Open Access Research

Sign-changing solution for a third-order boundary-value problem in ordered Banach space with lattice structure

Xiuli Lin* and Zengqin Zhao

Author Affiliations

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

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Boundary Value Problems 2014, 2014:132  doi:10.1186/1687-2770-2014-132


The electronic version of this article is the complete one and can be found online at: http://www.boundaryvalueproblems.com/content/2014/1/132


Received:11 December 2013
Accepted:7 April 2014
Published:22 May 2014

© 2014 Lin and Zhao; 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

In this paper, the sign-changing solution of a third-order two-point boundary-value problem is considered. By calculating the eigenvalues and the algebraic multiplicity of the linear problem and using a new fixed point theorem in an ordered Banach space with lattice structure, we give some conditions to guarantee the existence for a sign-changing solution.

Keywords:
third-order boundary-value problem; sign-changing solution; Green’s function; fixed point; lattice

1 Introduction

In this paper, we consider the following nonlinear third-order two-point boundary-value problem

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/132/mathml/M1','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/132/mathml/M1">View MathML</a>

(1.1)

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/132/mathml/M2','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/132/mathml/M2">View MathML</a>.

The study on the existence of the sign-changing solutions for the boundary-value problem is very useful and interesting both in theory and in application. Recently, there has been much attention focused on the problem, especially to the two-point or multi-point boundary-value problem. For the second-order two-point or multi-point boundary-value problem, many beautiful results have been given on the existence and multiplicity of the sign-changing solutions (see [1-5] and the references therein). For example, Xu and Sun [1] obtained an existence result of the sign-changing solutions for the second-order three-point boundary-value problem

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/132/mathml/M3','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/132/mathml/M3">View MathML</a>

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/132/mathml/M4','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/132/mathml/M4">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/132/mathml/M5','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/132/mathml/M5">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/132/mathml/M6','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/132/mathml/M6">View MathML</a>. Xu [2] considered the sign-changing solutions for the second-order multi-point boundary-value problem

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/132/mathml/M7','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/132/mathml/M7">View MathML</a>

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/132/mathml/M8','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/132/mathml/M8">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/132/mathml/M9','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/132/mathml/M9">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/132/mathml/M10','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/132/mathml/M10">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/132/mathml/M6','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/132/mathml/M6">View MathML</a>. In [4], Zhang and Sun obtained the existence and multiplicity of the sign-changing solutions for the integral boundary-value problem

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/132/mathml/M12','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/132/mathml/M12">View MathML</a>

(1.2)

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/132/mathml/M6','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/132/mathml/M6">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/132/mathml/M14','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/132/mathml/M14">View MathML</a> is nonnegative with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/132/mathml/M15','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/132/mathml/M15">View MathML</a>. For the integral boundary-value problem (1.2), Li and Liu [5] also obtained the existence and multiplicity of the sign-changing solutions in ordered Banach space with the lattice structure.

For the third-order boundary-value problem, the existence and multiplicity of solutions have also been discussed in many papers (see [6-11] and the references therein). However, the research on the sign-changing solutions has been proceeded slowly. For the problem (1.1), Yao and Feng [10,11] established several existence results for the solutions including the positive solutions using the lower and upper solutions and a maximum principle, respectively. To our knowledge, however, there are fewer papers considered the sign-changing solutions of the problem (1.1). Motivated by the work mentioned above, using the eigenvalues of linear operator, we give an existence result for the sign-changing solutions of the problem (1.1).

The main contribution of this paper are as follows: (a) for the sign-changing solutions of the problem (1.1), to our knowledge, there is no result using the eigenvalues of the linear operator until now; (b) we obtain the eigenvalues and the algebraic multiplicity of the linear problem corresponding the problem (1.1), which is one of the key points that we can use to prove our main result; (c) some conditions are given to guarantee the existence for a sign-changing solution of the problem (1.1).

2 Notations and preliminaries

The following results will be used throughout the paper.

Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/132/mathml/M16','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/132/mathml/M16">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/132/mathml/M17','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/132/mathml/M17">View MathML</a>. Then E is a real Banach space with the norm <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/132/mathml/M18','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/132/mathml/M18">View MathML</a>. Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/132/mathml/M19','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/132/mathml/M19">View MathML</a>, and P is a normal solid cone of E, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/132/mathml/M20','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/132/mathml/M20">View MathML</a>.

Let the operators K, F, A be defined by

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/132/mathml/M21','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/132/mathml/M21">View MathML</a>

(2.1)

and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/132/mathml/M22','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/132/mathml/M22">View MathML</a>, respectively, where

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/132/mathml/M23','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/132/mathml/M23">View MathML</a>

(2.2)

Remark 1 (1) <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/132/mathml/M24','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/132/mathml/M24">View MathML</a> are completely continuous. (2) <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/132/mathml/M25','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/132/mathml/M25">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/132/mathml/M26','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/132/mathml/M26">View MathML</a>. In fact, since it is obvious in the other case, we only need to prove the case <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/132/mathml/M27','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/132/mathml/M27">View MathML</a>. Now we suppose that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/132/mathml/M27','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/132/mathml/M27">View MathML</a>. Then

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/132/mathml/M29','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/132/mathml/M29">View MathML</a>

Definition 2.1[12]

We call E a lattice under the partial ordering ≤, if <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/132/mathml/M30','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/132/mathml/M30">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/132/mathml/M31','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/132/mathml/M31">View MathML</a> exist for arbitrary <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/132/mathml/M32','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/132/mathml/M32">View MathML</a>.

Remark 2<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/132/mathml/M16','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/132/mathml/M16">View MathML</a> is a lattice under the partial ordering ≤ that is deduced by the cone <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/132/mathml/M34','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/132/mathml/M34">View MathML</a> of E.

Definition 2.2[12]

Let E be a Banach space with a cone <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/132/mathml/M35','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/132/mathml/M35">View MathML</a> be a nonlinear operator. We call that A is a unilaterally asymptotically linear operator along <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/132/mathml/M36','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/132/mathml/M36">View MathML</a>, if there exists a bounded linear operator L such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/132/mathml/M37','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/132/mathml/M37">View MathML</a>

L is said to be the derived operator of A along <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/132/mathml/M38','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/132/mathml/M38">View MathML</a> and will be denoted by <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/132/mathml/M39','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/132/mathml/M39">View MathML</a>. Similarly, we can also define a unilaterally asymptotically linear operator along <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/132/mathml/M40','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/132/mathml/M40">View MathML</a>. Specially, if <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/132/mathml/M41','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/132/mathml/M41">View MathML</a>, We call that A is a unilaterally asymptotically linear operator along P and −P.

Definition 2.3[12]

Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/132/mathml/M42','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/132/mathml/M42">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/132/mathml/M43','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/132/mathml/M43">View MathML</a> be a nonlinear operator. A is said to be quasi-additive on lattice, if there exists <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/132/mathml/M44','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/132/mathml/M44">View MathML</a> such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/132/mathml/M45','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/132/mathml/M45">View MathML</a>

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/132/mathml/M46','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/132/mathml/M46">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/132/mathml/M47','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/132/mathml/M47">View MathML</a>.

Remark 3 It is easy to see that the operators F and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/132/mathml/M22','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/132/mathml/M22">View MathML</a> defined by (2.1) are both quasi-additive on the lattice <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/132/mathml/M49','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/132/mathml/M49">View MathML</a>.

Let us list some conditions and preliminary lemmas to be used in this paper.

(H1) <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/132/mathml/M50','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/132/mathml/M50">View MathML</a> is strictly increasing in u, and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/132/mathml/M51','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/132/mathml/M51">View MathML</a> for all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/132/mathml/M52','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/132/mathml/M52">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/132/mathml/M53','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/132/mathml/M53">View MathML</a>.

