Open Access Research

Positive solutions for singular boundary value problems involving integral conditions

Liang-Gen Hu

Author Affiliations

Department of Mathematics, Ningbo University, Ningbo, 315211, P.R. China

Boundary Value Problems 2012, 2012:72  doi:10.1186/1687-2770-2012-72


The electronic version of this article is the complete one and can be found online at: http://www.boundaryvalueproblems.com/content/2012/1/72


Received:20 December 2011
Accepted:1 June 2012
Published:5 July 2012

© 2012 Hu; 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 are interested in the following singular boundary value problem:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/72/mathml/M1','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/72/mathml/M1">View MathML</a>

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/72/mathml/M2','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/72/mathml/M2">View MathML</a> is a parameter and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/72/mathml/M3','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/72/mathml/M3">View MathML</a> is the Stieltjes integral. The function <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/72/mathml/M4','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/72/mathml/M4">View MathML</a> and w may be singular at <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/72/mathml/M5','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/72/mathml/M5">View MathML</a> and/or <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/72/mathml/M6','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/72/mathml/M6">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/72/mathml/M7','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/72/mathml/M7">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/72/mathml/M8','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/72/mathml/M8">View MathML</a>. Some a priori estimates and the existence, multiplicity and nonexistence of positive solutions are obtained. Our proofs are based on the method of global continuous theorem, the lower-upper solutions methods and fixed point index theory. Furthermore, we also discuss the interval of parameter μ such that the problem has a positive solution.

Keywords:
singularity; global continuous theorem; solution of boundedness; fixed point index; positive solution

1 Introduction

We are concerned with the second order nonlocal boundary value problem:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/72/mathml/M9','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/72/mathml/M9">View MathML</a>

(1.1)

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/72/mathml/M2','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/72/mathml/M2">View MathML</a> is a parameter and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/72/mathml/M3','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/72/mathml/M3">View MathML</a> is a Stieltjes integral. The function <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/72/mathml/M4','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/72/mathml/M4">View MathML</a> and w may be singular at <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/72/mathml/M5','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/72/mathml/M5">View MathML</a> and/or <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/72/mathml/M6','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/72/mathml/M6">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/72/mathml/M7','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/72/mathml/M7">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/72/mathml/M16','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/72/mathml/M16">View MathML</a>.

Integral boundary conditions and multi-point boundary conditions for differential equations come from many areas of applied mathematics and physics [1-7]. Recently, singular boundary value problems have been extensively considered in a lot of literature [1,2,5,8], since they model many physical phenomena including gas diffusion through porous media, nonlinear diffusion generated by nonlinear sources, chemically reacting systems as well as concentration in chemical or biological problems. In all these problems, positive solutions are very meaningful.

In [1,2], Webb and Infante considered the existence of positive solutions of nonlinear boundary value problem:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/72/mathml/M17','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/72/mathml/M17">View MathML</a>

where hg are nonnegative functions, subjected to the nonlocal boundary conditions

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/72/mathml/M18','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/72/mathml/M18">View MathML</a>

Here <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/72/mathml/M19','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/72/mathml/M19">View MathML</a> is a linear functional on <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/72/mathml/M20','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/72/mathml/M20">View MathML</a> given by

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/72/mathml/M21','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/72/mathml/M21">View MathML</a>

involving a Stieltjes integral with a signed measure, that is, A has bounded variation. They dealt with many boundary conditions given in the literature in a unified way by utilizing the fixed point index theory in cones.

Recently, many researchers were interested in the global structure of positive solutions for the nonlinear boundary value problem (see, e.g., [3,6,7]). In 2009, Ma and An [3] considered the problem (1.1). Assume that

(A0) <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/72/mathml/M22','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/72/mathml/M22">View MathML</a> is nondecreasing and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/72/mathml/M23','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/72/mathml/M23">View MathML</a> is not a constant on <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/72/mathml/M24','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/72/mathml/M24">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/72/mathml/M25','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/72/mathml/M25">View MathML</a> with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/72/mathml/M26','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/72/mathml/M26">View MathML</a>, and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/72/mathml/M27','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/72/mathml/M27">View MathML</a> for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/72/mathml/M28','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/72/mathml/M28">View MathML</a> (for the definition of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/72/mathml/M29','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/72/mathml/M29">View MathML</a>, see (2.1) below).

