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Existence of solutions for second-order three-point integral boundary value problems at resonance

Hongliang Liu and Zigen Ouyang*

Author Affiliations

School of Nuclear Science and Technology, School of Mathematics and Physics, University of South China, Hengyang, 421001, P.R. China

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Boundary Value Problems 2013, 2013:197  doi:10.1186/1687-2770-2013-197

The electronic version of this article is the complete one and can be found online at: http://www.boundaryvalueproblems.com/content/2013/1/197


Received:30 May 2013
Accepted:13 August 2013
Published:3 September 2013

© 2013 Liu and Ouyang; 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

A class of second-order three-point integral boundary value problems at resonance is investigated in this paper. Using intermediate value theorems, we obtain a sufficient condition for the existence of the solution for the equation. An example is given to demonstrate our main results.

MSC: 34B10, 34B16, 34B18.

Keywords:
integral boundary value problem; resonance; fixed point theorem; intermediate value theorem

1 Introduction

We are interested in the existence of the solutions for the following second-order three-point integral boundary value problems at resonance:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/197/mathml/M1','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/197/mathml/M1">View MathML</a>

(1.1)

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/197/mathml/M2','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/197/mathml/M2">View MathML</a>

(1.2)

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/197/mathml/M3','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/197/mathml/M3">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/197/mathml/M4','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/197/mathml/M4">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/197/mathml/M5','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/197/mathml/M5">View MathML</a>.

In the last few decades, many authors have studied the multi-point boundary value problems for linear and nonlinear ordinary differential equations by using various methods, such as Leray-Schauder fixed point theorem, coincidence degree theory, Krasnosel’skii fixed point theorem, the shooting method and Leggett-Williams fixed point theorem. We refer the readers to [1-10] and references therein. Also, there are a lot of papers dealing with the resonant case for multi-point boundary value problems, see [11-17].

In [18], Infante and Zima studied the existence of solutions for the following n-point boundary value problem with resonance:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/197/mathml/M6','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/197/mathml/M6">View MathML</a>

(1.3)

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/197/mathml/M7','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/197/mathml/M7">View MathML</a>

(1.4)

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/197/mathml/M8','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/197/mathml/M8">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/197/mathml/M9','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/197/mathml/M9">View MathML</a>. Using the Leggett-Williams norm-type theorem, they obtained the existence of a positive solution for problem (1.3)-(1.4).

Problem (1.1)-(1.2) with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/197/mathml/M10','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/197/mathml/M10">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/197/mathml/M11','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/197/mathml/M11">View MathML</a> was studied by Tariboon and Sitthiwirattham in [19]. They obtained the existence of at least one positive solution. In this paper, we are interested in the existence of the solution for problem (1.1)-(1.2) under the condition <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/197/mathml/M4','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/197/mathml/M4">View MathML</a>, which is a resonant case.

In this paper, using some properties of the Green function <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/197/mathml/M13','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/197/mathml/M13">View MathML</a> and intermediate value theorems, we establish a sufficient condition for the existence of positive solutions of problem (1.1)-(1.2).

The rest of the paper is organized as follows. The main results for problem (1.1)-(1.2) under the condition <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/197/mathml/M4','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/197/mathml/M4">View MathML</a> are given in Section 2. In Section 3, we give some lemmas for our results. We prove our main result in Section 4, and finally an example is given to illustrate our result.

2 Some lemmas and main results

In this section, we first introduce some lemmas which will be useful in the proof of our main results.

Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/197/mathml/M15','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/197/mathml/M15">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/197/mathml/M16','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/197/mathml/M16">View MathML</a> equipped with the norm

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/197/mathml/M17','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/197/mathml/M17">View MathML</a>

then Ω is a Banach space.

Lemma 2.1[20]

LetXbe a Banach space with<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/197/mathml/M18','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/197/mathml/M18">View MathML</a>closed and convex. Assume thatUis a relatively open subset ofCwith<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/197/mathml/M19','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/197/mathml/M19">View MathML</a>and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/197/mathml/M20','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/197/mathml/M20">View MathML</a>is completely continuous. Then either

(i) Thas a fixed point in<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/197/mathml/M21','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/197/mathml/M21">View MathML</a>, or

(ii) there exist<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/197/mathml/M22','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/197/mathml/M22">View MathML</a>and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/197/mathml/M23','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/197/mathml/M23">View MathML</a>with<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/197/mathml/M24','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/197/mathml/M24">View MathML</a>.

