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This article is part of the series A Tribute to Professor Ivan Kiguradze.

Open Access Research

Multiple monotone positive solutions for higher order differential equations with integral boundary conditions

Xinan Hao* and Lishan Liu

Author Affiliations

School of Mathematical Sciences, Qufu Normal University, Qufu, Shandong, 273165, P.R. China

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Boundary Value Problems 2014, 2014:74  doi:10.1186/1687-2770-2014-74

The electronic version of this article is the complete one and can be found online at: http://www.boundaryvalueproblems.com/content/2014/1/74


Received:17 January 2014
Accepted:19 March 2014
Published:28 March 2014

© 2014 Hao and Liu; 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

This paper investigates the higher order differential equations with nonlocal boundary conditions

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/74/mathml/M1','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/74/mathml/M1">View MathML</a>

The existence results of multiple monotone positive solutions are obtained by means of fixed point index theory for operators in a cone.

MSC: 34B10, 34B18.

Keywords:
monotone positive solutions; multiplicity; higher order differential equations; nonlocal boundary conditions

1 Introduction

In this paper, we are concerned with the existence of multiple monotone positive solutions for the higher order differential equation

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/74/mathml/M2','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/74/mathml/M2">View MathML</a>

(1.1)

subject to the following integral boundary conditions:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/74/mathml/M3','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/74/mathml/M3">View MathML</a>

(1.2)

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/74/mathml/M4','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/74/mathml/M4">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/74/mathml/M5','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/74/mathml/M5">View MathML</a> is continuous in which <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/74/mathml/M6','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/74/mathml/M6">View MathML</a>, A and B are right continuous on <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/74/mathml/M7','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/74/mathml/M7">View MathML</a>, left continuous at <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/74/mathml/M8','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/74/mathml/M8">View MathML</a>, and nondecreasing on <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/74/mathml/M9','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/74/mathml/M9">View MathML</a>, with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/74/mathml/M10','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/74/mathml/M10">View MathML</a>; <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/74/mathml/M11','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/74/mathml/M11">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/74/mathml/M12','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/74/mathml/M12">View MathML</a> denote the Riemann-Stieltjes integrals of v with respect to A and B, respectively.

Boundary value problems (BVPs for short) for nonlinear differential equations arise in many areas of applied mathematics and physics. Many authors have discussed the existence of positive solutions for second order or higher order differential equations with boundary conditions defined at a finite number of points, for instance, [1-12]. In [2], Graef and Yang considered the following nth-order multi-point BVP:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/74/mathml/M13','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/74/mathml/M13">View MathML</a>

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/74/mathml/M4','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/74/mathml/M4">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/74/mathml/M15','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/74/mathml/M15">View MathML</a> is a parameter, g and f are continuous functions, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/74/mathml/M16','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/74/mathml/M16">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/74/mathml/M17','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/74/mathml/M17">View MathML</a> for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/74/mathml/M18','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/74/mathml/M18">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/74/mathml/M19','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/74/mathml/M19">View MathML</a>. The authors obtained the existence and nonexistence results of positive solutions by using Krasnosel’skii’s fixed point theorem in cones. In [5], we studied the following second order m-point nonhomogeneous BVP:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/74/mathml/M20','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/74/mathml/M20">View MathML</a>

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/74/mathml/M21','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/74/mathml/M21">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/74/mathml/M22','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/74/mathml/M22">View MathML</a> (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/74/mathml/M23','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/74/mathml/M23">View MathML</a>), <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/74/mathml/M24','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/74/mathml/M24">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/74/mathml/M25','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/74/mathml/M25">View MathML</a>. The authors obtained the existence, nonexistence and multiplicity of positive solutions by using the Krasnosel’skii-Guo fixed point theorem, the upper-lower solutions method and topological degree theory.

Boundary value problems with integral boundary conditions for ordinary differential equations represent a very interesting and important class of problems and arise in the study of various physical, biological and chemical processes [13-15], such as heat conduction, chemical engineering, underground water flow, thermo-elasticity, and plasma physics. They include two, three, multi-point and nonlocal BVPs as special cases. The existence and multiplicity of positive solutions for such problems have received a great deal of attention, see [16-32] and the references therein. In [17], Feng, Ji and Ge considered the existence and multiplicity of positive solutions for a class of nonlinear boundary value problems of second order differential equations with integral boundary conditions in ordered Banach spaces

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/74/mathml/M26','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/74/mathml/M26">View MathML</a>

The arguments are based upon a specially constructed cone and fixed point theory in a cone for strict set contraction operators.