(H2) <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/132/mathml/M54','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/132/mathml/M54">View MathML</a> uniformly on <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/132/mathml/M55','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/132/mathml/M55">View MathML</a>. There exists a positive integer <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/132/mathml/M56','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/132/mathml/M56">View MathML</a> such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/132/mathml/M57','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/132/mathml/M57">View MathML</a>

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/132/mathml/M58','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/132/mathml/M58">View MathML</a> are the positive solutions of the equation

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/132/mathml/M59','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/132/mathml/M59">View MathML</a>

(2.3)

(H3) <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/132/mathml/M60','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/132/mathml/M60">View MathML</a> uniformly on <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/132/mathml/M55','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/132/mathml/M55">View MathML</a>, and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/132/mathml/M62','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/132/mathml/M62">View MathML</a>.

Lemma 2.1For any<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/132/mathml/M63','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/132/mathml/M63">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/132/mathml/M64','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/132/mathml/M64">View MathML</a>is a solution of the following problem:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/132/mathml/M65','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/132/mathml/M65">View MathML</a>

if and only if<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/132/mathml/M66','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/132/mathml/M66">View MathML</a>is a solution of the integral equation

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/132/mathml/M67','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/132/mathml/M67">View MathML</a>

where<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/132/mathml/M68','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/132/mathml/M68">View MathML</a>is defined by (2.2).

Proof On the one hand, integrating the equation

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/132/mathml/M69','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/132/mathml/M69">View MathML</a>

over <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/132/mathml/M70','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/132/mathml/M70">View MathML</a> for three times, we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/132/mathml/M71','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/132/mathml/M71">View MathML</a>

Then

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/132/mathml/M72','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/132/mathml/M72">View MathML</a>

Combining them with boundary condition <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/132/mathml/M73','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/132/mathml/M73">View MathML</a>, we conclude that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/132/mathml/M74','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/132/mathml/M74">View MathML</a>

Therefore,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/132/mathml/M75','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/132/mathml/M75">View MathML</a>

On the other hand, since

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/132/mathml/M76','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/132/mathml/M76">View MathML</a>

therefore,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/132/mathml/M77','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/132/mathml/M77">View MathML</a>

and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/132/mathml/M78','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/132/mathml/M78">View MathML</a>

Moreover, we get <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/132/mathml/M79','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/132/mathml/M79">View MathML</a>. □

Remark 4 Considering Lemma 2.1, we find that u is a solution of the problem (1.1) if and only if u is a fixed point of the operator <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/132/mathml/M22','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/132/mathml/M22">View MathML</a>.

From the following lemma, we can obtain the eigenvalues and the algebraic multiplicity of the linear operator K.

Lemma 2.2The eigenvalues of the linear operatorKare

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/132/mathml/M81','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/132/mathml/M81">View MathML</a>

and the algebraic multiplicity of each positive eigenvalue<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/132/mathml/M82','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/132/mathml/M82">View MathML</a>is equal to 1, where<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/132/mathml/M83','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/132/mathml/M83">View MathML</a>are the positive solutions of (2.3).

Proof Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/132/mathml/M84','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/132/mathml/M84">View MathML</a> be a positive eigenvalue of the linear operator K, and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/132/mathml/M85','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/132/mathml/M85">View MathML</a> be an eigenfunction corresponding to eigenvalue <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/132/mathml/M84','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/132/mathml/M84">View MathML</a>. By Lemma 2.1, we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/132/mathml/M87','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/132/mathml/M87">View MathML</a>

(2.4)

The auxiliary equation of the differential equation (2.4) has roots −μ, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/132/mathml/M88','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/132/mathml/M88">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/132/mathml/M89','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/132/mathml/M89">View MathML</a>, where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/132/mathml/M90','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/132/mathml/M90">View MathML</a>. Thus the general solution of (2.4) is

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/132/mathml/M91','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/132/mathml/M91">View MathML</a>

Then

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/132/mathml/M92','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/132/mathml/M92">View MathML</a>

Applying the condition <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/132/mathml/M93','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/132/mathml/M93">View MathML</a>, we obtain <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/132/mathml/M94','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/132/mathml/M94">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/132/mathml/M95','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/132/mathml/M95">View MathML</a>, where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/132/mathml/M96','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/132/mathml/M96">View MathML</a>.

