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Bifurcation from interval and positive solutions for a class of fourth-order two-point boundary value problem

Wenguo Shen* and Tao He

Author Affiliations

Department of Basic Courses, Lanzhou Institute of Technology, Lanzhou, 730050, People’s Republic of China

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Boundary Value Problems 2013, 2013:170  doi:10.1186/1687-2770-2013-170

The electronic version of this article is the complete one and can be found online at: http://www.boundaryvalueproblems.com/content/2013/1/170


Received:14 November 2012
Accepted:8 July 2013
Published:22 July 2013

© 2013 Shen and He; licensee Springer

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

Abstract

We consider the fourth-order two-point boundary value problem <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/170/mathml/M1','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/170/mathml/M1">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/170/mathml/M2','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/170/mathml/M2">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/170/mathml/M3','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/170/mathml/M3">View MathML</a>, which is not necessarily linearizable. We give conditions on the parameters k, l and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/170/mathml/M4','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/170/mathml/M4">View MathML</a> that guarantee the existence of positive solutions. The proof of our main result is based upon topological degree theory and global bifurcation techniques.

MSC: 34B15.

Keywords:
topological degree; fourth-order ordinary differential equation; bifurcation; positive solution; eigenvalue

1 Introduction

The deformations of an elastic beam in an equilibrium state with fixed both endpoints can be described by the fourth-order boundary value problem

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/170/mathml/M5','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/170/mathml/M5">View MathML</a>

(1.1)

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/170/mathml/M6','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/170/mathml/M6">View MathML</a> is continuous, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/170/mathml/M7','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/170/mathml/M7">View MathML</a> is a parameter and l is a given constant. Since problem (1.1) cannot transform into a system of second-order equations, the treatment method of the second-order system does not apply to it. Thus, the existing literature on problem (1.1) is limited. When <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/170/mathml/M8','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/170/mathml/M8">View MathML</a>, the existence of positive solutions of problem (1.1) has been studied by several authors, see [1-5]. Especially, when <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/170/mathml/M9','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/170/mathml/M9">View MathML</a>, Xu and Han [6] studied the existence of nodal solutions of problem (1.1) by applying disconjugate operator theory and bifurcation techniques.

Recently, motivated by [6], when k, l satisfy (A1), Shen [7] studied the existence of nodal solutions of a general fourth-order boundary value problem by applying disconjugate operator theory [8,9] and Rabinowitz’s global bifurcation theorem

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/170/mathml/M10','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/170/mathml/M10">View MathML</a>

(1.2)

where

(A1) one of following conditions holds:

(i) k, l satisfying <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/170/mathml/M11','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/170/mathml/M11">View MathML</a> are given constants with

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/170/mathml/M12','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/170/mathml/M12">View MathML</a>

(1.3)

(ii) k, l satisfying <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/170/mathml/M13','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/170/mathml/M13">View MathML</a> are given constants with

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/170/mathml/M14','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/170/mathml/M14">View MathML</a>

(1.4)

In this paper, we consider bifurcation from interval and positive solutions for problem (1.2). In order to prove our main result, condition (A1) and the following weaker conditions are satisfied throughout this paper:

(H1) <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/170/mathml/M15','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/170/mathml/M15">View MathML</a> is continuous and there exist functions <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/170/mathml/M16','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/170/mathml/M16">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/170/mathml/M17','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/170/mathml/M17">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/170/mathml/M18','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/170/mathml/M18">View MathML</a>, and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/170/mathml/M19','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/170/mathml/M19">View MathML</a> such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/170/mathml/M20','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/170/mathml/M20">View MathML</a>

(1.5)

for some functions <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/170/mathml/M21','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/170/mathml/M21">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/170/mathml/M22','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/170/mathml/M22">View MathML</a> defined on <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/170/mathml/M23','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/170/mathml/M23">View MathML</a> with

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/170/mathml/M24','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/170/mathml/M24">View MathML</a>

(1.6)

uniformly for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/170/mathml/M25','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/170/mathml/M25">View MathML</a>, and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/170/mathml/M26','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/170/mathml/M26">View MathML</a>

(1.7)

for some functions <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/170/mathml/M27','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/170/mathml/M27">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/170/mathml/M28','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/170/mathml/M28">View MathML</a> defined on <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/170/mathml/M23','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/170/mathml/M23">View MathML</a> with

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/170/mathml/M30','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/170/mathml/M30">View MathML</a>

(1.8)

uniformly for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/170/mathml/M25','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/170/mathml/M25">View MathML</a>.