(A1) <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/72/mathml/M30','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/72/mathml/M30">View MathML</a> is continuous and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/72/mathml/M31','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/72/mathml/M31">View MathML</a> on any subinterval of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/72/mathml/M32','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/72/mathml/M32">View MathML</a>, and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/72/mathml/M33','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/72/mathml/M33">View MathML</a>, where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/72/mathml/M34','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/72/mathml/M34">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/72/mathml/M35','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/72/mathml/M35">View MathML</a>.

(A2) <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/72/mathml/M36','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/72/mathml/M36">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/72/mathml/M37','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/72/mathml/M37">View MathML</a> for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/72/mathml/M38','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/72/mathml/M38">View MathML</a>.

(A3) <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/72/mathml/M39','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/72/mathml/M39">View MathML</a>, where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/72/mathml/M40','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/72/mathml/M40">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/72/mathml/M41','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/72/mathml/M41">View MathML</a>.

They obtained the following main result:

Theorem 1.1 ([3], Theorem 4.1])

Assume that (A0)-(A3) hold. Then there exists a component<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/72/mathml/M42','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/72/mathml/M42">View MathML</a>inwhich joins<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/72/mathml/M43','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/72/mathml/M43">View MathML</a>to<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/72/mathml/M44','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/72/mathml/M44">View MathML</a>, and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/72/mathml/M45','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/72/mathml/M45">View MathML</a>

for some<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/72/mathml/M46','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/72/mathml/M46">View MathML</a>. Moreover, there exists<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/72/mathml/M47','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/72/mathml/M47">View MathML</a>such that (1.1) has at least two positive solutions for<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/72/mathml/M48','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/72/mathml/M48">View MathML</a>. Here<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/72/mathml/M42','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/72/mathml/M42">View MathML</a>joins<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/72/mathml/M50','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/72/mathml/M50">View MathML</a>to<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/72/mathml/M51','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/72/mathml/M51">View MathML</a>such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/72/mathml/M52','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/72/mathml/M52">View MathML</a>

Here, ∑ is the closure of the set of positive solutions of (1.1) on <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/72/mathml/M53','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/72/mathml/M53">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/72/mathml/M54','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/72/mathml/M54">View MathML</a>, and the component of a set M is a maximal connected subset of M.

A natural problem arises: How can we consider the global structure of positive solutions for the case <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/72/mathml/M55','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/72/mathml/M55">View MathML</a>?

In this paper, we first obtain the global structure of positive solutions by the use of global continuous theorem, and some a priori estimates. Applying the analysis technique, we construct the lower and upper solutions. Those combined with the fixed point index theory, the existence, multiplicity and nonexistence of positive solutions to (1.1) in the case <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/72/mathml/M55','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/72/mathml/M55">View MathML</a> are investigated. Finally, we discuss the interval of parameter μ such that the problem (1.1) has positive solutions. The proof of the method which is based on the construction of some bounds of the solution together with global continuous theorem and fixed point index is of independent interest, and is different from the other papers.

This paper is arranged as follows. We will give some hypotheses and lemmas in Section 2. In Section 3, new criteria of the existence, multiplicity and nonexistence of a positive solution are obtained. Moreover, an example is given to illustrate our result.

2 Preliminaries and lemmas

Let X denote the Banach space <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/72/mathml/M20','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/72/mathml/M20">View MathML</a> with the maximum norm

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/72/mathml/M58','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/72/mathml/M58">View MathML</a>

Define <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/72/mathml/M59','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/72/mathml/M59">View MathML</a>, then K is a cone. Let

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/72/mathml/M60','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/72/mathml/M60">View MathML</a>

(2.1)

Denote

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/72/mathml/M61','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/72/mathml/M61">View MathML</a>

(2.2)

Throughout this paper, we suppose that the following conditions hold:

(H0) <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/72/mathml/M22','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/72/mathml/M22">View MathML</a> is nondecreasing, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/72/mathml/M63','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/72/mathml/M63">View MathML</a> on <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/72/mathml/M24','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/72/mathml/M24">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/72/mathml/M26','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/72/mathml/M26">View MathML</a> with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/72/mathml/M25','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/72/mathml/M25">View MathML</a>.