Applying the second condition <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/132/mathml/M97','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/132/mathml/M97">View MathML</a>, that is,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/132/mathml/M98','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/132/mathml/M98">View MathML</a>

(2.5)

Considering (2.3), we see that μ is one of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/132/mathml/M99','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/132/mathml/M99">View MathML</a> , that is,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/132/mathml/M100','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/132/mathml/M100">View MathML</a>

are eigenvalues of the linear operator K and the eigenfunction corresponding to the eigenvalue <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/132/mathml/M101','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/132/mathml/M101">View MathML</a> is

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/132/mathml/M102','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/132/mathml/M102">View MathML</a>

(2.6)

where C is a nonzero constant.

Next we prove that the algebraic multiplicity of the eigenvalue <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/132/mathml/M82','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/132/mathml/M82">View MathML</a> is 1. From (2.6), any two eigenfunctions corresponding to the same eigenvalue <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/132/mathml/M104','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/132/mathml/M104">View MathML</a> are merely nonzero constant multiples of each other, that is,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/132/mathml/M105','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/132/mathml/M105">View MathML</a>

Now we show that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/132/mathml/M106','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/132/mathml/M106">View MathML</a>

Obviously, we only need to show that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/132/mathml/M107','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/132/mathml/M107">View MathML</a>

In fact, for any <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/132/mathml/M108','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/132/mathml/M108">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/132/mathml/M109','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/132/mathml/M109">View MathML</a> is an eigenfunction of linear operator K corresponding to the eigenvalue <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/132/mathml/M101','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/132/mathml/M101">View MathML</a> if <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/132/mathml/M111','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/132/mathml/M111">View MathML</a>. Considering (2.6), there exists a nonzero constant σ such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/132/mathml/M112','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/132/mathml/M112">View MathML</a>

By direct computation, we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/132/mathml/M113','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/132/mathml/M113">View MathML</a>

(2.7)

It is easy to see that the solution for the corresponding homogeneous equation of (2.7) is of the form

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/132/mathml/M114','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/132/mathml/M114">View MathML</a>

Then, by an ordinary differential equation method, we see that the general solution of (2.7) is of the form

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/132/mathml/M115','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/132/mathml/M115">View MathML</a>

where

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/132/mathml/M116','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/132/mathml/M116">View MathML</a>

is the special solution of the equation

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/132/mathml/M117','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/132/mathml/M117">View MathML</a>

and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/132/mathml/M118','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/132/mathml/M118">View MathML</a>

is the special solution of the equation

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/132/mathml/M119','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/132/mathml/M119">View MathML</a>

Then

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/132/mathml/M120','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/132/mathml/M120">View MathML</a>

Applying the condition <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/132/mathml/M93','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/132/mathml/M93">View MathML</a>, we obtain <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/132/mathml/M94','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/132/mathml/M94">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/132/mathml/M95','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/132/mathml/M95">View MathML</a>. From the condition <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/132/mathml/M97','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/132/mathml/M97">View MathML</a>, we obtain

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/132/mathml/M125','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/132/mathml/M125">View MathML</a>

(2.8)

From (2.5), we have <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/132/mathml/M126','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/132/mathml/M126">View MathML</a>. Thus it follows from (2.8) that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/132/mathml/M127','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/132/mathml/M127">View MathML</a>

which implies that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/132/mathml/M128','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/132/mathml/M128">View MathML</a>

That is

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/132/mathml/M129','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/132/mathml/M129">View MathML</a>

which is a contradiction of

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/132/mathml/M130','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/132/mathml/M130">View MathML</a>

Therefore, the algebraic multiplicity of the eigenvalue <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/132/mathml/M82','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/132/mathml/M82">View MathML</a> is 1. □

Lemma 2.3Suppose that (H1) holds and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/132/mathml/M132','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/132/mathml/M132">View MathML</a>is a solution of the (1.1), then<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/132/mathml/M133','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/132/mathml/M133">View MathML</a>. Similarly, if<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/132/mathml/M134','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/132/mathml/M134">View MathML</a>is a solution of the (1.1), then<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/132/mathml/M135','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/132/mathml/M135">View MathML</a>.

