce the following space:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/120/mathml/M35','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/120/mathml/M35">View MathML</a>

with the norm <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/120/mathml/M36','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/120/mathml/M36">View MathML</a>. Then <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/120/mathml/M37','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/120/mathml/M37">View MathML</a> is also a Banach space.

A function <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/120/mathml/M38','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/120/mathml/M38">View MathML</a> is called a solution of (1.1) if it satisfies the differential equation

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/120/mathml/M39','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/120/mathml/M39">View MathML</a>

and the function y satisfies the conditions <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/120/mathml/M40','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/120/mathml/M40">View MathML</a>, and the Dirichlet boundary conditions <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/120/mathml/M41','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/120/mathml/M41">View MathML</a>.

Lemma 2.1 (see [21])

Ifyis a solution of the integral equation

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/120/mathml/M42','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/120/mathml/M42">View MathML</a>

(2.1)

thenyis a solution of (1.1), where<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/120/mathml/M43','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/120/mathml/M43">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/120/mathml/M44','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/120/mathml/M44">View MathML</a>. Note that if<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/120/mathml/M3','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/120/mathml/M3">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/120/mathml/M46','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/120/mathml/M46">View MathML</a>, then<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/120/mathml/M47','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/120/mathml/M47">View MathML</a>is a completely continuous operator, and the existence of positive solutions for (1.1) is equivalent to that of positive fixed points ofA.

Remark 2.1 By (2.1), we easily find y is concave on <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/120/mathml/M30','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/120/mathml/M30">View MathML</a>. Indeed,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/120/mathml/M49','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/120/mathml/M49">View MathML</a>

implies y is concave on <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/120/mathml/M30','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/120/mathml/M30">View MathML</a>. Furthermore, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/120/mathml/M51','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/120/mathml/M51">View MathML</a> (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/120/mathml/M17','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/120/mathml/M17">View MathML</a>) leads to <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/120/mathml/M53','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/120/mathml/M53">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/120/mathml/M54','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/120/mathml/M54">View MathML</a>.

Let P be a cone in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/120/mathml/M55','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/120/mathml/M55">View MathML</a> which is defined as

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/120/mathml/M56','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/120/mathml/M56">View MathML</a>

In what follows, we prove that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/120/mathml/M57','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/120/mathml/M57">View MathML</a>.

Lemma 2.2<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/120/mathml/M57','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/120/mathml/M57">View MathML</a>.

Proof We easily see that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/120/mathml/M59','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/120/mathml/M59">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/120/mathml/M44','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/120/mathml/M44">View MathML</a>. Consequently, on the one hand, we find

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/120/mathml/M61','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/120/mathml/M61">View MathML</a>

On the other hand,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/120/mathml/M62','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/120/mathml/M62">View MathML</a>

Therefore, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/120/mathml/M63','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/120/mathml/M63">View MathML</a>, for any <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/120/mathml/M25','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/120/mathml/M25">View MathML</a>, as required. This completes the proof. □

We denote <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/120/mathml/M65','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/120/mathml/M65">View MathML</a> for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/120/mathml/M66','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/120/mathml/M66">View MathML</a> in the sequel.

Lemma 2.3 (see [30])

Suppose<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/120/mathml/M67','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/120/mathml/M67">View MathML</a>is a completely continuous operator and has no fixed points on<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/120/mathml/M68','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/120/mathml/M68">View MathML</a>.

1. If<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/120/mathml/M69','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/120/mathml/M69">View MathML</a>for all<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/120/mathml/M70','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/120/mathml/M70">View MathML</a>, then<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/120/mathml/M71','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/120/mathml/M71">View MathML</a>, whereiis fixed point index onP.

2. If<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/120/mathml/M72','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/120/mathml/M72">View MathML</a>for all<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/120/mathml/M70','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/120/mathml/M70">View MathML</a>, then<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/120/mathml/M74','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/120/mathml/M74">View MathML</a>.