(H1) <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/72/mathml/M4','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/72/mathml/M4">View MathML</a> and w may be singular at <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/72/mathml/M5','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/72/mathml/M5">View MathML</a> and/or 1, satisfying

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/72/mathml/M69','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/72/mathml/M69">View MathML</a>

(2.3)

(H2) <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/72/mathml/M70','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/72/mathml/M70">View MathML</a> (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/72/mathml/M71','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/72/mathml/M71">View MathML</a> obviously holds).

(H3) <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/72/mathml/M72','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/72/mathml/M72">View MathML</a>.

Remark 2.1 It is easy to see from (H0) that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/72/mathml/M73','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/72/mathml/M73">View MathML</a>. Therefore, if we assume that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/72/mathml/M74','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/72/mathml/M74">View MathML</a>, then (2.3) obviously holds.

Define an operator <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/72/mathml/M75','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/72/mathml/M75">View MathML</a> as follows:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/72/mathml/M76','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/72/mathml/M76">View MathML</a>

Assume that the conditions (H0)-(H2) hold, then it is easy to verify that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/72/mathml/M77','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/72/mathml/M77">View MathML</a> is well defined and completely continuous.

Lemma 2.1 ([9] Global continuation theorem)

LetXbe a Banach space and letKbe an order cone inX. Consider the equation

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/72/mathml/M78','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/72/mathml/M78">View MathML</a>

(2.4)

where<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/72/mathml/M79','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/72/mathml/M79">View MathML</a>and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/72/mathml/M80','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/72/mathml/M80">View MathML</a>. Suppose that<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/72/mathml/M81','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/72/mathml/M81">View MathML</a>is completely continuous and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/72/mathml/M82','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/72/mathml/M82">View MathML</a>for all<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/72/mathml/M83','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/72/mathml/M83">View MathML</a>, then<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/72/mathml/M84','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/72/mathml/M84">View MathML</a>, the component of the solution set of (2.4) containing<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/72/mathml/M85','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/72/mathml/M85">View MathML</a>is unbounded.

Remark 2.2

(1) We note that u is a positive solution of the problem (1.1) if and only if <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/72/mathml/M86','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/72/mathml/M86">View MathML</a> on K.

(2) If <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/72/mathml/M87','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/72/mathml/M87">View MathML</a> for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/72/mathml/M88','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/72/mathml/M88">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/72/mathml/M89','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/72/mathml/M89">View MathML</a> for all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/72/mathml/M83','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/72/mathml/M83">View MathML</a>, then we get from Lemma 2.1 that there exists an unbounded continuum <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/72/mathml/M91','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/72/mathml/M91">View MathML</a> emanating from <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/72/mathml/M92','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/72/mathml/M92">View MathML</a> in the closure of the set of positive solutions (1.1) in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/72/mathml/M93','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/72/mathml/M93">View MathML</a>.

Lemma 2.2 ([10])

LetXbe a Banach space, Kan order cone inXand<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/72/mathml/M94','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/72/mathml/M94">View MathML</a>an open bounded set inXwith<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/72/mathml/M95','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/72/mathml/M95">View MathML</a>. Suppose that<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/72/mathml/M96','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/72/mathml/M96">View MathML</a>is a completely continuous operator. If<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/72/mathml/M97','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/72/mathml/M97">View MathML</a>, for all<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/72/mathml/M98','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/72/mathml/M98">View MathML</a>and all<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/72/mathml/M99','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/72/mathml/M99">View MathML</a>, then<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/72/mathml/M100','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/72/mathml/M100">View MathML</a>.

3 Main results

Lemma 3.1Let (H0)-(H3) hold and let<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/72/mathml/M101','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/72/mathml/M101">View MathML</a>with<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/72/mathml/M102','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/72/mathml/M102">View MathML</a>. Then there exists a constant<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/72/mathml/M103','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/72/mathml/M103">View MathML</a>such that for all<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/72/mathml/M104','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/72/mathml/M104">View MathML</a>and all possible positive solutions<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/72/mathml/M105','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/72/mathml/M105">View MathML</a>of (1.1), the inequality<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/72/mathml/M106','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/72/mathml/M106">View MathML</a>holds.