Motivated by the works mentioned above, in this paper, we consider the existence of multiple monotone positive solutions for BVP (1.1) and (1.2). In comparison with previous works, our paper has several new features. Firstly, we consider higher order boundary value problems, and we allow the nonlinearity f to contain derivatives of the unknown function <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/74/mathml/M27','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/74/mathml/M27">View MathML</a> up to <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/74/mathml/M28','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/74/mathml/M28">View MathML</a> order. Secondly, we discuss the boundary value problem with integral boundary conditions, i.e., BVP (1.1) and (1.2), which includes two-point, three-point, multi-point and nonlocal boundary value problems as special cases. Thirdly, we consider the existence of multiple monotone positive solutions. To our knowledge, few papers have considered the monotone positive solutions for a higher order differential equation with integral boundary conditions. We shall emphasize here that with these new features our work improves and generalizes the results of [2] and some other known results to some degree. In this work we shall also utilize the following fixed point theorem in cones.

Lemma 1.1 ([33,34])

LetKbe a cone in a Banach spaceE. LetDbe an open bounded subset ofEwith<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/74/mathml/M29','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/74/mathml/M29">View MathML</a>and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/74/mathml/M30','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/74/mathml/M30">View MathML</a>. Assume that<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/74/mathml/M31','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/74/mathml/M31">View MathML</a>is a compact operator such that<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/74/mathml/M32','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/74/mathml/M32">View MathML</a>for<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/74/mathml/M33','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/74/mathml/M33">View MathML</a>. Then the following results hold.

(1) If<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/74/mathml/M34','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/74/mathml/M34">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/74/mathml/M33','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/74/mathml/M33">View MathML</a>, then<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/74/mathml/M36','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/74/mathml/M36">View MathML</a>.

(2) If there exists<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/74/mathml/M37','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/74/mathml/M37">View MathML</a>such that<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/74/mathml/M38','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/74/mathml/M38">View MathML</a>for all<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/74/mathml/M33','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/74/mathml/M33">View MathML</a>and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/74/mathml/M15','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/74/mathml/M15">View MathML</a>, then<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/74/mathml/M41','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/74/mathml/M41">View MathML</a>.

(3) LetUbe open inEsuch that<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/74/mathml/M42','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/74/mathml/M42">View MathML</a>. If<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/74/mathml/M36','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/74/mathml/M36">View MathML</a>and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/74/mathml/M44','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/74/mathml/M44">View MathML</a>, thenAhas a fixed point in<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/74/mathml/M45','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/74/mathml/M45">View MathML</a>. The same result holds if<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/74/mathml/M41','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/74/mathml/M41">View MathML</a>and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/74/mathml/M47','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/74/mathml/M47">View MathML</a>.

2 Preliminary lemmas

Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/74/mathml/M48','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/74/mathml/M48">View MathML</a>, then E is a Banach space with the norm <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/74/mathml/M49','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/74/mathml/M49">View MathML</a> for each <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/74/mathml/M50','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/74/mathml/M50">View MathML</a>.

We make the following assumptions:

(<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/74/mathml/M51','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/74/mathml/M51">View MathML</a>) <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/74/mathml/M52','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/74/mathml/M52">View MathML</a> is continuous.

(<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/74/mathml/M53','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/74/mathml/M53">View MathML</a>) <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/74/mathml/M54','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/74/mathml/M54">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/74/mathml/M55','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/74/mathml/M55">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/74/mathml/M56','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/74/mathml/M56">View MathML</a>, where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/74/mathml/M57','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/74/mathml/M57">View MathML</a>,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/74/mathml/M58','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/74/mathml/M58">View MathML</a>

Lemma 2.1Assume that (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/74/mathml/M53','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/74/mathml/M53">View MathML</a>) holds. Then, for any<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/74/mathml/M60','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/74/mathml/M60">View MathML</a>, the BVP

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/74/mathml/M61','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/74/mathml/M61">View MathML</a>

(2.1)

has a unique solutionuthat can be expressed in the form

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/74/mathml/M62','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/74/mathml/M62">View MathML</a>

(2.2)

where

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/74/mathml/M63','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/74/mathml/M63">View MathML</a>

(2.3)

Proof Firstly, we prove that if u is a solution of BVP (2.1), then it will take the form of (2.2). Now, integrating differential equation (2.1) from 0 to t twice, we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/74/mathml/M64','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/74/mathml/M64">View MathML</a>