(H2) <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/170/mathml/M32','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/170/mathml/M32">View MathML</a> for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/170/mathml/M25','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/170/mathml/M25">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/170/mathml/M34','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/170/mathml/M34">View MathML</a>.

(H3) There exists a function <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/170/mathml/M35','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/170/mathml/M35">View MathML</a> with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/170/mathml/M36','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/170/mathml/M36">View MathML</a> in any subinterval of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/170/mathml/M37','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/170/mathml/M37">View MathML</a> such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/170/mathml/M38','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/170/mathml/M38">View MathML</a>

(1.9)

It is the purpose of this paper to study the existence of positive solutions of (1.2) under conditions (A1), (H1), (H2) and (H3). The main tool we use is the following global bifurcation theorem for the problem which is not necessarily linearizable.

Theorem A (Rabinowitz [10])

LetVbe a real reflexive Banach space. Let<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/170/mathml/M39','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/170/mathml/M39">View MathML</a>be completely continuous such that<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/170/mathml/M40','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/170/mathml/M40">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/170/mathml/M41','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/170/mathml/M41">View MathML</a>. Let<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/170/mathml/M42','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/170/mathml/M42">View MathML</a> (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/170/mathml/M43','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/170/mathml/M43">View MathML</a>) be such that<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/170/mathml/M44','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/170/mathml/M44">View MathML</a>is an isolated solution of the following equation:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/170/mathml/M45','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/170/mathml/M45">View MathML</a>

(1.10)

for<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/170/mathml/M46','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/170/mathml/M46">View MathML</a>and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/170/mathml/M47','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/170/mathml/M47">View MathML</a>, where<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/170/mathml/M48','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/170/mathml/M48">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/170/mathml/M49','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/170/mathml/M49">View MathML</a>are not bifurcation points of (1.10). Furthermore, assume that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/170/mathml/M50','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/170/mathml/M50">View MathML</a>

(1.11)

where<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/170/mathml/M51','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/170/mathml/M51">View MathML</a>is an isolating neighborhood of the trivial solution. Let

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/170/mathml/M52','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/170/mathml/M52">View MathML</a>

Then there exists a continuum (i.e., a closed connected set) ofcontaining<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/170/mathml/M55','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/170/mathml/M55">View MathML</a>, and either

(i) is unbounded in<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/170/mathml/M57','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/170/mathml/M57">View MathML</a>, or

(ii) <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/170/mathml/M58','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/170/mathml/M58">View MathML</a>.

Remark 1.1 For other results on the existence and multiplicity of positive solutions and nodal solutions for boundary value problems of fourth-order ordinary differential equations based on bifurcation techniques, see [11-20].

2 Hypotheses and lemmas

Let

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/170/mathml/M59','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/170/mathml/M59">View MathML</a>

(2.1)

Theorem 2.1 (see [[7], Theorem 2.4])

Let (A1) hold. Then

(i) <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/170/mathml/M60','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/170/mathml/M60">View MathML</a>is disconjugate on<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/170/mathml/M37','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/170/mathml/M37">View MathML</a>, and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/170/mathml/M62','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/170/mathml/M62">View MathML</a>has a factorization

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/170/mathml/M63','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/170/mathml/M63">View MathML</a>

(2.2)

where<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/170/mathml/M64','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/170/mathml/M64">View MathML</a>with<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/170/mathml/M65','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/170/mathml/M65">View MathML</a> (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/170/mathml/M66','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/170/mathml/M66">View MathML</a>);

(ii) <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/170/mathml/M3','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/170/mathml/M3">View MathML</a>if and only if

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/170/mathml/M68','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/170/mathml/M68">View MathML</a>

(2.3)

where

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/170/mathml/M69','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/170/mathml/M69">View MathML</a>

(2.4)

Theorem 2.2 (see [[7], Theorem 2.7])

Let (A1) hold and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/170/mathml/M70','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/170/mathml/M70">View MathML</a>with<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/170/mathml/M71','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/170/mathml/M71">View MathML</a>on any subinterval of<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/170/mathml/M37','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/170/mathml/M37">View MathML</a>. Then

(i) the problem

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/170/mathml/M73','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/170/mathml/M73">View MathML</a>

(2.5)

has an infinite sequence of positive eigenvalues

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/170/mathml/M74','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/170/mathml/M74">View MathML</a>

(2.6)

(ii) <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/170/mathml/M75','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/170/mathml/M75">View MathML</a>as<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/170/mathml/M76','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/170/mathml/M76">View MathML</a>;

(iii) to each eigenvalue<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/170/mathml/M77','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/170/mathml/M77">View MathML</a>, there corresponds an essential unique eigenfunction<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/170/mathml/M78','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/170/mathml/M78">View MathML</a>which has exactly<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/170/mathml/M79','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/170/mathml/M79">View MathML</a>simple zeros in<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/170/mathml/M80','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/170/mathml/M80">View MathML</a>and is positive near 0;

(iv) given an arbitrary subinterval of<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/170/mathml/M37','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/170/mathml/M37">View MathML</a>, an eigenfunction that belongs to a sufficiently large eigenvalue changes its sign in that subinterval;

(v) for each<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/170/mathml/M82','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/170/mathml/M82">View MathML</a>, the algebraic multiplicity of<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/170/mathml/M77','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/170/mathml/M77">View MathML</a>is 1.