Lemma 2.4 (see [30])

If<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/120/mathml/M75','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/120/mathml/M75">View MathML</a>is a completely continuous operator. If there exists<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/120/mathml/M76','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/120/mathml/M76">View MathML</a>such that<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/120/mathml/M77','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/120/mathml/M77">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/120/mathml/M78','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/120/mathml/M78">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/120/mathml/M79','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/120/mathml/M79">View MathML</a>, then<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/120/mathml/M80','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/120/mathml/M80">View MathML</a>.

Lemma 2.5 (see [30])

If<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/120/mathml/M81','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/120/mathml/M81">View MathML</a>and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/120/mathml/M75','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/120/mathml/M75">View MathML</a>is a completely continuous operator. If<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/120/mathml/M83','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/120/mathml/M83">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/120/mathml/M84','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/120/mathml/M84">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/120/mathml/M85','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/120/mathml/M85">View MathML</a>, then<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/120/mathml/M86','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/120/mathml/M86">View MathML</a>.

Lemma 2.6Let<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/120/mathml/M87','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/120/mathml/M87">View MathML</a>. Then

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/120/mathml/M88','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/120/mathml/M88">View MathML</a>

(2.2)

Lemma 2.7 (Jensen’s inequalities)

Let<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/120/mathml/M89','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/120/mathml/M89">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/120/mathml/M90','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/120/mathml/M90">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/120/mathml/M91','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/120/mathml/M91">View MathML</a> (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/120/mathml/M92','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/120/mathml/M92">View MathML</a>), and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/120/mathml/M93','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/120/mathml/M93">View MathML</a>. Then

3 Main results

Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/120/mathml/M95','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/120/mathml/M95">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/120/mathml/M96','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/120/mathml/M96">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/120/mathml/M97','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/120/mathml/M97">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/120/mathml/M98','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/120/mathml/M98">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/120/mathml/M99','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/120/mathml/M99">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/120/mathml/M100','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/120/mathml/M100">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/120/mathml/M101','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/120/mathml/M101">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/120/mathml/M102','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/120/mathml/M102">View MathML</a>. We now list our hypotheses.

(H1) There is a <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/120/mathml/M103','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/120/mathml/M103">View MathML</a> such that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/120/mathml/M104','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/120/mathml/M104">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/120/mathml/M105','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/120/mathml/M105">View MathML</a> imply

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/120/mathml/M106','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/120/mathml/M106">View MathML</a>

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/120/mathml/M107','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/120/mathml/M107">View MathML</a> satisfy

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/120/mathml/M108','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/120/mathml/M108">View MathML</a>

(H2) There exist <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/120/mathml/M109','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/120/mathml/M109">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/120/mathml/M110','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/120/mathml/M110">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/120/mathml/M111','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/120/mathml/M111">View MathML</a> satisfying

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/120/mathml/M112','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/120/mathml/M112">View MathML</a>

such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/120/mathml/M113','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/120/mathml/M113">View MathML</a>

(3.1)

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/120/mathml/M114','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/120/mathml/M114">View MathML</a>.

(H3) There exist <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/120/mathml/M115','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/120/mathml/M115">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/120/mathml/M116','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/120/mathml/M116">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/120/mathml/M117','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/120/mathml/M117">View MathML</a> satisfying

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/120/mathml/M118','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/120/mathml/M118">View MathML</a>

such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/120/mathml/M119','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/120/mathml/M119">View MathML</a>

(3.2)

(H4) There is a <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/120/mathml/M103','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/120/mathml/M103">View MathML</a> such that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/120/mathml/M121','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/120/mathml/M121">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/120/mathml/M105','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/120/mathml/M105">View MathML</a> imply

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/120/mathml/M123','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/120/mathml/M123">View MathML</a>

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/120/mathml/M124','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/120/mathml/M124">View MathML</a> satisfy

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/120/mathml/M125','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/120/mathml/M125">View MathML</a>

(H5) There exist <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/120/mathml/M109','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/120/mathml/M109">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/120/mathml/M127','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/120/mathml/M127">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/120/mathml/M128','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/120/mathml/M128">View MathML</a> satisfying

