This article is part of the series Jean Mawhin’s Achievements in Nonlinear Analysis.

Open Access Research

Eigenvalue criteria for existence of positive solutions of impulsive differential equations with non-separated boundary conditions

Ruixi Liang1* and Jianhua Shen2

Author Affiliations

1 Department of Mathematics and Statistics, Central South University, Changsha, Hunan, 410075, P.R. China

2 Department of Mathematics, Hangzhou Normal University, Hangzhou, Zhejiang, 310036, P.R. China

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Boundary Value Problems 2013, 2013:3  doi:10.1186/1687-2770-2013-3


The electronic version of this article is the complete one and can be found online at: http://www.boundaryvalueproblems.com/content/2013/1/3


Received:29 August 2012
Accepted:28 December 2012
Published:14 January 2013

© 2013 Liang and Shen; 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

In this paper, we discuss the existence of positive solutions for second-order differential equations subject to nonlinear impulsive conditions and non-separated periodic boundary value conditions. Our criteria for the existence of positive solutions will be expressed in terms of the first eigenvalue of the corresponding nonimpulsive problem. The main tool of study is a fixed point theorem in a cone.

MSC: 34B37, 34B18.

Keywords:
impulsive differential equation; positive solution; fixed point theorem; non-separated periodic boundary value condition

1 Introduction

Let ω be a fixed positive number. In this paper, we are concerned with the existence of positive solutions for the following boundary value problem (BVP) with impulses:

(1.1a)

(1.1b)

(1.1c)

Here, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/3/mathml/M4','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/3/mathml/M4">View MathML</a> denotes the quasi-derivative of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/3/mathml/M5','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/3/mathml/M5">View MathML</a>. The condition (1.1c) is called a non-separated periodic boundary value condition for (1.1a).

We assume throughout, and with further mention, that the following conditions hold.

(H1) Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/3/mathml/M6','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/3/mathml/M6">View MathML</a>, and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/3/mathml/M7','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/3/mathml/M7">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/3/mathml/M8','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/3/mathml/M8">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/3/mathml/M9','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/3/mathml/M9">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/3/mathml/M10','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/3/mathml/M10">View MathML</a>. <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/3/mathml/M11','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/3/mathml/M11">View MathML</a>, where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/3/mathml/M12','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/3/mathml/M12">View MathML</a> (respectively <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/3/mathml/M13','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/3/mathml/M13">View MathML</a>) denotes the right limit (respectively, the left limit) of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/3/mathml/M14','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/3/mathml/M14">View MathML</a> at <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/3/mathml/M15','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/3/mathml/M15">View MathML</a>.

(H)

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/3/mathml/M16','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/3/mathml/M16">View MathML</a>

A function <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/3/mathml/M5','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/3/mathml/M5">View MathML</a> defined on <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/3/mathml/M18','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/3/mathml/M18">View MathML</a> is called a solution of BVP (1.1) ((1.1a)-(1.1c)) if its first derivative <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/3/mathml/M19','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/3/mathml/M19">View MathML</a> exists for each <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/3/mathml/M20','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/3/mathml/M20">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/3/mathml/M21','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/3/mathml/M21">View MathML</a> is absolutely continuous on each close subinterval of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/3/mathml/M22','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/3/mathml/M22">View MathML</a>, there exist finite values <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/3/mathml/M23','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/3/mathml/M23">View MathML</a>, the impulse conditions (1.1b) and the boundary conditions (1.1c) are satisfied, and the equation (1.1a) is satisfied almost everywhere on <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/3/mathml/M22','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/3/mathml/M22">View MathML</a>.

For the case of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/3/mathml/M25','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/3/mathml/M25">View MathML</a> (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/3/mathml/M26','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/3/mathml/M26">View MathML</a>), the problem (1.1) is related to a non-separated periodic boundary value problem of ODE. Atici and Guseinov [1] have proved the existence of a positive and twin positive solutions to BVP (1.1) by applying a fixed point theorem for the completely continuous operators in cones. In [2], Graef and Kong studied the following periodic boundary value problem:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/3/mathml/M27','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/3/mathml/M27">View MathML</a>

(1.2)

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/3/mathml/M28','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/3/mathml/M28">View MathML</a>. Based upon the properties of Green’s function obtained in [1], the authors extended and improved the work of [1] by using topological degree theory. They derived new criteria for the existence of non-trivial solutions, positive solutions and negative solutions of the problem (1.2) when f is a sign-changing function and not necessarily bounded from below even over <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/3/mathml/M29','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/3/mathml/M29">View MathML</a>. Very recently, He et al.[3] studied BVP (1.1) without impulses and generalized the results of [1,4] via the fixed point index theory. The problem (1.2) in the case of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/3/mathml/M30','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/3/mathml/M30">View MathML</a>, the usual periodic boundary value problem, has been extensively investigated; see [4-7] for some results.