Proof Suppose on the contrary that there exist a sequence <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/72/mathml/M107','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/72/mathml/M107">View MathML</a> and a sequence <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/72/mathml/M108','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/72/mathml/M108">View MathML</a> of the positive solutions of (1.1) corresponding to <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/72/mathml/M109','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/72/mathml/M109">View MathML</a> such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/72/mathml/M110','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/72/mathml/M110">View MathML</a>

Denote <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/72/mathml/M111','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/72/mathml/M111">View MathML</a>. From the concavity of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/72/mathml/M112','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/72/mathml/M112">View MathML</a>, it follows that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/72/mathml/M113','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/72/mathml/M113">View MathML</a>

(3.1)

Choose <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/72/mathml/M114','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/72/mathml/M114">View MathML</a>, where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/72/mathml/M115','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/72/mathml/M115">View MathML</a>. Then we find from (H3) that there exists a constant <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/72/mathml/M116','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/72/mathml/M116">View MathML</a> such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/72/mathml/M117','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/72/mathml/M117">View MathML</a>

(3.2)

Since <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/72/mathml/M118','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/72/mathml/M118">View MathML</a>, we get

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/72/mathml/M119','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/72/mathml/M119">View MathML</a>

This, together with (3.1) and (3.2), implies

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/72/mathml/M120','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/72/mathml/M120">View MathML</a>

(3.3)

Put <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/72/mathml/M121','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/72/mathml/M121">View MathML</a>, for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/72/mathml/M122','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/72/mathml/M122">View MathML</a>. Hence, we know from (3.3) that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/72/mathml/M123','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/72/mathml/M123">View MathML</a>

On the other hand, multiplying (1.1) by ψ and integrating by parts, we obtain that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/72/mathml/M124','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/72/mathml/M124">View MathML</a>

leads to

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/72/mathml/M125','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/72/mathml/M125">View MathML</a>

i.e.,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/72/mathml/M126','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/72/mathml/M126">View MathML</a>

This is a contradiction. □

Lemma 3.2Assume that the hypotheses (H0)-(H3) hold. Then there exists<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/72/mathml/M127','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/72/mathml/M127">View MathML</a>such that if the problem (1.1) has a positive solution for parameterμ, then<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/72/mathml/M128','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/72/mathml/M128">View MathML</a>.

Proof Let u be a positive solution of (1.1) corresponding to μ. From the hypotheses (H2) and (H3), it follows that there exists <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/72/mathml/M129','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/72/mathml/M129">View MathML</a> such that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/72/mathml/M130','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/72/mathml/M130">View MathML</a> for all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/72/mathml/M131','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/72/mathml/M131">View MathML</a>. Consequently, we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/72/mathml/M132','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/72/mathml/M132">View MathML</a>

(3.4)

Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/72/mathml/M133','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/72/mathml/M133">View MathML</a> be the first eigenvalue of

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/72/mathml/M134','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/72/mathml/M134">View MathML</a>

and let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/72/mathml/M135','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/72/mathml/M135">View MathML</a> be the positive eigenfunction corresponding to <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/72/mathml/M133','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/72/mathml/M133">View MathML</a> (see [8]). It is easy to see that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/72/mathml/M137','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/72/mathml/M137">View MathML</a>. Multiplying (3.4) by <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/72/mathml/M135','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/72/mathml/M135">View MathML</a> and integrating by parts, we get that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/72/mathml/M139','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/72/mathml/M139">View MathML</a>

implies

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/72/mathml/M140','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/72/mathml/M140">View MathML</a>

This completes the proof. □

Lemma 3.3Assume that (H0)-(H3) hold. Then we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/72/mathml/M141','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/72/mathml/M141">View MathML</a>

Proof Suppose this fails, that is, there exists <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/72/mathml/M142','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/72/mathml/M142">View MathML</a> such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/72/mathml/M143','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/72/mathml/M143">View MathML</a>

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/72/mathml/M144','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/72/mathml/M144">View MathML</a> is a positive constant. Since <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/72/mathml/M145','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/72/mathml/M145">View MathML</a>, we get that for all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/72/mathml/M146','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/72/mathml/M146">View MathML</a>

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/72/mathml/M147','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/72/mathml/M147">View MathML</a>

where

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/72/mathml/M148','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/72/mathml/M148">View MathML</a>

Hence, we find that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/72/mathml/M149','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/72/mathml/M149">View MathML</a>

Thus, it implies that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/72/mathml/M150','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/72/mathml/M150">View MathML</a>. This is contradiction. □

Theorem 3.1Assume that the conditions (H0)-(H3) hold and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/72/mathml/M151','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/72/mathml/M151">View MathML</a>. Then there exists a constant<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/72/mathml/M152','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/72/mathml/M152">View MathML</a>such that the problem (1.1) has at least two positive solutions for<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/72/mathml/M153','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/72/mathml/M153">View MathML</a>, and at least one positive solution for<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/72/mathml/M154','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/72/mathml/M154">View MathML</a>, and no positive solution for<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/72/mathml/M155','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/72/mathml/M155">View MathML</a>.