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/120/mathml/M129','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/120/mathml/M129">View MathML</a>

such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/120/mathml/M130','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/120/mathml/M130">View MathML</a>

(3.3)

(H6) There exist <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/120/mathml/M115','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/120/mathml/M115">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/120/mathml/M132','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/120/mathml/M132">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/120/mathml/M133','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/120/mathml/M133">View MathML</a> satisfying

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/120/mathml/M134','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/120/mathml/M134">View MathML</a>

such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/120/mathml/M135','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/120/mathml/M135">View MathML</a>

(3.4)

Theorem 3.1Suppose that (H1)-(H3) are satisfied. Then (1.1) has at least two positive solutions.

Proof If <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/120/mathml/M70','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/120/mathml/M70">View MathML</a>, it follows from (H1) that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/120/mathml/M137','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/120/mathml/M137">View MathML</a>

Now Lemma 2.3 yields

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/120/mathml/M138','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/120/mathml/M138">View MathML</a>

(3.5)

Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/120/mathml/M139','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/120/mathml/M139">View MathML</a>. Then for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/120/mathml/M140','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/120/mathml/M140">View MathML</a>, we find

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/120/mathml/M141','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/120/mathml/M141">View MathML</a>

(3.6)

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/120/mathml/M142','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/120/mathml/M142">View MathML</a>. Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/120/mathml/M143','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/120/mathml/M143">View MathML</a>, where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/120/mathml/M144','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/120/mathml/M144">View MathML</a>. Next, from (H2), we prove <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/120/mathml/M145','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/120/mathml/M145">View MathML</a>. Indeed, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/120/mathml/M146','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/120/mathml/M146">View MathML</a> implies <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/120/mathml/M147','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/120/mathml/M147">View MathML</a>. Lemma 2.6, together with this, leads to

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/120/mathml/M148','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/120/mathml/M148">View MathML</a>

(3.7)

Multiply both sides of the above by <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/120/mathml/M149','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/120/mathml/M149">View MathML</a> and integrate over <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/120/mathml/M30','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/120/mathml/M30">View MathML</a> and use (2.2) to obtain

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/120/mathml/M151','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/120/mathml/M151">View MathML</a>

(3.8)

Combining this and (3.1), we get

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/120/mathml/M152','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/120/mathml/M152">View MathML</a>

(3.9)

In what follows, we will distinguish three cases.

Case 1. <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/120/mathml/M153','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/120/mathml/M153">View MathML</a>. By (H2), we know <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/120/mathml/M154','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/120/mathml/M154">View MathML</a>. (3.9) implies

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/120/mathml/M155','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/120/mathml/M155">View MathML</a>

Therefore, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/120/mathml/M51','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/120/mathml/M51">View MathML</a> (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/120/mathml/M17','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/120/mathml/M17">View MathML</a>), and then <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/120/mathml/M53','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/120/mathml/M53">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/120/mathml/M54','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/120/mathml/M54">View MathML</a> by Remark 2.1, which contradicts <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/120/mathml/M140','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/120/mathml/M140">View MathML</a>.

Case 2. <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/120/mathml/M161','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/120/mathml/M161">View MathML</a>. Equation (3.9) implies

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/120/mathml/M162','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/120/mathml/M162">View MathML</a>

and thus <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/120/mathml/M163','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/120/mathml/M163">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/120/mathml/M54','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/120/mathml/M54">View MathML</a>, which also contradicts <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/120/mathml/M140','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/120/mathml/M140">View MathML</a>.