On the other hand, impulsive differential equations are a basic tool to study processes that are subjected to abrupt changes in their state. There has been a significant development in the last two decades. Boundary problems of second-order differential equations with impulse have received considerable attention and much literature has been published; see, for instance, [8-17] and their references. However, there are fewer results about positive solutions for second-order impulsive differential equations. To our best knowledge, there is no result about nonlinear impulsive differential equations with non-separated periodic boundary conditions.

Motivated by the work above, in this paper we study the existence of positive solutions for the boundary value problem (1.1). By using fixed point theorems in a cone, criteria are established under some conditions on <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/3/mathml/M31','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/3/mathml/M31">View MathML</a> concerning the first eigenvalue corresponding to the relevant linear operator. More important, the impulsive terms are different from those of papers [8,9].

2 Preliminaries

In this section, we collect some preliminary results that will be used in the subsequent section. We denote by <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/3/mathml/M32','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/3/mathml/M32">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/3/mathml/M33','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/3/mathml/M33">View MathML</a> the unique solutions of the corresponding homogeneous equation

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/3/mathml/M34','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/3/mathml/M34">View MathML</a>

(2.1)

under the initial boundary conditions

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/3/mathml/M35','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/3/mathml/M35">View MathML</a>

(2.2)

Put <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/3/mathml/M36','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/3/mathml/M36">View MathML</a>, then by [[1], Lemma 2.3], <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/3/mathml/M37','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/3/mathml/M37">View MathML</a>.

Definition 2.1 For two differential functions y and z, we define their Wronskian by

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/3/mathml/M38','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/3/mathml/M38">View MathML</a>

Theorem 2.1The Wronskian of any two solutions for equations (2.1) is constant. Especially, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/3/mathml/M39','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/3/mathml/M39">View MathML</a>.

Proof Suppose that y and z are two solutions of (2.1), then

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/3/mathml/M40','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/3/mathml/M40">View MathML</a>

therefore, the Wronskian is constant. Further, from the initial conditions (2.2), we have <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/3/mathml/M41','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/3/mathml/M41">View MathML</a>. The proof is complete. □

Consider the following equation:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/3/mathml/M42','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/3/mathml/M42">View MathML</a>

(2.3)

From Theorem 2.5 in [1], equation (2.3) has a Green function <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/3/mathml/M43','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/3/mathml/M43">View MathML</a> for all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/3/mathml/M44','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/3/mathml/M44">View MathML</a>, which has the following properties:

(<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/3/mathml/M45','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/3/mathml/M45">View MathML</a>) <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/3/mathml/M46','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/3/mathml/M46">View MathML</a> is continuous in t and s for all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/3/mathml/M47','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/3/mathml/M47">View MathML</a>.

(<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/3/mathml/M48','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/3/mathml/M48">View MathML</a>) If <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/3/mathml/M49','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/3/mathml/M49">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/3/mathml/M50','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/3/mathml/M50">View MathML</a>, then <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/3/mathml/M51','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/3/mathml/M51">View MathML</a>.

(<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/3/mathml/M52','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/3/mathml/M52">View MathML</a>)

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/3/mathml/M53','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/3/mathml/M53">View MathML</a>

Combining with Theorem 2.1, we can also prove that

(<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/3/mathml/M54','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/3/mathml/M54">View MathML</a>)

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/3/mathml/M55','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/3/mathml/M55">View MathML</a>

Remark 1 From paper [1], we can get <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/3/mathml/M46','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/3/mathml/M46">View MathML</a> when <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/3/mathml/M57','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/3/mathml/M57">View MathML</a> (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/3/mathml/M58','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/3/mathml/M58">View MathML</a>) and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/3/mathml/M59','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/3/mathml/M59">View MathML</a>,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/3/mathml/M60','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/3/mathml/M60">View MathML</a>

Especially, in the case of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/3/mathml/M61','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/3/mathml/M61">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/3/mathml/M62','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/3/mathml/M62">View MathML</a> (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/3/mathml/M58','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/3/mathml/M58">View MathML</a>), Green’s function <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/3/mathml/M46','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/3/mathml/M46">View MathML</a> has the form

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/3/mathml/M65','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/3/mathml/M65">View MathML</a>

Define an operator

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/3/mathml/M66','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/3/mathml/M66">View MathML</a>

then it is easy to check that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/3/mathml/M67','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/3/mathml/M67">View MathML</a> is a completely continuous operator. By virtue of the Krein-Rutman theorem, the authors in [3] got the following result.