Proof Define

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/72/mathml/M156','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/72/mathml/M156">View MathML</a>

(3.5)

From Lemma 2.1 and Remark 2.2, we can find that there exists an unbounded continuum <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/72/mathml/M91','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/72/mathml/M91">View MathML</a> emanating from <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/72/mathml/M92','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/72/mathml/M92">View MathML</a> in the closure of the set of positive solutions in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/72/mathml/M93','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/72/mathml/M93">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/72/mathml/M82','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/72/mathml/M82">View MathML</a> for all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/72/mathml/M83','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/72/mathml/M83">View MathML</a>. Meanwhile, Lemma 3.1 and Lemma 3.3 respectively imply that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/72/mathml/M105','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/72/mathml/M105">View MathML</a> is bounded (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/72/mathml/M2','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/72/mathml/M2">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/72/mathml/M150','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/72/mathml/M150">View MathML</a>) and unbounded (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/72/mathml/M2','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/72/mathml/M2">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/72/mathml/M166','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/72/mathml/M166">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/72/mathml/M167','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/72/mathml/M167">View MathML</a>). Therefore, we conclude that the set of (3.5) is nonempty. Those combined with Lemma 3.2 follows that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/72/mathml/M152','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/72/mathml/M152">View MathML</a> is well defined and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/72/mathml/M169','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/72/mathml/M169">View MathML</a>. From the definition of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/72/mathml/M170','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/72/mathml/M170">View MathML</a>, it is easy to see that the problem (1.1) has at least two positive solutions for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/72/mathml/M171','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/72/mathml/M171">View MathML</a>. Again, since the continuum is a compact connected set and T is a completely continuous operator, the problem (1.1) has at least one positive solution at <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/72/mathml/M154','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/72/mathml/M154">View MathML</a>.

Next, we only show that the problem (1.1) has no positive solution for any <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/72/mathml/M155','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/72/mathml/M155">View MathML</a>. Suppose on the contrary that there exists some <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/72/mathml/M174','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/72/mathml/M174">View MathML</a> (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/72/mathml/M175','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/72/mathml/M175">View MathML</a>) such that the problem (1.1) has a positive solution <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/72/mathml/M176','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/72/mathml/M176">View MathML</a> corresponding to <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/72/mathml/M174','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/72/mathml/M174">View MathML</a>. Then we will prove that the problem (1.1) has at least two positive solutions for any <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/72/mathml/M178','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/72/mathml/M178">View MathML</a> which contradicts the definition of (3.5).

For the sake of obtaining the contradiction, we divide the proof into four steps.

Step 1. Constructing a modified boundary value problem.

Choose arbitrarily a constant <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/72/mathml/M179','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/72/mathml/M179">View MathML</a>. Since f is uniformly continuous on <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/72/mathml/M180','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/72/mathml/M180">View MathML</a>, there exists a constant <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/72/mathml/M181','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/72/mathml/M181">View MathML</a> such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/72/mathml/M182','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/72/mathml/M182">View MathML</a>

(3.6)

where

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/72/mathml/M183','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/72/mathml/M183">View MathML</a>

Denote <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/72/mathml/M184','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/72/mathml/M184">View MathML</a>, for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/72/mathml/M185','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/72/mathml/M185">View MathML</a>. Then we claim that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/72/mathml/M186','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/72/mathml/M186">View MathML</a>

(3.7)

and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/72/mathml/M187','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/72/mathml/M187">View MathML</a>

(3.8)

Indeed, using (3.6), we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/72/mathml/M188','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/72/mathml/M188">View MathML</a>

Define a set <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/72/mathml/M189','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/72/mathml/M189">View MathML</a>. Then the set <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/72/mathml/M94','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/72/mathml/M94">View MathML</a> is bounded and open in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/72/mathml/M20','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/72/mathml/M20">View MathML</a>. Now, we construct the modified second-order boundary value problem:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/72/mathml/M192','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/72/mathml/M192">View MathML</a>

(3.9)

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/72/mathml/M193','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/72/mathml/M193">View MathML</a> is defined by

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/72/mathml/M194','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/72/mathml/M194">View MathML</a>

Step 2. We will show that if u is a positive solution of (3.9), then <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/72/mathml/M195','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/72/mathml/M195">View MathML</a>.