Theorem 2.3 (see [[7], Theorem 2.8]) (Maximum principle)

Let (A1) hold. Let<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/170/mathml/M84','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/170/mathml/M84">View MathML</a>with<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/170/mathml/M85','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/170/mathml/M85">View MathML</a>on<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/170/mathml/M37','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/170/mathml/M37">View MathML</a>and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/170/mathml/M87','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/170/mathml/M87">View MathML</a>in<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/170/mathml/M37','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/170/mathml/M37">View MathML</a>. If<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/170/mathml/M89','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/170/mathml/M89">View MathML</a>satisfies

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/170/mathml/M90','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/170/mathml/M90">View MathML</a>

(2.7)

then<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/170/mathml/M91','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/170/mathml/M91">View MathML</a>on<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/170/mathml/M80','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/170/mathml/M80">View MathML</a>.

Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/170/mathml/M93','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/170/mathml/M93">View MathML</a> with the norm <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/170/mathml/M94','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/170/mathml/M94">View MathML</a>. Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/170/mathml/M95','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/170/mathml/M95">View MathML</a> with its usual norm <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/170/mathml/M96','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/170/mathml/M96">View MathML</a>. By a positive solution of (1.2), we mean x is a solution of (1.2) with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/170/mathml/M91','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/170/mathml/M91">View MathML</a> (i.e., <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/170/mathml/M98','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/170/mathml/M98">View MathML</a> in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/170/mathml/M80','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/170/mathml/M80">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/170/mathml/M100','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/170/mathml/M100">View MathML</a>).

Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/170/mathml/M101','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/170/mathml/M101">View MathML</a> with the inner product <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/170/mathml/M102','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/170/mathml/M102">View MathML</a> and the norm <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/170/mathml/M103','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/170/mathml/M103">View MathML</a>. Further, define the linear operator <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/170/mathml/M104','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/170/mathml/M104">View MathML</a>

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/170/mathml/M105','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/170/mathml/M105">View MathML</a>

(2.8)

with

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/170/mathml/M106','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/170/mathml/M106">View MathML</a>

(2.9)

Then <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/170/mathml/M107','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/170/mathml/M107">View MathML</a> is a closed operator and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/170/mathml/M108','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/170/mathml/M108">View MathML</a> is completely continuous.

Lemma 2.4Let<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/170/mathml/M109','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/170/mathml/M109">View MathML</a>be the first eigenfunction of (2.5). Then, for all<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/170/mathml/M110','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/170/mathml/M110">View MathML</a>, we get

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/170/mathml/M111','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/170/mathml/M111">View MathML</a>

(2.10)

Proof Obviously, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/170/mathml/M112','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/170/mathml/M112">View MathML</a>, we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/170/mathml/M113','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/170/mathml/M113">View MathML</a>

Integrating by parts, we obtain

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/170/mathml/M114','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/170/mathml/M114">View MathML</a>

 □

Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/170/mathml/M115','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/170/mathml/M115">View MathML</a> be the closure of the set of positive solutions of the problem

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/170/mathml/M116','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/170/mathml/M116">View MathML</a>

(2.11)

We extend the function f to a continuous function <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/170/mathml/M117','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/170/mathml/M117">View MathML</a> defined on <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/170/mathml/M118','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/170/mathml/M118">View MathML</a> by

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/170/mathml/M119','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/170/mathml/M119">View MathML</a>

(2.12)

Then <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/170/mathml/M120','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/170/mathml/M120">View MathML</a> for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/170/mathml/M121','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/170/mathml/M121">View MathML</a>. For <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/170/mathml/M122','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/170/mathml/M122">View MathML</a>, let x be an arbitrary solution of the problem

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/170/mathml/M123','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/170/mathml/M123">View MathML</a>

(2.13)

Since <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/170/mathml/M124','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/170/mathml/M124">View MathML</a> for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/170/mathml/M25','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/170/mathml/M25">View MathML</a>, we have <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/170/mathml/M98','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/170/mathml/M98">View MathML</a> for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/170/mathml/M25','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/170/mathml/M25">View MathML</a>. Thus x is a nonnegative solution of (2.11), and the closure of the set of nontrivial solutions <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/170/mathml/M128','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/170/mathml/M128">View MathML</a> of (2.13) in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/170/mathml/M129','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/170/mathml/M129">View MathML</a> is exactly Σ.

Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/170/mathml/M130','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/170/mathml/M130">View MathML</a> be the Nemytskii operator associated with the function <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/170/mathml/M117','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/170/mathml/M117">View MathML</a>

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/170/mathml/M132','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/170/mathml/M132">View MathML</a>

(2.14)

Then (2.13), with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/170/mathml/M122','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/170/mathml/M122">View MathML</a>, is equivalent to the operator equation

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/170/mathml/M134','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/170/mathml/M134">View MathML</a>

(2.15)

In the following, we shall apply the Leray-Schauder degree theory, mainly to the mapping <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/170/mathml/M135','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/170/mathml/M135">View MathML</a>,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/170/mathml/M136','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/170/mathml/M136">View MathML</a>

(2.16)

For <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/170/mathml/M137','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/170/mathml/M137">View MathML</a>, let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/170/mathml/M138','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/170/mathml/M138">View MathML</a>, and let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/170/mathml/M139','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/170/mathml/M139">View MathML</a> denote the degree of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/170/mathml/M140','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/170/mathml/M140">View MathML</a> on <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/170/mathml/M141','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/170/mathml/M141">View MathML</a> with respect to 0.

Lemma 2.5Let<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/170/mathml/M142','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/170/mathml/M142">View MathML</a>be a compact interval with<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/170/mathml/M143','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/170/mathml/M143">View MathML</a>. Then there exists a number<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/170/mathml/M144','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/170/mathml/M144">View MathML</a>with the property

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/170/mathml/M145','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/170/mathml/M145">View MathML</a>

(2.17)

Proof Suppose to the contrary that there exist sequences <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/170/mathml/M146','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/170/mathml/M146">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/170/mathml/M147','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/170/mathml/M147">View MathML</a> in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/170/mathml/M148','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/170/mathml/M148">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/170/mathml/M149','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/170/mathml/M149">View MathML</a> in E, such that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/170/mathml/M150','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/170/mathml/M150">View MathML</a> for all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/170/mathml/M151','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/170/mathml/M151">View MathML</a>, then <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/170/mathml/M152','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/170/mathml/M152">View MathML</a> in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/170/mathml/M37','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/170/mathml/M37">View MathML</a>.

Set <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/170/mathml/M154','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/170/mathml/M154">View MathML</a>. Then <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/170/mathml/M155','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/170/mathml/M155">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/170/mathml/M156','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/170/mathml/M156">View MathML</a>. Now, from condition (H1), we have the following:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/170/mathml/M157','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/170/mathml/M157">View MathML</a>

(2.18)

and, accordingly,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/170/mathml/M158','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/170/mathml/M158">View MathML</a>

(2.19)

Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/170/mathml/M159','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/170/mathml/M159">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/170/mathml/M160','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/170/mathml/M160">View MathML</a> denote the nonnegative eigenfunctions corresponding to <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/170/mathml/M161','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/170/mathml/M161">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/170/mathml/M162','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/170/mathml/M162">View MathML</a>, respectively. Then we have, from the first inequality in (2.19),

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/170/mathml/M163','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/170/mathml/M163">View MathML</a>

(2.20)

From Lemma 2.4, we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/170/mathml/M164','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/170/mathml/M164">View MathML</a>

(2.21)

Since <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/170/mathml/M149','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/170/mathml/M149">View MathML</a> in E, from (1.6) we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/170/mathml/M166','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/170/mathml/M166">View MathML</a>

(2.22)

By the fact that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/170/mathml/M156','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/170/mathml/M156">View MathML</a>, we conclude that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/170/mathml/M168','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/170/mathml/M168">View MathML</a> in E. Thus,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/170/mathml/M169','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/170/mathml/M169">View MathML</a>

(2.23)

Combining this and (2.21) and letting <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/170/mathml/M170','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/170/mathml/M170">View MathML</a> in (2.20), we get

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/170/mathml/M171','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/170/mathml/M171">View MathML</a>

(2.24)

and consequently

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/170/mathml/M172','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/170/mathml/M172">View MathML</a>

(2.25)

Similarly, we deduce from the second inequality in (2.19) that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/170/mathml/M173','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/170/mathml/M173">View MathML</a>

(2.26)

Thus, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/170/mathml/M174','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/170/mathml/M174">View MathML</a>. This contradicts <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/170/mathml/M175','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/170/mathml/M175">View MathML</a>. □

Corollary 2.6For<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/170/mathml/M176','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/170/mathml/M176">View MathML</a>and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/170/mathml/M177','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/170/mathml/M177">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/170/mathml/M178','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/170/mathml/M178">View MathML</a>.