(2.4)

Letting <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/74/mathml/M8','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/74/mathml/M8">View MathML</a> in (2.4), we get

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/74/mathml/M66','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/74/mathml/M66">View MathML</a>

(2.5)

Substituting the boundary conditions of (2.1) and (2.5) into (2.4) yields

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/74/mathml/M67','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/74/mathml/M67">View MathML</a>

(2.6)

and, consequently,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/74/mathml/M68','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/74/mathml/M68">View MathML</a>

Solving the above two equations for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/74/mathml/M69','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/74/mathml/M69">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/74/mathml/M70','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/74/mathml/M70">View MathML</a>, we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/74/mathml/M71','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/74/mathml/M71">View MathML</a>

and so

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/74/mathml/M72','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/74/mathml/M72">View MathML</a>

(2.7)

Hence, (2.2) follows from (2.6) and (2.7).

Next we prove that the u given by (2.2) satisfies the differential equation and boundary conditions of (2.1). From (2.2), we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/74/mathml/M73','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/74/mathml/M73">View MathML</a>

(2.8)

Direct differentiation of (2.8) gives <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/74/mathml/M74','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/74/mathml/M74">View MathML</a>. Also, from (2.2) we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/74/mathml/M75','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/74/mathml/M75">View MathML</a>

and, similarly,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/74/mathml/M76','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/74/mathml/M76">View MathML</a>

Therefore, by solving the above two equations with the double integrals as unknowns, we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/74/mathml/M77','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/74/mathml/M77">View MathML</a>

(2.9)

and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/74/mathml/M78','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/74/mathml/M78">View MathML</a>

(2.10)

Hence (2.6) follows from (2.2), (2.9) and (2.10), and thus <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/74/mathml/M79','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/74/mathml/M79">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/74/mathml/M80','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/74/mathml/M80">View MathML</a>. This completes the proof. □

Defining

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/74/mathml/M81','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/74/mathml/M81">View MathML</a>

then <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/74/mathml/M82','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/74/mathml/M82">View MathML</a> is the Green function of BVP (1.1) and (1.2). Moreover, solving BVP (1.1) and (1.2) is equivalent to finding a solution of the following integral equation:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/74/mathml/M83','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/74/mathml/M83">View MathML</a>

Remark 2.1 If (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/74/mathml/M53','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/74/mathml/M53">View MathML</a>) holds, then for any <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/74/mathml/M85','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/74/mathml/M85">View MathML</a>, it is easy to testify that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/74/mathml/M86','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/74/mathml/M86">View MathML</a>

(2.11)

Lemma 2.2Let<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/74/mathml/M87','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/74/mathml/M87">View MathML</a>, then for any<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/74/mathml/M88','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/74/mathml/M88">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/74/mathml/M89','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/74/mathml/M89">View MathML</a>, we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/74/mathml/M90','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/74/mathml/M90">View MathML</a>

Proof It is easy to show that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/74/mathml/M91','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/74/mathml/M91">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/74/mathml/M92','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/74/mathml/M92">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/74/mathml/M93','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/74/mathml/M93">View MathML</a>. For <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/74/mathml/M88','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/74/mathml/M88">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/74/mathml/M89','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/74/mathml/M89">View MathML</a>, we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/74/mathml/M96','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/74/mathml/M96">View MathML</a>

For any <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/74/mathml/M97','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/74/mathml/M97">View MathML</a>, we define <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/74/mathml/M98','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/74/mathml/M98">View MathML</a>. From Lemma 2.2, we know that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/74/mathml/M99','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/74/mathml/M99">View MathML</a>

(2.12)

 □

Lemma 2.3Assume that (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/74/mathml/M53','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/74/mathml/M53">View MathML</a>) holds. If<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/74/mathml/M101','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/74/mathml/M101">View MathML</a>satisfies the boundary conditions (1.2) and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/74/mathml/M102','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/74/mathml/M102">View MathML</a>

then

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/74/mathml/M103','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/74/mathml/M103">View MathML</a>

(2.13)

Proof Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/74/mathml/M104','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/74/mathml/M104">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/74/mathml/M105','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/74/mathml/M105">View MathML</a>, then we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/74/mathml/M106','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/74/mathml/M106">View MathML</a>

For <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/74/mathml/M105','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/74/mathml/M105">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/74/mathml/M108','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/74/mathml/M108">View MathML</a> implies that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/74/mathml/M109','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/74/mathml/M109">View MathML</a>

Thus

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/74/mathml/M110','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/74/mathml/M110">View MathML</a>

i.e.,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/74/mathml/M111','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/74/mathml/M111">View MathML</a>

On the other hand,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/74/mathml/M112','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/74/mathml/M112">View MathML</a>

and so

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/74/mathml/M113','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/74/mathml/M113">View MathML</a>

i.e., <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/74/mathml/M114','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/74/mathml/M114">View MathML</a>, therefore <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/74/mathml/M115','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/74/mathml/M115">View MathML</a>, and so <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/74/mathml/M116','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/74/mathml/M116">View MathML</a>.