Let u be a positive solution of (3.9), then we claim that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/72/mathml/M196','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/72/mathml/M196">View MathML</a>

(3.10)

Suppose this fails, that is, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/72/mathml/M197','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/72/mathml/M197">View MathML</a>. Clearly, we only show that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/72/mathml/M198','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/72/mathml/M198">View MathML</a>, for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/72/mathml/M199','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/72/mathml/M199">View MathML</a>. Comparing the boundary conditions (3.8) and (3.9), the only following three cases need to be considered. Case I. There exists <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/72/mathml/M200','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/72/mathml/M200">View MathML</a> such that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/72/mathml/M201','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/72/mathml/M201">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/72/mathml/M202','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/72/mathml/M202">View MathML</a>, for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/72/mathml/M203','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/72/mathml/M203">View MathML</a> and some <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/72/mathml/M204','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/72/mathml/M204">View MathML</a>; Case II. There exists <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/72/mathml/M205','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/72/mathml/M205">View MathML</a> such that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/72/mathml/M206','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/72/mathml/M206">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/72/mathml/M207','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/72/mathml/M207">View MathML</a> for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/72/mathml/M208','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/72/mathml/M208">View MathML</a>. Case III. There exists <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/72/mathml/M209','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/72/mathml/M209">View MathML</a> such that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/72/mathml/M210','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/72/mathml/M210">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/72/mathml/M211','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/72/mathml/M211">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/72/mathml/M212','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/72/mathml/M212">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/72/mathml/M213','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/72/mathml/M213">View MathML</a>. See the three Figures 1, 2 and 3.

thumbnailFigure 1. Case I.

thumbnailFigure 2. Case II.

thumbnailFigure 3. Case III.

Case I. From (3.7), it implies that there exists a constant <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/72/mathml/M214','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/72/mathml/M214">View MathML</a> such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/72/mathml/M215','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/72/mathml/M215">View MathML</a>

(3.11)

Since f is uniformly continuous on <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/72/mathml/M180','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/72/mathml/M180">View MathML</a>, there exists a <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/72/mathml/M217','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/72/mathml/M217">View MathML</a> such that if <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/72/mathml/M218','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/72/mathml/M218">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/72/mathml/M219','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/72/mathml/M219">View MathML</a>, then we get

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/72/mathml/M220','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/72/mathml/M220">View MathML</a>

(3.12)

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/72/mathml/M221','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/72/mathml/M221">View MathML</a>. From the assumption of Case I, it follows that there exists a subinterval <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/72/mathml/M222','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/72/mathml/M222">View MathML</a> with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/72/mathml/M223','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/72/mathml/M223">View MathML</a> such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/72/mathml/M224','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/72/mathml/M224">View MathML</a>

and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/72/mathml/M225','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/72/mathml/M225">View MathML</a>

This together with (3.11) and (3.12) leads to

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/72/mathml/M226','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/72/mathml/M226">View MathML</a>

This is a contradiction.

Case II. Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/72/mathml/M227','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/72/mathml/M227">View MathML</a>, for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/72/mathml/M185','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/72/mathml/M185">View MathML</a>. Then we have that for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/72/mathml/M208','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/72/mathml/M208">View MathML</a>

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/72/mathml/M230','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/72/mathml/M230">View MathML</a>

(3.13)

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/72/mathml/M231','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/72/mathml/M231">View MathML</a>

(3.14)

Obviously, we have <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/72/mathml/M232','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/72/mathml/M232">View MathML</a>. From (3.13), it implies that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/72/mathml/M233','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/72/mathml/M233">View MathML</a>, for any <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/72/mathml/M234','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/72/mathml/M234">View MathML</a>. Hence, the function <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/72/mathml/M235','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/72/mathml/M235">View MathML</a> is strictly increasing in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/72/mathml/M236','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/72/mathml/M236">View MathML</a>. From (3.8) and the boundary condition (3.9), we find

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/72/mathml/M237','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/72/mathml/M237">View MathML</a>

We obtain a contradiction. In particular, if <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/72/mathml/M238','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/72/mathml/M238">View MathML</a>, then we obtain

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/72/mathml/M239','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/72/mathml/M239">View MathML</a>

This contradicts (3.13).