Proof Lemma 2.5, applied to the interval <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/170/mathml/M179','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/170/mathml/M179">View MathML</a>, guarantees the existence of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/170/mathml/M144','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/170/mathml/M144">View MathML</a> such that for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/170/mathml/M177','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/170/mathml/M177">View MathML</a>,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/170/mathml/M182','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/170/mathml/M182">View MathML</a>

(2.27)

Hence, for any <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/170/mathml/M177','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/170/mathml/M177">View MathML</a>,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/170/mathml/M184','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/170/mathml/M184">View MathML</a>

(2.28)

which ends the proof. □

Lemma 2.7Suppose<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/170/mathml/M185','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/170/mathml/M185">View MathML</a>. Then there exists<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/170/mathml/M186','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/170/mathml/M186">View MathML</a>such that<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/170/mathml/M187','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/170/mathml/M187">View MathML</a>with<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/170/mathml/M188','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/170/mathml/M188">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/170/mathml/M189','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/170/mathml/M189">View MathML</a>,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/170/mathml/M190','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/170/mathml/M190">View MathML</a>

(2.29)

where<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/170/mathml/M160','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/170/mathml/M160">View MathML</a>is the nonnegative eigenfunction corresponding to<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/170/mathml/M162','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/170/mathml/M162">View MathML</a>.

Proof We assume to the contrary that there exist <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/170/mathml/M193','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/170/mathml/M193">View MathML</a> and a sequence <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/170/mathml/M147','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/170/mathml/M147">View MathML</a>, with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/170/mathml/M195','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/170/mathml/M195">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/170/mathml/M196','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/170/mathml/M196">View MathML</a> in E, such that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/170/mathml/M197','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/170/mathml/M197">View MathML</a> for all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/170/mathml/M151','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/170/mathml/M151">View MathML</a>. As

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/170/mathml/M199','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/170/mathml/M199">View MathML</a>

(2.30)

and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/170/mathml/M200','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/170/mathml/M200">View MathML</a> in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/170/mathml/M80','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/170/mathml/M80">View MathML</a>, it follows that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/170/mathml/M202','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/170/mathml/M202">View MathML</a>

(2.31)

Notice that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/170/mathml/M203','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/170/mathml/M203">View MathML</a> has a unique decomposition

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/170/mathml/M204','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/170/mathml/M204">View MathML</a>

(2.32)

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/170/mathml/M205','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/170/mathml/M205">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/170/mathml/M206','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/170/mathml/M206">View MathML</a>. Since <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/170/mathml/M152','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/170/mathml/M152">View MathML</a> on <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/170/mathml/M37','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/170/mathml/M37">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/170/mathml/M209','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/170/mathml/M209">View MathML</a>, we have from (2.32) that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/170/mathml/M210','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/170/mathml/M210">View MathML</a>.

Choose <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/170/mathml/M211','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/170/mathml/M211">View MathML</a> such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/170/mathml/M212','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/170/mathml/M212">View MathML</a>

(2.33)

By (H1), there exists <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/170/mathml/M213','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/170/mathml/M213">View MathML</a> such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/170/mathml/M214','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/170/mathml/M214">View MathML</a>

(2.34)

Therefore, for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/170/mathml/M25','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/170/mathml/M25">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/170/mathml/M216','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/170/mathml/M216">View MathML</a>,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/170/mathml/M217','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/170/mathml/M217">View MathML</a>

(2.35)

Since <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/170/mathml/M196','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/170/mathml/M196">View MathML</a>, there exists <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/170/mathml/M219','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/170/mathml/M219">View MathML</a> such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/170/mathml/M220','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/170/mathml/M220">View MathML</a>

(2.36)

and consequently

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/170/mathml/M221','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/170/mathml/M221">View MathML</a>

(2.37)

Applying Lemma 2.4 and (2.37), it follows that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/170/mathml/M222','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/170/mathml/M222">View MathML</a>

(2.38)

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/170/mathml/M223','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/170/mathml/M223">View MathML</a>

(2.39)

Thus,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/170/mathml/M224','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/170/mathml/M224">View MathML</a>

(2.40)

This contradicts (2.33). □

Corollary 2.8For<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/170/mathml/M185','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/170/mathml/M185">View MathML</a>and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/170/mathml/M226','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/170/mathml/M226">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/170/mathml/M227','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/170/mathml/M227">View MathML</a>.