Proof The proof is obvious. □

Lemma 2.4Suppose that (H1)-(H3) hold. Then the operatorAis Fréchet differentiable atθand ∞, and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/132/mathml/M136','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/132/mathml/M136">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/132/mathml/M137','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/132/mathml/M137">View MathML</a>.

Proof Since (H3): <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/132/mathml/M138','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/132/mathml/M138">View MathML</a> uniformly on <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/132/mathml/M55','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/132/mathml/M55">View MathML</a>. That is, for any <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/132/mathml/M140','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/132/mathml/M140">View MathML</a>, there exists <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/132/mathml/M141','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/132/mathml/M141">View MathML</a> such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/132/mathml/M142','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/132/mathml/M142">View MathML</a>

From (H1), it is easy to see that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/132/mathml/M143','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/132/mathml/M143">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/132/mathml/M144','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/132/mathml/M144">View MathML</a>. Then, for any <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/132/mathml/M145','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/132/mathml/M145">View MathML</a> with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/132/mathml/M146','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/132/mathml/M146">View MathML</a>, we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/132/mathml/M147','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/132/mathml/M147">View MathML</a>

Then

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/132/mathml/M148','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/132/mathml/M148">View MathML</a>

Thus,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/132/mathml/M149','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/132/mathml/M149">View MathML</a>

which means <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/132/mathml/M150','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/132/mathml/M150">View MathML</a>.

Since (H2): <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/132/mathml/M151','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/132/mathml/M151">View MathML</a> uniformly on <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/132/mathml/M55','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/132/mathml/M55">View MathML</a>. That is, for any <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/132/mathml/M140','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/132/mathml/M140">View MathML</a>, there exists <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/132/mathml/M154','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/132/mathml/M154">View MathML</a> such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/132/mathml/M155','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/132/mathml/M155">View MathML</a>

Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/132/mathml/M156','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/132/mathml/M156">View MathML</a>. Then

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/132/mathml/M157','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/132/mathml/M157">View MathML</a>

Thus,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/132/mathml/M158','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/132/mathml/M158">View MathML</a>

Then

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/132/mathml/M159','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/132/mathml/M159">View MathML</a>

Therefore, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/132/mathml/M137','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/132/mathml/M137">View MathML</a>. □

Remark 5 Suppose (H2) holds. Similar to Lemma 2.4, we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/132/mathml/M161','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/132/mathml/M161">View MathML</a>

Lemma 2.5[13]

Suppose thatEis an ordered Banach space with a lattice structure, Pis a normal solid cone inE, and the nonlinear operatorAis quasi-additive on the lattice. Assume that

(i) Ais strongly increasing onPandP;

(ii) both<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/132/mathml/M162','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/132/mathml/M162">View MathML</a>and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/132/mathml/M163','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/132/mathml/M163">View MathML</a>exist with<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/132/mathml/M164','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/132/mathml/M164">View MathML</a>and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/132/mathml/M165','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/132/mathml/M165">View MathML</a>; 1 is not an eigenvalue of<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/132/mathml/M162','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/132/mathml/M162">View MathML</a>or<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/132/mathml/M163','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/132/mathml/M163">View MathML</a>corresponding a positive eigenvector;

(iii) <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/132/mathml/M168','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/132/mathml/M168">View MathML</a>; the Fréchet derivative<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/132/mathml/M169','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/132/mathml/M169">View MathML</a>ofAatθis strongly positive, and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/132/mathml/M170','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/132/mathml/M170">View MathML</a>;

(iv) the Fréchet derivative<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/132/mathml/M171','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/132/mathml/M171">View MathML</a>ofAatexists; 1 is not an eigenvalue of<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/132/mathml/M171','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/132/mathml/M171">View MathML</a>; the sumβof the algebraic multiplicities for all eigenvalues of<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/132/mathml/M171','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/132/mathml/M171">View MathML</a>lying in the interval<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/132/mathml/M174','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/132/mathml/M174">View MathML</a>is an even number.