Case 3. <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/120/mathml/M166','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/120/mathml/M166">View MathML</a>. Since <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/120/mathml/M167','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/120/mathml/M167">View MathML</a>, we have by (3.6) and (3.9),

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/120/mathml/M168','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/120/mathml/M168">View MathML</a>

Therefore,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/120/mathml/M169','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/120/mathml/M169">View MathML</a>

which contradicts (H2). So, we have <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/120/mathml/M170','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/120/mathml/M170">View MathML</a> for all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/120/mathml/M171','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/120/mathml/M171">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/120/mathml/M172','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/120/mathml/M172">View MathML</a>. Now, by virtue of Lemma 2.4, we obtain

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/120/mathml/M173','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/120/mathml/M173">View MathML</a>

(3.10)

On the other hand, by (H3), we prove <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/120/mathml/M174','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/120/mathml/M174">View MathML</a> is bounded in P. By (3.2) together with (3.8), we obtain

(3.11)

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/120/mathml/M176','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/120/mathml/M176">View MathML</a>. Now we distinguish the following two cases.

Case 1. <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/120/mathml/M177','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/120/mathml/M177">View MathML</a>. (H3) implies

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/120/mathml/M178','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/120/mathml/M178">View MathML</a>

Combining this and (3.11), we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/120/mathml/M179','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/120/mathml/M179">View MathML</a>

Therefore,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/120/mathml/M180','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/120/mathml/M180">View MathML</a>

Case 2. <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/120/mathml/M181','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/120/mathml/M181">View MathML</a>. (3.11) implies

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/120/mathml/M182','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/120/mathml/M182">View MathML</a>

and thus

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/120/mathml/M183','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/120/mathml/M183">View MathML</a>

Therefore, we obtain the boundedness of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/120/mathml/M174','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/120/mathml/M174">View MathML</a>, as claimed. Taking <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/120/mathml/M185','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/120/mathml/M185">View MathML</a>, we have <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/120/mathml/M170','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/120/mathml/M170">View MathML</a> for all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/120/mathml/M187','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/120/mathml/M187">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/120/mathml/M172','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/120/mathml/M172">View MathML</a>. Now, by virtue of Lemma 2.4, we obtain

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/120/mathml/M189','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/120/mathml/M189">View MathML</a>

(3.12)

Combining (3.5), (3.10), and (3.12), we arrive at

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/120/mathml/M190','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/120/mathml/M190">View MathML</a>

Now A has at least two fixed points, one on <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/120/mathml/M191','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/120/mathml/M191">View MathML</a> and the other on <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/120/mathml/M192','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/120/mathml/M192">View MathML</a>. Hence (1.1) has at least two positive solutions. The proof is completed. □

Theorem 3.2Suppose that (H4)-(H6) are satisfied. Then (1.1) has at least two positive solutions.

Proof If <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/120/mathml/M70','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/120/mathml/M70">View MathML</a>, then we find

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/120/mathml/M194','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/120/mathml/M194">View MathML</a>

(3.13)

By (H4),

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/120/mathml/M195','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/120/mathml/M195">View MathML</a>

so that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/120/mathml/M196','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/120/mathml/M196">View MathML</a>

Now Lemma 2.3 yields

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/120/mathml/M197','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/120/mathml/M197">View MathML</a>

(3.14)

Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/120/mathml/M139','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/120/mathml/M139">View MathML</a>. Then for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/120/mathml/M140','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/120/mathml/M140">View MathML</a>, we find

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/120/mathml/M200','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/120/mathml/M200">View MathML</a>

(3.15)

Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/120/mathml/M201','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/120/mathml/M201">View MathML</a>. Next, from (H5), we prove <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/120/mathml/M202','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/120/mathml/M202">View MathML</a>. Indeed, if <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/120/mathml/M203','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/120/mathml/M203">View MathML</a>, we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/120/mathml/M204','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/120/mathml/M204">View MathML</a>

Multiply both sides of the above by <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/120/mathml/M149','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/120/mathml/M149">View MathML</a> and integrate over <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/120/mathml/M30','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/120/mathml/M30">View MathML</a> and use (2.2) to obtain

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/120/mathml/M207','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/120/mathml/M207">View MathML</a>

(3.16)

Combining this and (3.3), we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/120/mathml/M208','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/120/mathml/M208">View MathML</a>