Case III. Since <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/72/mathml/M240','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/72/mathml/M240">View MathML</a>, for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/72/mathml/M241','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/72/mathml/M241">View MathML</a>, we have that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/72/mathml/M242','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/72/mathml/M242">View MathML</a>, for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/72/mathml/M241','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/72/mathml/M241">View MathML</a>. Again since <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/72/mathml/M244','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/72/mathml/M244">View MathML</a>, we know from the maximum principle that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/72/mathml/M245','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/72/mathml/M245">View MathML</a>, for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/72/mathml/M246','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/72/mathml/M246">View MathML</a>. This contradicts the assumption of Case III.

Therefore, we conclude that the claim (3.10) holds.

Step 3. <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/72/mathml/M247','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/72/mathml/M247">View MathML</a> (the definition of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/72/mathml/M248','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/72/mathml/M248">View MathML</a> see below).

Using (2.2), we define an operator <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/72/mathml/M249','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/72/mathml/M249">View MathML</a> by

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/72/mathml/M250','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/72/mathml/M250">View MathML</a>

Then <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/72/mathml/M251','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/72/mathml/M251">View MathML</a> is completely continuous and u is a positive solution of (3.9) if and only if <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/72/mathml/M252','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/72/mathml/M252">View MathML</a> on K. From the definition of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/72/mathml/M253','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/72/mathml/M253">View MathML</a>, it implies that there exists <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/72/mathml/M254','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/72/mathml/M254">View MathML</a> such that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/72/mathml/M255','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/72/mathml/M255">View MathML</a>, for all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/72/mathml/M83','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/72/mathml/M83">View MathML</a>. Consequently, we get from Lemma 2.2 that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/72/mathml/M257','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/72/mathml/M257">View MathML</a>

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/72/mathml/M258','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/72/mathml/M258">View MathML</a>. Applying the conclusion of Step 2 and the excision property of fixed point index, we find that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/72/mathml/M259','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/72/mathml/M259">View MathML</a>

(3.15)

Step 4. We conclude that the problem (1.1) has at least two positive solutions corresponding to μ.

Since the problem (1.1) is equivalent to the problem (3.9) on <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/72/mathml/M260','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/72/mathml/M260">View MathML</a>, we get that the problem (1.1) has a positive solution in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/72/mathml/M260','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/72/mathml/M260">View MathML</a>. Without loss of generality, we may suppose that T has no fixed point on <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/72/mathml/M262','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/72/mathml/M262">View MathML</a> (otherwise the proof is completed). Then <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/72/mathml/M263','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/72/mathml/M263">View MathML</a> is well defined and (3.15) implies

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/72/mathml/M264','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/72/mathml/M264">View MathML</a>

(3.16)

On the other hand, from Lemma 3.2, we choose <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/72/mathml/M265','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/72/mathml/M265">View MathML</a> such that the problem (1.1) has no positive solution in K. By apriori estimate in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/72/mathml/M266','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/72/mathml/M266">View MathML</a>, there exists <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/72/mathml/M267','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/72/mathml/M267">View MathML</a> such that for all possible positive solutions <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/72/mathml/M268','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/72/mathml/M268">View MathML</a> of (1.1) with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/72/mathml/M269','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/72/mathml/M269">View MathML</a>, we know that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/72/mathml/M270','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/72/mathml/M270">View MathML</a>. Define <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/72/mathml/M271','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/72/mathml/M271">View MathML</a> by

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/72/mathml/M272','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/72/mathml/M272">View MathML</a>

Then it is easy to verify that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/72/mathml/M273','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/72/mathml/M273">View MathML</a> is completely continuous on <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/72/mathml/M274','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/72/mathml/M274">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/72/mathml/M275','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/72/mathml/M275">View MathML</a> for all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/72/mathml/M276','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/72/mathml/M276">View MathML</a>. From the property of homotopy invariance, it follows that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/72/mathml/M277','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/72/mathml/M277">View MathML</a>

Hence, by the additivity property and (3.16), we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/72/mathml/M278','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/72/mathml/M278">View MathML</a>

Then we conclude that the problem (1.1) has at least two positive solutions corresponding to μ. □