Lemma 2.1The spectral radius<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/3/mathml/M68','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/3/mathml/M68">View MathML</a>andThas a positive eigenfunction corresponding to its first eigenvalue<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/3/mathml/M69','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/3/mathml/M69">View MathML</a>.

In what follows, we denote the positive eigenfunction corresponding to <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/3/mathml/M70','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/3/mathml/M70">View MathML</a> by ϕ and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/3/mathml/M71','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/3/mathml/M71">View MathML</a>. Define a mapping Φ and a cone K in a Banach space <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/3/mathml/M72','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/3/mathml/M72">View MathML</a> by

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/3/mathml/M73','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/3/mathml/M73">View MathML</a>

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/3/mathml/M74','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/3/mathml/M74">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/3/mathml/M75','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/3/mathml/M75">View MathML</a>.

Lemma 2.2The fixed point of the mapping Φ is a solution of (1.1).

Proof Clearly, Φu is continuous in t. For <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/3/mathml/M76','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/3/mathml/M76">View MathML</a>,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/3/mathml/M77','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/3/mathml/M77">View MathML</a>

Using (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/3/mathml/M52','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/3/mathml/M52">View MathML</a>) and (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/3/mathml/M54','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/3/mathml/M54">View MathML</a>), we have <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/3/mathml/M80','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/3/mathml/M80">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/3/mathml/M81','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/3/mathml/M81">View MathML</a> and

which implies that the fixed point of Φ is the solution of (1.1). The proof is complete. □

The proofs of the main theorems of this paper are based on fixed point theory. The following two well-known lemmas in [18] are needed in our argument.

Lemma 2.3[18]

LetXbe a Banach space andKbe a cone inX. Suppose<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/3/mathml/M83','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/3/mathml/M83">View MathML</a>and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/3/mathml/M84','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/3/mathml/M84">View MathML</a>are open subsets ofXsuch that<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/3/mathml/M85','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/3/mathml/M85">View MathML</a>, and suppose that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/3/mathml/M86','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/3/mathml/M86">View MathML</a>

is a completely continuous operator such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/3/mathml/M87','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/3/mathml/M87">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/3/mathml/M88','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/3/mathml/M88">View MathML</a>for<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/3/mathml/M89','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/3/mathml/M89">View MathML</a>and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/3/mathml/M90','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/3/mathml/M90">View MathML</a>, and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/3/mathml/M88','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/3/mathml/M88">View MathML</a>for<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/3/mathml/M92','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/3/mathml/M92">View MathML</a>and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/3/mathml/M93','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/3/mathml/M93">View MathML</a>, or

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/3/mathml/M94','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/3/mathml/M94">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/3/mathml/M88','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/3/mathml/M88">View MathML</a>for<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/3/mathml/M92','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/3/mathml/M92">View MathML</a>and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/3/mathml/M90','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/3/mathml/M90">View MathML</a>, and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/3/mathml/M88','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/3/mathml/M88">View MathML</a>for<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/3/mathml/M89','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/3/mathml/M89">View MathML</a>and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/3/mathml/M93','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/3/mathml/M93">View MathML</a>.

Then Φ has a fixed point in<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/3/mathml/M101','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/3/mathml/M101">View MathML</a>.

Lemma 2.4[18]

LetXbe a Banach space andKbe a cone inX. Suppose<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/3/mathml/M83','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/3/mathml/M83">View MathML</a>and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/3/mathml/M84','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/3/mathml/M84">View MathML</a>are open subsets ofXsuch that<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/3/mathml/M85','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/3/mathml/M85">View MathML</a>, and suppose that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/3/mathml/M105','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/3/mathml/M105">View MathML</a>

is a completely continuous operator such that

There exists<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/3/mathml/M106','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/3/mathml/M106">View MathML</a>such that<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/3/mathml/M107','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/3/mathml/M107">View MathML</a>for<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/3/mathml/M89','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/3/mathml/M89">View MathML</a>and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/3/mathml/M109','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/3/mathml/M109">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/3/mathml/M110','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/3/mathml/M110">View MathML</a>for<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/3/mathml/M92','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/3/mathml/M92">View MathML</a>, or

There exists<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/3/mathml/M106','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/3/mathml/M106">View MathML</a>such that<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/3/mathml/M107','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/3/mathml/M107">View MathML</a>for<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/3/mathml/M92','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/3/mathml/M92">View MathML</a>and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/3/mathml/M109','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/3/mathml/M109">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/3/mathml/M110','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/3/mathml/M110">View MathML</a>for<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/3/mathml/M89','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/3/mathml/M89">View MathML</a>.