Now, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/74/mathml/M116','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/74/mathml/M116">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/74/mathml/M118','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/74/mathml/M118">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/74/mathml/M119','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/74/mathml/M119">View MathML</a> is concave downward, so we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/74/mathml/M120','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/74/mathml/M120">View MathML</a>

(2.14)

From (2.14) and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/74/mathml/M121','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/74/mathml/M121">View MathML</a>, we obtain (2.13). This completes the proof of Lemma 2.3. □

Remark 2.2 From Lemma 2.3, if u is a positive solution of BVP (1.1) and (1.2), then u is nondecreasing on <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/74/mathml/M9','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/74/mathml/M9">View MathML</a>, i.e., u is a monotone positive solution of BVP (1.1) and (1.2).

Let

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/74/mathml/M123','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/74/mathml/M123">View MathML</a>

Obviously, K is a cone in E. For any <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/74/mathml/M124','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/74/mathml/M124">View MathML</a>, let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/74/mathml/M125','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/74/mathml/M125">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/74/mathml/M126','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/74/mathml/M126">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/74/mathml/M127','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/74/mathml/M127">View MathML</a>. Define an operator <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/74/mathml/M128','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/74/mathml/M128">View MathML</a> as follows:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/74/mathml/M129','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/74/mathml/M129">View MathML</a>

(2.15)

Then u is a solution of BVP (1.1) and (1.2) if and only if u solves the operator equation <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/74/mathml/M130','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/74/mathml/M130">View MathML</a>.

Lemma 2.4Suppose that (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/74/mathml/M51','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/74/mathml/M51">View MathML</a>) and (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/74/mathml/M53','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/74/mathml/M53">View MathML</a>) hold, then<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/74/mathml/M133','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/74/mathml/M133">View MathML</a>is completely continuous.

Proof For all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/74/mathml/M134','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/74/mathml/M134">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/74/mathml/M105','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/74/mathml/M105">View MathML</a>, by (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/74/mathml/M51','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/74/mathml/M51">View MathML</a>), (2.11), (2.12) and (2.15), we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/74/mathml/M137','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/74/mathml/M137">View MathML</a>

and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/74/mathml/M138','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/74/mathml/M138">View MathML</a>

Thus, further from the first inequality of (2.12), we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/74/mathml/M139','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/74/mathml/M139">View MathML</a>

Hence, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/74/mathml/M140','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/74/mathml/M140">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/74/mathml/M141','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/74/mathml/M141">View MathML</a>.

Next by standard methods and the Ascoli-Arzela theorem, one can prove that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/74/mathml/M133','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/74/mathml/M133">View MathML</a> is completely continuous. So this is omitted. □

Let

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/74/mathml/M143','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/74/mathml/M143">View MathML</a>

Proceeding as for the proof of Lemma 2.5 in [33], we have the following.

Lemma 2.5<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/74/mathml/M144','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/74/mathml/M144">View MathML</a>has the following properties:

(a) <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/74/mathml/M144','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/74/mathml/M144">View MathML</a>is open relative toK;

(b) <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/74/mathml/M146','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/74/mathml/M146">View MathML</a>;

(c) <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/74/mathml/M147','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/74/mathml/M147">View MathML</a>if and only if<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/74/mathml/M148','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/74/mathml/M148">View MathML</a>;

(d) if<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/74/mathml/M147','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/74/mathml/M147">View MathML</a>, then<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/74/mathml/M150','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/74/mathml/M150">View MathML</a>for<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/74/mathml/M151','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/74/mathml/M151">View MathML</a>.

Now for convenience we introduce the following notations:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/74/mathml/M152','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/74/mathml/M152">View MathML</a>

To prove our main results, we need the following lemmas.