ThenAhas at least three nontrivial fixed points containing one sign-changing fixed point.

3 Main result

We state the main result of this paper.

Theorem 3.1Suppose that (H1)-(H3) hold. Then the problem (1.1) has at least three solutions including a sign-changing solution.

We need only to prove that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/132/mathml/M22','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/132/mathml/M22">View MathML</a> satisfies the four conditions of Lemma 2.5.

Proof Noticing

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/132/mathml/M176','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/132/mathml/M176">View MathML</a>

(i) A is strongly increasing on P and −P. In fact, from (H1) and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/132/mathml/M177','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/132/mathml/M177">View MathML</a>, we see that A is strongly increasing on P. Similarly, A is strongly increasing on −P.

(ii) From <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/132/mathml/M178','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/132/mathml/M178">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/132/mathml/M179','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/132/mathml/M179">View MathML</a> and Lemma 2.2, we find that 1 is not an eigenvalue of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/132/mathml/M162','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/132/mathml/M162">View MathML</a> or <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/132/mathml/M163','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/132/mathml/M163">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/132/mathml/M182','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/132/mathml/M182">View MathML</a>.

(iii) From <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/132/mathml/M150','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/132/mathml/M150">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/132/mathml/M184','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/132/mathml/M184">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/132/mathml/M177','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/132/mathml/M177">View MathML</a>, Lemma 2.2 and (H1), we have <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/132/mathml/M186','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/132/mathml/M186">View MathML</a> is strongly positive, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/132/mathml/M187','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/132/mathml/M187">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/132/mathml/M188','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/132/mathml/M188">View MathML</a>.

(iv) Since <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/132/mathml/M137','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/132/mathml/M137">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/132/mathml/M190','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/132/mathml/M190">View MathML</a> and Lemma 2.2, the condition (iv) of Lemma 2.5 is satisfied.

Therefore, from Lemma 2.5, we see that A has at least three nontrivial fixed points including one sign-changing fixed point. Then, the problem (1.1) has at least three solutions, including one sign-changing solution. □

Example 3.1 Consider the following third-order boundary-value problem

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/132/mathml/M191','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/132/mathml/M191">View MathML</a>

(3.1)

where

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/132/mathml/M192','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/132/mathml/M192">View MathML</a>

By simple calculations, we have <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/132/mathml/M193','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/132/mathml/M193">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/132/mathml/M194','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/132/mathml/M194">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/132/mathml/M195','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/132/mathml/M195">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/132/mathml/M196','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/132/mathml/M196">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/132/mathml/M197','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/132/mathml/M197">View MathML</a>. Then it is easy to see that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/132/mathml/M198','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/132/mathml/M198">View MathML</a> satisfies the conditions (H1)-(H3). Therefore, the boundary-value problem (3.1) has at least three solutions, including one sign-changing solution.

Competing interests

The authors declare that they have no competing interests.

Authors’ contributions

The authors declare that the study was realized in collaboration with the same responsibility. All authors read and approved the final manuscript.

Acknowledgements

The authors are highly grateful for the referees’ careful reading and comments on this paper. The research is supported by Program for Scientific research innovation team in Colleges and universities of Shandong Province, the Doctoral Program Foundation of Education Ministry of China (20133705110003), the Natural Science Foundation of Shandong Province of China (ZR2011AM008, ZR2012AM010, ZR2012AQ024).

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