Then Φ has a fixed point in<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/3/mathml/M118','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/3/mathml/M118">View MathML</a>.

3 Main results

Recalling that δ was defined after Lemma 2.1, for convenience, we introduce the following notations. Assume that the constant <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/3/mathml/M119','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/3/mathml/M119">View MathML</a> and γ is some positive function on J,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/3/mathml/M120','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/3/mathml/M120">View MathML</a>

Theorem 3.1Assume that there exist positive constantsα, βsuch that<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/3/mathml/M121','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/3/mathml/M121">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/3/mathml/M122','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/3/mathml/M122">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/3/mathml/M123','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/3/mathml/M123">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/3/mathml/M124','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/3/mathml/M124">View MathML</a>and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/3/mathml/M125','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/3/mathml/M125">View MathML</a>

(3.1)

Then (1.1) has at least one positive solutionusuch that<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/3/mathml/M126','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/3/mathml/M126">View MathML</a>.

Proof Clearly, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/3/mathml/M127','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/3/mathml/M127">View MathML</a>, let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/3/mathml/M128','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/3/mathml/M128">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/3/mathml/M129','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/3/mathml/M129">View MathML</a>. Define the open sets

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/3/mathml/M130','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/3/mathml/M130">View MathML</a>

Then <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/3/mathml/M131','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/3/mathml/M131">View MathML</a> is completely continuous. By (3.1) and the definition of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/3/mathml/M132','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/3/mathml/M132">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/3/mathml/M133','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/3/mathml/M133">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/3/mathml/M134','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/3/mathml/M134">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/3/mathml/M135','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/3/mathml/M135">View MathML</a>, there exists <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/3/mathml/M136','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/3/mathml/M136">View MathML</a> such that

(3.2)

(3.3)

and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/3/mathml/M139','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/3/mathml/M139">View MathML</a>

(3.4)

Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/3/mathml/M140','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/3/mathml/M140">View MathML</a>. We show that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/3/mathml/M141','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/3/mathml/M141">View MathML</a>

(3.5)

If not, there exist <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/3/mathml/M142','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/3/mathml/M142">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/3/mathml/M143','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/3/mathml/M143">View MathML</a> such that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/3/mathml/M144','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/3/mathml/M144">View MathML</a>. Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/3/mathml/M145','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/3/mathml/M145">View MathML</a>. Noting that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/3/mathml/M146','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/3/mathml/M146">View MathML</a> for any <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/3/mathml/M147','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/3/mathml/M147">View MathML</a>, we obtain that for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/3/mathml/M147','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/3/mathml/M147">View MathML</a>,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/3/mathml/M149','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/3/mathml/M149">View MathML</a>

which implies that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/3/mathml/M150','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/3/mathml/M150">View MathML</a>, a contradiction.

On the other hand, for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/3/mathml/M151','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/3/mathml/M151">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/3/mathml/M152','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/3/mathml/M152">View MathML</a>, we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/3/mathml/M153','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/3/mathml/M153">View MathML</a>

From Lemma 2.4 it follows that Φ has a fixed point <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/3/mathml/M154','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/3/mathml/M154">View MathML</a>. Furthermore, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/3/mathml/M155','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/3/mathml/M155">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/3/mathml/M156','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/3/mathml/M156">View MathML</a>, which means that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/3/mathml/M5','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/3/mathml/M5">View MathML</a> is a positive solution of Eq. (1.1). The proof is complete. □

In the next theorem, we make use of the eigenvalue <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/3/mathml/M70','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/3/mathml/M70">View MathML</a> and the corresponding eigenfunction ϕ introduced in Lemma 2.1.

Theorem 3.2Assume that there exist positive constantsα, βsuch that<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/3/mathml/M159','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/3/mathml/M159">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/3/mathml/M160','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/3/mathml/M160">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/3/mathml/M123','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/3/mathml/M123">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/3/mathml/M124','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/3/mathml/M124">View MathML</a>and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/3/mathml/M163','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/3/mathml/M163">View MathML</a>

(3.6)

here<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/3/mathml/M164','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/3/mathml/M164">View MathML</a>onJ. Then (1.1) has at least one positive solutionusuch that<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/3/mathml/M165','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/3/mathml/M165">View MathML</a>.