Lemma 2.6Assume that (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/74/mathml/M51','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/74/mathml/M51">View MathML</a>), (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/74/mathml/M53','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/74/mathml/M53">View MathML</a>) hold andfsatisfies

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/74/mathml/M155','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/74/mathml/M155">View MathML</a>

(2.16)

then<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/74/mathml/M156','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/74/mathml/M156">View MathML</a>.

Proof For <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/74/mathml/M157','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/74/mathml/M157">View MathML</a>, we have <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/74/mathml/M158','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/74/mathml/M158">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/74/mathml/M159','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/74/mathml/M159">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/74/mathml/M105','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/74/mathml/M105">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/74/mathml/M161','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/74/mathml/M161">View MathML</a>. Then by (2.16) we have, for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/74/mathml/M105','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/74/mathml/M105">View MathML</a>,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/74/mathml/M163','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/74/mathml/M163">View MathML</a>

This implies that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/74/mathml/M164','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/74/mathml/M164">View MathML</a> for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/74/mathml/M157','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/74/mathml/M157">View MathML</a>. By the point (1) in Lemma 1.1, we have <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/74/mathml/M156','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/74/mathml/M156">View MathML</a>. □

Lemma 2.7Assume that (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/74/mathml/M51','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/74/mathml/M51">View MathML</a>), (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/74/mathml/M53','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/74/mathml/M53">View MathML</a>) hold andfsatisfies

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/74/mathml/M169','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/74/mathml/M169">View MathML</a>

(2.17)

then<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/74/mathml/M170','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/74/mathml/M170">View MathML</a>.

Proof Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/74/mathml/M171','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/74/mathml/M171">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/74/mathml/M105','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/74/mathml/M105">View MathML</a>, then <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/74/mathml/M173','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/74/mathml/M173">View MathML</a> with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/74/mathml/M174','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/74/mathml/M174">View MathML</a>. Next we prove that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/74/mathml/M175','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/74/mathml/M175">View MathML</a>

In fact, if not, then there exist <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/74/mathml/M176','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/74/mathml/M176">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/74/mathml/M177','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/74/mathml/M177">View MathML</a> such that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/74/mathml/M178','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/74/mathml/M178">View MathML</a>. By (2.17) and the point (d) in Lemma 2.5, we have, for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/74/mathml/M179','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/74/mathml/M179">View MathML</a>,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/74/mathml/M180','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/74/mathml/M180">View MathML</a>

This implies that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/74/mathml/M181','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/74/mathml/M181">View MathML</a>, and so by the point (c) in Lemma 2.5, this is a contradiction. It follows from the point (2) of Lemma 1.1 that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/74/mathml/M170','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/74/mathml/M170">View MathML</a>. □

3 Main results

In the following, we shall give the main results on the existence of multiple positive solutions of BVP (1.1) and (1.2).

Theorem 3.1Suppose that (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/74/mathml/M51','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/74/mathml/M51">View MathML</a>) and (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/74/mathml/M53','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/74/mathml/M53">View MathML</a>) are satisfied. In addition, assume that one of the following conditions holds.

(<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/74/mathml/M185','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/74/mathml/M185">View MathML</a>) There exist<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/74/mathml/M186','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/74/mathml/M186">View MathML</a>with<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/74/mathml/M187','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/74/mathml/M187">View MathML</a>and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/74/mathml/M188','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/74/mathml/M188">View MathML</a>such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/74/mathml/M189','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/74/mathml/M189">View MathML</a>

(<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/74/mathml/M190','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/74/mathml/M190">View MathML</a>) There exist<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/74/mathml/M186','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/74/mathml/M186">View MathML</a>with<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/74/mathml/M192','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/74/mathml/M192">View MathML</a>such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/74/mathml/M193','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/74/mathml/M193">View MathML</a>

Then BVP (1.1) and (1.2) has two nondecreasing positive solutions<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/74/mathml/M194','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/74/mathml/M194">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/74/mathml/M195','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/74/mathml/M195">View MathML</a>inK. Moreover, if in (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/74/mathml/M185','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/74/mathml/M185">View MathML</a>) <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/74/mathml/M197','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/74/mathml/M197">View MathML</a>is replaced by<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/74/mathml/M198','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/74/mathml/M198">View MathML</a>, then BVP (1.1) and (1.2) has a third nondecreasing positive solution<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/74/mathml/M199','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/74/mathml/M199">View MathML</a>.