Proof Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/170/mathml/M228','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/170/mathml/M228">View MathML</a>, where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/170/mathml/M229','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/170/mathml/M229">View MathML</a> is the number asserted in Lemma 2.7. As <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/170/mathml/M140','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/170/mathml/M140">View MathML</a> is bounded in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/170/mathml/M231','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/170/mathml/M231">View MathML</a>, there exists <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/170/mathml/M232','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/170/mathml/M232">View MathML</a> such that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/170/mathml/M233','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/170/mathml/M233">View MathML</a> for all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/170/mathml/M234','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/170/mathml/M234">View MathML</a>. By Lemma 2.7, one has

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/170/mathml/M235','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/170/mathml/M235">View MathML</a>

(2.41)

Hence

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/170/mathml/M236','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/170/mathml/M236">View MathML</a>

(2.42)

 □

Now, using Theorem A, we may prove the following.

Proposition 2.9<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/170/mathml/M237','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/170/mathml/M237">View MathML</a>is a bifurcation interval from the trivial solution for (2.15). There exists an unbounded componentCof a positive solution of (2.15), which meets<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/170/mathml/M238','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/170/mathml/M238">View MathML</a>. Moreover,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/170/mathml/M239','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/170/mathml/M239">View MathML</a>

(2.43)

Proof For fixed <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/170/mathml/M151','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/170/mathml/M151">View MathML</a> with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/170/mathml/M241','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/170/mathml/M241">View MathML</a>, let us take that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/170/mathml/M242','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/170/mathml/M242">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/170/mathml/M243','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/170/mathml/M243">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/170/mathml/M244','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/170/mathml/M244">View MathML</a>. It is easy to check that for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/170/mathml/M245','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/170/mathml/M245">View MathML</a>, all of the conditions of Theorem A are satisfied. So, there exists a connected component <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/170/mathml/M246','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/170/mathml/M246">View MathML</a> of solutions of (2.15) containing <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/170/mathml/M247','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/170/mathml/M247">View MathML</a>, and either

(i) <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/170/mathml/M246','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/170/mathml/M246">View MathML</a> is unbounded, or

(ii) <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/170/mathml/M249','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/170/mathml/M249">View MathML</a>.

By Lemma 2.5, the case (ii) cannot occur. Thus <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/170/mathml/M246','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/170/mathml/M246">View MathML</a> is unbounded bifurcated from <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/170/mathml/M247','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/170/mathml/M247">View MathML</a> in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/170/mathml/M252','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/170/mathml/M252">View MathML</a>. Furthermore, we have from Lemma 2.5 that for any closed interval <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/170/mathml/M253','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/170/mathml/M253">View MathML</a>, if <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/170/mathml/M254','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/170/mathml/M254">View MathML</a>, then <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/170/mathml/M255','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/170/mathml/M255">View MathML</a> in E is impossible. So, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/170/mathml/M246','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/170/mathml/M246">View MathML</a> must be bifurcated from <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/170/mathml/M238','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/170/mathml/M238">View MathML</a> in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/170/mathml/M258','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/170/mathml/M258">View MathML</a>. □

3 Main results

Theorem 3.1Let (A1), (H1), (H2), (H3) hold. Assume that either

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/170/mathml/M259','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/170/mathml/M259">View MathML</a>

(3.1)

or

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/170/mathml/M260','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/170/mathml/M260">View MathML</a>

(3.2)

then problem (1.2) has at least one positive solution.

Proof of Theorem 3.1 It is clear that any solution of (2.15) of the form <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/170/mathml/M261','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/170/mathml/M261">View MathML</a> yields a solution x of (1.2). We will show that C crosses the hyperplane <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/170/mathml/M262','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/170/mathml/M262">View MathML</a> in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/170/mathml/M258','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/170/mathml/M258">View MathML</a>. To do this, it is enough to show that C joins <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/170/mathml/M238','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/170/mathml/M238">View MathML</a> to <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/170/mathml/M265','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/170/mathml/M265">View MathML</a>. Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/170/mathml/M266','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/170/mathml/M266">View MathML</a> satisfy

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/170/mathml/M267','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/170/mathml/M267">View MathML</a>

(3.3)

We note that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/170/mathml/M268','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/170/mathml/M268">View MathML</a> for all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/170/mathml/M151','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/170/mathml/M151">View MathML</a> since <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/170/mathml/M270','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/170/mathml/M270">View MathML</a> is the only solution of (2.15) for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/170/mathml/M271','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/170/mathml/M271">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/170/mathml/M272','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/170/mathml/M272">View MathML</a>.