Proof Obviously, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/3/mathml/M127','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/3/mathml/M127">View MathML</a>, put <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/3/mathml/M167','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/3/mathml/M167">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/3/mathml/M129','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/3/mathml/M129">View MathML</a>. Define the open sets

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/3/mathml/M169','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/3/mathml/M169">View MathML</a>

At first, we show that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/3/mathml/M170','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/3/mathml/M170">View MathML</a>. For any <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/3/mathml/M154','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/3/mathml/M154">View MathML</a>, from (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/3/mathml/M48','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/3/mathml/M48">View MathML</a>), we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/3/mathml/M173','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/3/mathml/M173">View MathML</a>

On the other hand,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/3/mathml/M174','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/3/mathml/M174">View MathML</a>

It is easy to check that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/3/mathml/M175','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/3/mathml/M175">View MathML</a> is completely continuous.

Next, we show that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/3/mathml/M176','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/3/mathml/M176">View MathML</a>

(3.7)

If not, there exist <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/3/mathml/M177','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/3/mathml/M177">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/3/mathml/M178','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/3/mathml/M178">View MathML</a> such that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/3/mathml/M179','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/3/mathml/M179">View MathML</a>. Hence,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/3/mathml/M180','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/3/mathml/M180">View MathML</a>

(3.8)

Multiplying the first equation of (3.8) by ϕ and integrating from 0 to ω, we obtain that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/3/mathml/M181','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/3/mathml/M181">View MathML</a>

(3.9)

One can find that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/3/mathml/M182','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/3/mathml/M182">View MathML</a>

(3.10)

Substituting (3.10) into (3.9), we get

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/3/mathml/M183','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/3/mathml/M183">View MathML</a>

Noting that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/3/mathml/M184','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/3/mathml/M184">View MathML</a>, therefore,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/3/mathml/M185','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/3/mathml/M185">View MathML</a>

which implies that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/3/mathml/M186','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/3/mathml/M186">View MathML</a>

a contradiction.

Finally, we show that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/3/mathml/M187','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/3/mathml/M187">View MathML</a>

Since <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/3/mathml/M31','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/3/mathml/M31">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/3/mathml/M189','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/3/mathml/M189">View MathML</a> are negative for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/3/mathml/M190','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/3/mathml/M190">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/3/mathml/M147','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/3/mathml/M147">View MathML</a>, the condition (3.6) implies that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/3/mathml/M192','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/3/mathml/M192">View MathML</a>. Hence, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/3/mathml/M193','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/3/mathml/M193">View MathML</a> for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/3/mathml/M194','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/3/mathml/M194">View MathML</a> and for any <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/3/mathml/M194','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/3/mathml/M194">View MathML</a>,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/3/mathml/M196','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/3/mathml/M196">View MathML</a>

Suppose that there exist <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/3/mathml/M197','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/3/mathml/M197">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/3/mathml/M198','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/3/mathml/M198">View MathML</a> such that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/3/mathml/M179','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/3/mathml/M179">View MathML</a>, that is,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/3/mathml/M200','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/3/mathml/M200">View MathML</a>

(3.11)

Multiplying the first equation of (3.11) by ϕ and integrating from 0 to ω, we obtain that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/3/mathml/M201','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/3/mathml/M201">View MathML</a>

(3.12)

One can get that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/3/mathml/M202','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/3/mathml/M202">View MathML</a>

(3.13)

Substituting (3.13) into (3.12), we get

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/3/mathml/M203','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/3/mathml/M203">View MathML</a>

Noting that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/3/mathml/M184','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/3/mathml/M184">View MathML</a>, therefore,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/3/mathml/M205','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/3/mathml/M205">View MathML</a>

It is impossible for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/3/mathml/M206','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/3/mathml/M206">View MathML</a>. When <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/3/mathml/M207','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/3/mathml/M207">View MathML</a>,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/3/mathml/M208','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/3/mathml/M208">View MathML</a>

a contradiction.