Proof Assume that (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/74/mathml/M185','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/74/mathml/M185">View MathML</a>) holds. We show that either T has a fixed point <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/74/mathml/M201','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/74/mathml/M201">View MathML</a> or in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/74/mathml/M202','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/74/mathml/M202">View MathML</a>. If <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/74/mathml/M203','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/74/mathml/M203">View MathML</a> for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/74/mathml/M204','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/74/mathml/M204">View MathML</a>, by Lemmas 2.6 and 2.7, we have <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/74/mathml/M205','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/74/mathml/M205">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/74/mathml/M206','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/74/mathml/M206">View MathML</a>, and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/74/mathml/M207','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/74/mathml/M207">View MathML</a>. By Lemma 2.5(b), we have <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/74/mathml/M208','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/74/mathml/M208">View MathML</a> since <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/74/mathml/M209','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/74/mathml/M209">View MathML</a>. It follows from Lemma 1.1(3) that T has a fixed point <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/74/mathml/M210','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/74/mathml/M210">View MathML</a>. Similarly, T has a fixed point <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/74/mathml/M211','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/74/mathml/M211">View MathML</a>. The proof is similar when (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/74/mathml/M190','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/74/mathml/M190">View MathML</a>) holds. □

Corollary 3.1If there exists<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/74/mathml/M124','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/74/mathml/M124">View MathML</a>such that one of the following conditions holds:

(<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/74/mathml/M214','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/74/mathml/M214">View MathML</a>) <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/74/mathml/M215','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/74/mathml/M215">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/74/mathml/M216','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/74/mathml/M216">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/74/mathml/M217','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/74/mathml/M217">View MathML</a>for<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/74/mathml/M147','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/74/mathml/M147">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/74/mathml/M219','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/74/mathml/M219">View MathML</a>,

(<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/74/mathml/M220','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/74/mathml/M220">View MathML</a>) <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/74/mathml/M221','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/74/mathml/M221">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/74/mathml/M222','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/74/mathml/M222">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/74/mathml/M217','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/74/mathml/M217">View MathML</a>for<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/74/mathml/M157','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/74/mathml/M157">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/74/mathml/M225','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/74/mathml/M225">View MathML</a>,

then BVP (1.1) and (1.2) has at least two nondecreasing positive solutions inK.

Proof We show that (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/74/mathml/M214','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/74/mathml/M214">View MathML</a>) implies (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/74/mathml/M185','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/74/mathml/M185">View MathML</a>). It is easy to verify that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/74/mathml/M228','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/74/mathml/M228">View MathML</a> implies that there exists <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/74/mathml/M229','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/74/mathml/M229">View MathML</a> such that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/74/mathml/M230','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/74/mathml/M230">View MathML</a>. Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/74/mathml/M231','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/74/mathml/M231">View MathML</a>, by <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/74/mathml/M232','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/74/mathml/M232">View MathML</a>, there exists <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/74/mathml/M233','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/74/mathml/M233">View MathML</a> such that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/74/mathml/M234','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/74/mathml/M234">View MathML</a> for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/74/mathml/M105','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/74/mathml/M105">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/74/mathml/M236','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/74/mathml/M236">View MathML</a>. Let

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/74/mathml/M237','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/74/mathml/M237">View MathML</a>

then for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/74/mathml/M238','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/74/mathml/M238">View MathML</a>, we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/74/mathml/M239','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/74/mathml/M239">View MathML</a>

This implies that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/74/mathml/M240','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/74/mathml/M240">View MathML</a> and (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/74/mathml/M185','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/74/mathml/M185">View MathML</a>) holds. Similarly, (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/74/mathml/M220','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/74/mathml/M220">View MathML</a>) implies (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/74/mathml/M190','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/74/mathml/M190">View MathML</a>). This completes the proof. □

Remark 3.1 We establish the multiplicity of monotone positive solutions for a higher order differential equation with integral boundary conditions, and we allow the nonlinearity f to contain derivatives of the unknown function <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/74/mathml/M27','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/74/mathml/M27">View MathML</a> up to <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/74/mathml/M28','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/74/mathml/M28">View MathML</a> order, so our work improves and generalizes the results of [2] to some degree.

Competing interests

The authors declare that they have no competing interests.

Authors’ contributions

XH wrote the first manuscript and LL corrected and improved the final version. Both authors read and approved the final manuscript.

Acknowledgements

Research supported by the National Natural Science Foundation of China (11371221, 11201260), the Specialized Research Fund for the Doctoral Program of Higher Education of China (20123705120004, 20123705110001), a Project of Shandong Province Higher Educational Science and Technology Program (J11LA06) and Foundation of Qufu Normal University (BSQD20100103).

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