Remark 3.1

(i) From the hypotheses (H2) and (H3), it implies that there exists <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/72/mathml/M279','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/72/mathml/M279">View MathML</a> such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/72/mathml/M280','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/72/mathml/M280">View MathML</a>

(3.17)

Let f attain its maximum at the point <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/72/mathml/M281','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/72/mathml/M281">View MathML</a> of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/72/mathml/M282','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/72/mathml/M282">View MathML</a>. If <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/72/mathml/M283','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/72/mathml/M283">View MathML</a>, then adopting the similar method as in [11], Theorem 1], we get that for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/72/mathml/M284','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/72/mathml/M284">View MathML</a>, the problem (1.1) has at least two positive solutions <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/72/mathml/M285','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/72/mathml/M285">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/72/mathml/M286','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/72/mathml/M286">View MathML</a> such that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/72/mathml/M287','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/72/mathml/M287">View MathML</a> by the use of compression of conical shells in [12], Corollary 20.1]. Consequently, we know that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/72/mathml/M288','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/72/mathml/M288">View MathML</a>.

(ii) If u is a positive solution of the equation (1.1) corresponding to μ, then we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/72/mathml/M289','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/72/mathml/M289">View MathML</a>

i.e., from (3.17),

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/72/mathml/M290','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/72/mathml/M290">View MathML</a>

Therefore, we get that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/72/mathml/M291','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/72/mathml/M291">View MathML</a>.

Corollary 3.1Assume that (H1)-(H3) hold. Consider the followingm-point boundary value problem

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/72/mathml/M292','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/72/mathml/M292">View MathML</a>

(3.18)

whereμis a positive parameter, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/72/mathml/M293','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/72/mathml/M293">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/72/mathml/M294','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/72/mathml/M294">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/72/mathml/M295','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/72/mathml/M295">View MathML</a>and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/72/mathml/M296','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/72/mathml/M296">View MathML</a>. Then there exists a constant<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/72/mathml/M297','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/72/mathml/M297">View MathML</a>such that the problem (3.18) has at least two positive solutions for<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/72/mathml/M298','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/72/mathml/M298">View MathML</a>, and at least one positive solution for<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/72/mathml/M299','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/72/mathml/M299">View MathML</a>, and no positive solution for<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/72/mathml/M300','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/72/mathml/M300">View MathML</a>.

Proof In the boundary condition of (1.1), if we let

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/72/mathml/M301','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/72/mathml/M301">View MathML</a>

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/72/mathml/M302','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/72/mathml/M302">View MathML</a> is the characteristic function on <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/72/mathml/M303','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/72/mathml/M303">View MathML</a>, i.e.,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/72/mathml/M304','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/72/mathml/M304">View MathML</a>

Then the boundary condition of (1.1) reduces to the m-point boundary condition of (3.18). Applying the method of Theorem 3.1, we get the conclusion. □

Example 3.1 We consider the following singular boundary value problem:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/72/mathml/M305','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/72/mathml/M305">View MathML</a>

(3.19)

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/72/mathml/M306','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/72/mathml/M306">View MathML</a>.

Computing yields

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/72/mathml/M307','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/72/mathml/M307">View MathML</a>

We find that for any <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/72/mathml/M308','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/72/mathml/M308">View MathML</a>,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/72/mathml/M309','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/72/mathml/M309">View MathML</a>

Thus, it implies that (2.3) holds. It is easy to verify that the conditions (H2) and (H3) hold. Therefore, by Theorem 3.1, we obtain that there exists a constant <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/72/mathml/M152','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/72/mathml/M152">View MathML</a> such that the problem (3.19) has at least two positive solutions for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/72/mathml/M153','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/72/mathml/M153">View MathML</a>, and at least one positive solution for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/72/mathml/M154','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/72/mathml/M154">View MathML</a>, and no positive solution for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/72/mathml/M155','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/72/mathml/M155">View MathML</a>.

Competing interests

The author declares that they have no competing interests.

Author’s contributions

The author typed, read and approved the final manuscript.

Acknowledgement

The author would like to thank the anonymous referees very much for helpful comments and suggestions which led to the improvement of presentation and quality of the work. The work was supported partly by NSCF of Tianyuan Youth Foundation (No. 11126125), K. C. Wong Magna Fund of Ningbo University, Subject Foundation of Ningbo University (No. xkl11044) and Hulan’s Excellent Doctor Foundation of Ningbo University.

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