From Lemma 2.3 it follows that Φ has a fixed point <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/3/mathml/M209','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/3/mathml/M209">View MathML</a>. Furthermore, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/3/mathml/M155','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/3/mathml/M155">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/3/mathml/M211','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/3/mathml/M211">View MathML</a>, which means that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/3/mathml/M5','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/3/mathml/M5">View MathML</a> is a positive solution of Eq. (1.1). The proof is complete. □

Corollary 3.1Assume that<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/3/mathml/M213','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/3/mathml/M213">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/3/mathml/M214','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/3/mathml/M214">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/3/mathml/M215','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/3/mathml/M215">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/3/mathml/M216','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/3/mathml/M216">View MathML</a>and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/3/mathml/M217','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/3/mathml/M217">View MathML</a>

or

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/3/mathml/M218','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/3/mathml/M218">View MathML</a>

here<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/3/mathml/M164','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/3/mathml/M164">View MathML</a>onJ. Then (1.1) has at least one positive solution.

Corollary 3.2Assume that there exists a constantαsuch that<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/3/mathml/M220','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/3/mathml/M220">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/3/mathml/M221','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/3/mathml/M221">View MathML</a> (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/3/mathml/M222','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/3/mathml/M222">View MathML</a>, αand ∞) and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/3/mathml/M223','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/3/mathml/M223">View MathML</a>

here<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/3/mathml/M224','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/3/mathml/M224">View MathML</a>onJ. Then there exists one open interval<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/3/mathml/M225','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/3/mathml/M225">View MathML</a>such that (1.1) has at least two positive solutions for<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/3/mathml/M226','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/3/mathml/M226">View MathML</a>.

Example 1 Consider the equation

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/3/mathml/M227','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/3/mathml/M227">View MathML</a>

(3.14)

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/3/mathml/M228','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/3/mathml/M228">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/3/mathml/M229','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/3/mathml/M229">View MathML</a> and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/3/mathml/M230','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/3/mathml/M230">View MathML</a>

here <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/3/mathml/M231','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/3/mathml/M231">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/3/mathml/M232','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/3/mathml/M232">View MathML</a>. Since <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/3/mathml/M233','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/3/mathml/M233">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/3/mathml/M234','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/3/mathml/M234">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/3/mathml/M235','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/3/mathml/M235">View MathML</a>, by Theorem 3.1, (3.14) has at least one positive solution for any <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/3/mathml/M236','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/3/mathml/M236">View MathML</a>.

Example 2 Consider the equation

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/3/mathml/M237','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/3/mathml/M237">View MathML</a>

(3.15)

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/3/mathml/M238','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/3/mathml/M238">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/3/mathml/M239','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/3/mathml/M239">View MathML</a>.

It is well known that, for the problem consisting of the equation <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/3/mathml/M240','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/3/mathml/M240">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/3/mathml/M241','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/3/mathml/M241">View MathML</a>, and the boundary condition

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/3/mathml/M242','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/3/mathml/M242">View MathML</a>

(3.16)

the first eigenvalue is 0 (see, for example, [[19], p.428]). It follows that the first eigenvalue is <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/3/mathml/M243','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/3/mathml/M243">View MathML</a> for the problem consisting of the equation

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/3/mathml/M244','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/3/mathml/M244">View MathML</a>

and the boundary condition (3.16). Meanwhile, we can obtain the positive eigenfunction <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/3/mathml/M245','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/3/mathml/M245">View MathML</a> corresponding to <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/3/mathml/M70','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/3/mathml/M70">View MathML</a>. It is also easy to check that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/3/mathml/M247','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/3/mathml/M247">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/3/mathml/M248','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/3/mathml/M248">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/3/mathml/M249','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/3/mathml/M249">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/3/mathml/M250','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/3/mathml/M250">View MathML</a> (here <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/3/mathml/M251','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/3/mathml/M251">View MathML</a>). So, the right-hand side of the inequality in Corollary 3.2 is obviously satisfied. Considering the monotonicity of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/3/mathml/M31','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/3/mathml/M31">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/3/mathml/M253','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/3/mathml/M253">View MathML</a>, we can choose a sufficiently small positive constant α such that the left-hand side of the inequality is true. Therefore, by a direct application of Corollary 3.2, there exists one open interval <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/3/mathml/M225','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/3/mathml/M225">View MathML</a> such that (3.15) has at least two positive solutions for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/3/mathml/M226','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/3/mathml/M226">View MathML</a>.

Competing interests

The authors declare that they have no competing interests.

Authors’ contributions

All authors contributed equally to the manuscript and read and approved the final manuscript.

Acknowledgements

The authors would like to thank anonymous referees very much for helpful comments and suggestions which led to the improvement of presentation and quality of work. This research was partially supported by the NNSF of China (No. 11001274, 11171085) and the Postdoctoral Science Foundation of Central South University and China (No. 2011M501280).

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