Case 1. <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/170/mathml/M273','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/170/mathml/M273">View MathML</a>.

In this case, we show that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/170/mathml/M274','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/170/mathml/M274">View MathML</a>

We divide the proof into two steps.

Step 1. We show that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/170/mathml/M275','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/170/mathml/M275">View MathML</a> is bounded.

Since <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/170/mathml/M266','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/170/mathml/M266">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/170/mathml/M277','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/170/mathml/M277">View MathML</a>. From (H3), we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/170/mathml/M278','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/170/mathml/M278">View MathML</a>

(3.4)

Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/170/mathml/M279','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/170/mathml/M279">View MathML</a> denote the nonnegative eigenfunction corresponding to <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/170/mathml/M280','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/170/mathml/M280">View MathML</a>.

From (3.4), we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/170/mathml/M281','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/170/mathml/M281">View MathML</a>

(3.5)

By Lemma 2.4, we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/170/mathml/M282','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/170/mathml/M282">View MathML</a>

(3.6)

Thus

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/170/mathml/M283','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/170/mathml/M283">View MathML</a>

(3.7)

Step 2. We show that C joins <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/170/mathml/M238','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/170/mathml/M238">View MathML</a> to <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/170/mathml/M285','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/170/mathml/M285">View MathML</a>.

From (3.3) and (3.7), we have that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/170/mathml/M286','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/170/mathml/M286">View MathML</a>. Notice that (2.15) is equivalent to the integral equation

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/170/mathml/M287','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/170/mathml/M287">View MathML</a>

(3.8)

which implies that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/170/mathml/M288','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/170/mathml/M288">View MathML</a>

(3.9)

We divide both of (3.9) by <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/170/mathml/M289','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/170/mathml/M289">View MathML</a> and set <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/170/mathml/M290','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/170/mathml/M290">View MathML</a>. Since <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/170/mathml/M291','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/170/mathml/M291">View MathML</a> is bounded in E, there exists a subsequence of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/170/mathml/M292','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/170/mathml/M292">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/170/mathml/M293','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/170/mathml/M293">View MathML</a>, with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/170/mathml/M294','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/170/mathml/M294">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/170/mathml/M295','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/170/mathml/M295">View MathML</a> on <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/170/mathml/M80','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/170/mathml/M80">View MathML</a>, such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/170/mathml/M297','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/170/mathml/M297">View MathML</a>

(3.10)

relabeling if necessary. Thus, (3.9) yields that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/170/mathml/M298','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/170/mathml/M298">View MathML</a>

(3.11)

which implies that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/170/mathml/M299','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/170/mathml/M299">View MathML</a>

(3.12)

Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/170/mathml/M300','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/170/mathml/M300">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/170/mathml/M301','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/170/mathml/M301">View MathML</a> denote the nonnegative eigenfunction corresponding to <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/170/mathml/M302','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/170/mathml/M302">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/170/mathml/M303','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/170/mathml/M303">View MathML</a>, respectively. Then we have, from the first inequality in (3.12),

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/170/mathml/M304','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/170/mathml/M304">View MathML</a>

From Lemma 2.4, integrating by parts, we obtain that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/170/mathml/M305','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/170/mathml/M305">View MathML</a>

and consequently

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/170/mathml/M306','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/170/mathml/M306">View MathML</a>

(3.13)

Similarly, we deduce from the second inequality in (3.12) that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/170/mathml/M307','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/170/mathml/M307">View MathML</a>

(3.14)

Thus

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/170/mathml/M308','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/170/mathml/M308">View MathML</a>

(3.15)

So, C joins <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/170/mathml/M238','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/170/mathml/M238">View MathML</a> to <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/170/mathml/M285','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/170/mathml/M285">View MathML</a>.

Case 2. <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/170/mathml/M311','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/170/mathml/M311">View MathML</a>.

In this case, if <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/170/mathml/M266','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/170/mathml/M266">View MathML</a> is such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/170/mathml/M313','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/170/mathml/M313">View MathML</a>

and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/170/mathml/M314','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/170/mathml/M314">View MathML</a>

then

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/170/mathml/M315','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/170/mathml/M315">View MathML</a>

(3.16)

and, moreover,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/170/mathml/M316','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/170/mathml/M316">View MathML</a>

(3.17)

Assume that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/170/mathml/M275','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/170/mathml/M275">View MathML</a> is bounded; applying a similar argument to that used in Step 2 of Case 1, after taking a subsequence and relabeling if necessary, we obtain

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/170/mathml/M318','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/170/mathml/M318">View MathML</a>

(3.18)

Again C joins <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/170/mathml/M238','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/170/mathml/M238">View MathML</a> to <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/170/mathml/M285','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/170/mathml/M285">View MathML</a> and the result follows. □

Competing interests

The authors declare that they have no competing interests.