Consequently,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/120/mathml/M209','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/120/mathml/M209">View MathML</a>

which contradicts (H5). This implies <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/120/mathml/M210','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/120/mathml/M210">View MathML</a>, and thus <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/120/mathml/M83','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/120/mathml/M83">View MathML</a> for all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/120/mathml/M140','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/120/mathml/M140">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/120/mathml/M213','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/120/mathml/M213">View MathML</a>. Now Lemma 2.5 yields

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/120/mathml/M214','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/120/mathml/M214">View MathML</a>

(3.17)

On the other hand, by (H6), we prove <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/120/mathml/M215','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/120/mathml/M215">View MathML</a> is bounded in P. By (3.4) together with (3.16), we obtain

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/120/mathml/M216','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/120/mathml/M216">View MathML</a>

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/120/mathml/M217','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/120/mathml/M217">View MathML</a>. Therefore,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/120/mathml/M218','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/120/mathml/M218">View MathML</a>

namely,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/120/mathml/M219','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/120/mathml/M219">View MathML</a>

This proves the boundedness of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/120/mathml/M215','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/120/mathml/M215">View MathML</a>, as required. Choosing <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/120/mathml/M221','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/120/mathml/M221">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/120/mathml/M222','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/120/mathml/M222">View MathML</a>, we have <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/120/mathml/M83','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/120/mathml/M83">View MathML</a> for all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/120/mathml/M187','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/120/mathml/M187">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/120/mathml/M225','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/120/mathml/M225">View MathML</a>. Now Lemma 2.5 yields

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/120/mathml/M226','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/120/mathml/M226">View MathML</a>

(3.18)

Combining (3.14), (3.17), and (3.18), we obtain

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/120/mathml/M227','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/120/mathml/M227">View MathML</a>

Hence A has at least two fixed points, one on <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/120/mathml/M228','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/120/mathml/M228">View MathML</a> and the other on <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/120/mathml/M229','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/120/mathml/M229">View MathML</a>, and thus (1.1) has at least two positive solutions. The proof is completed. □

4 An example

Let us consider the problem

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/120/mathml/M230','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/120/mathml/M230">View MathML</a>

(4.1)

Taking <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/120/mathml/M231','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/120/mathml/M231">View MathML</a> in (H1), <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/120/mathml/M232','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/120/mathml/M232">View MathML</a>, and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/120/mathml/M233','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/120/mathml/M233">View MathML</a> is chosen such that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/120/mathml/M234','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/120/mathml/M234">View MathML</a>. Set <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/120/mathml/M235','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/120/mathml/M235">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/120/mathml/M236','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/120/mathml/M236">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/120/mathml/M237','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/120/mathml/M237">View MathML</a>. Therefore, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/120/mathml/M238','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/120/mathml/M238">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/120/mathml/M239','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/120/mathml/M239">View MathML</a>, and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/120/mathml/M240','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/120/mathml/M240">View MathML</a>

As a result, (H1) holds. On the other hand, by simple computation, we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/120/mathml/M241','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/120/mathml/M241">View MathML</a>

Therefore,

(i) There exist <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/120/mathml/M109','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/120/mathml/M109">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/120/mathml/M243','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/120/mathml/M243">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/120/mathml/M244','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/120/mathml/M244">View MathML</a> such that (H2) holds.

(ii) There exist <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/120/mathml/M115','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/120/mathml/M115">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/120/mathml/M246','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/120/mathml/M246">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/120/mathml/M247','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/120/mathml/M247">View MathML</a> such that (H3) holds.

Consequently, the problem (4.1) has at least two positive solutions by Theorem 3.1.

Competing interests

The authors declare that they have no competing interests.

Authors’ contributions

KZ and JX obtained the results in a joint research. All the authors read and approved the final manuscript.

Acknowledgements

Research supported by the NNSF-China (10971046), Shandong and Hebei Provincial Natural Science Foundation (ZR2012AQ007, A2012402036), GIIFSDU (yzc12063), IIFSDU (2012TS020) and the Project of Shandong Province Higher Educational Science and Technology Program (J09LA55).

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