Authors’ contributions

WS conceived of the study, and participated in its design and coordination and helped to draft the manuscript. TH drafted the manuscript. All authors read and approved the final manuscript.

Acknowledgements

This work is supported by the NSF of Gansu Province (No. 1114-04).

References

  1. Agarwal, RP, Chow, YM: Iterative methods for a fourth-order boundary value problem. J. Comput. Appl. Math.. 10(2), 203–217 (1984). Publisher Full Text OpenURL

  2. Ma, R, Wu, HP: Positive solutions of a fourth-order two-point boundary value problem. Acta Math. Sci., Ser. A. 22(2), 244–249 (2002)

  3. Yao, Q: Positive solutions for eigenvalue problems of fourth-order elastic beam equations. Appl. Math. Lett.. 17(2), 237–243 (2004). Publisher Full Text OpenURL

  4. Yao, Q: Solvability of an elastic beam equation with Caratheodory function. Math. Appl.. 17(3), 389–392 (2004)

  5. Korman, P: Uniqueness and exact multiplicity of solutions for a class of fourth-order semilinear problems. Proc. R. Soc. Edinb. A. 134(1), 179–190 (2004). Publisher Full Text OpenURL

  6. Xu, J, Han, XL: Nodal solutions for a fourth-order two-point boundary value problem. Bound. Value Probl.. 2010, Article ID 570932 (2010)

  7. Shen, WG: Existence of nodal solutions of a nonlinear fourth-order two-point boundary value problem. Bound. Value Probl. (2012). BioMed Central Full Text OpenURL

  8. Elias, U: Eigenvalue problems for the equations <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/170/mathml/M321','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/170/mathml/M321">View MathML</a>. J. Differ. Equ.. 29(1), 28–57 (1978). Publisher Full Text OpenURL

  9. Elias, U: Oscillation Theory of Two-Term Differential Equations, Kluwer Academic, Dordrecht (1997)

  10. Rabinowitz, PH: Some aspects of nonlinear eigenvalue problems. Rocky Mt. J. Math.. 3, 161–202 (1973). Publisher Full Text OpenURL

  11. Ma, R: Existence of positive solutions of a fourth-order boundary value problem. Appl. Math. Comput.. 168(2), 1219–1231 (2005). Publisher Full Text OpenURL

  12. Ma, R: Nodal solutions for a fourth-order two-point boundary value problem. J. Math. Anal. Appl.. 314(1), 254–265 (2006). Publisher Full Text OpenURL

  13. Ma, R: Nodal solutions of boundary value problem of fourth-order ordinary differential equations. J. Math. Anal. Appl.. 319(2), 424–434 (2006). Publisher Full Text OpenURL

  14. Ma, R, Thompson, B: Nodal solutions for a nonlinear fourth-order eigenvalue problem. Acta Math. Sin. Engl. Ser.. 24(1), 27–34 (2008). Publisher Full Text OpenURL

  15. Ma, R, Xu, J: Bifurcation from interval and positive solutions of a fourth-order boundary value problem. Nonlinear Anal., Theory Methods Appl.. 72(1), 113–122 (2010). Publisher Full Text OpenURL

  16. Bai, Z, Wang, H: On positive solutions of some nonlinear fourth-order beam equations. J. Math. Anal. Appl.. 270(1), 357–368 (2006). Publisher Full Text OpenURL

  17. Ma, R, Gao, CH, Han, XL: On linear and nonlinear fourth-order eigenvalue problems with indefinite weight. Nonlinear Anal., Theory Methods Appl.. 74(18), 6965–6969 (2011). Publisher Full Text OpenURL

  18. Ma, R, Gao, CH: Nodal solutions of a nonlinear eigenvalue problem of the Euler-Bernoulli equation. J. Math. Anal. Appl.. 387(2), 1160–1166 (2012). Publisher Full Text OpenURL

  19. Ma, R, Chen, TL: Existence of positive solutions of fourth-order problems with integral boundary conditions. Bound. Value Probl.. 2011, Article ID 297578 (2011)

  20. Ma, R, Xu, L: Existence of positive solutions of a nonlinear fourth-order boundary value problem. Appl. Math. Lett.. 23(5), 537–543 (2010). Publisher Full Text OpenURL