Lemma 2.2Problem (1.1)-(1.2) is equivalent to the following integral equation:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/197/mathml/M25','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/197/mathml/M25">View MathML</a>

(2.1)

where

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/197/mathml/M26','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/197/mathml/M26">View MathML</a>

(2.2)

Proof Assume that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/197/mathml/M27','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/197/mathml/M27">View MathML</a> is a solution of problem (1.1)-(1.2), then it satisfies the following integral equation:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/197/mathml/M28','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/197/mathml/M28">View MathML</a>

(2.3)

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/197/mathml/M29','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/197/mathml/M29">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/197/mathml/M30','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/197/mathml/M30">View MathML</a> are constants. By the boundary value condition (1.2), we obtain

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/197/mathml/M31','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/197/mathml/M31">View MathML</a>

(2.4)

Combining (2.3) with (2.4), we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/197/mathml/M32','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/197/mathml/M32">View MathML</a>

(2.5)

According to (2.5) it is easy to see that (2.1) holds.

On the other hand, if <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/197/mathml/M27','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/197/mathml/M27">View MathML</a> is a solution of equation (2.1), deriving both sides of (2.1) two order, it is easy to show that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/197/mathml/M27','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/197/mathml/M27">View MathML</a> is also a solution of problem (1.1)-(1.2).

Therefore, problem (1.1)-(1.2) is equivalent to the integral equation (2.1) with the function <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/197/mathml/M13','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/197/mathml/M13">View MathML</a> defined in (2.2). The proof is completed. □

Lemma 2.3For any<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/197/mathml/M36','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/197/mathml/M36">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/197/mathml/M13','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/197/mathml/M13">View MathML</a>is continuous, and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/197/mathml/M38','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/197/mathml/M38">View MathML</a>for any<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/197/mathml/M39','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/197/mathml/M39">View MathML</a>.

Proof The continuity of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/197/mathml/M13','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/197/mathml/M13">View MathML</a> for any <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/197/mathml/M41','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/197/mathml/M41">View MathML</a> is obvious. Let

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/197/mathml/M42','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/197/mathml/M42">View MathML</a>

Here we only need to prove that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/197/mathml/M43','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/197/mathml/M43">View MathML</a> for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/197/mathml/M44','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/197/mathml/M44">View MathML</a>, the rest of the proof is similar. So, from the definition of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/197/mathml/M45','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/197/mathml/M45">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/197/mathml/M10','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/197/mathml/M10">View MathML</a> and the resonant condition <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/197/mathml/M4','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/197/mathml/M4">View MathML</a>, we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/197/mathml/M48','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/197/mathml/M48">View MathML</a>

for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/197/mathml/M49','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/197/mathml/M49">View MathML</a>. The proof is completed. □

Let

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/197/mathml/M50','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/197/mathml/M50">View MathML</a>

(2.6)

Then

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/197/mathml/M51','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/197/mathml/M51">View MathML</a>

(2.7)

Thus, problem (1.1)-(1.2) is equivalent to the following integral equation:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/197/mathml/M52','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/197/mathml/M52">View MathML</a>

(2.8)

By a simple computation, the new Green function <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/197/mathml/M53','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/197/mathml/M53">View MathML</a> has the following properties.

Lemma 2.4For any<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/197/mathml/M36','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/197/mathml/M36">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/197/mathml/M53','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/197/mathml/M53">View MathML</a>is continuous, and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/197/mathml/M56','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/197/mathml/M56">View MathML</a>for any<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/197/mathml/M39','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/197/mathml/M39">View MathML</a>. Furthermore,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/197/mathml/M58','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/197/mathml/M58">View MathML</a>

(2.9)

Lemma 2.5For any<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/197/mathml/M59','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/197/mathml/M59">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/197/mathml/M53','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/197/mathml/M53">View MathML</a>is nonincreasing with respect to<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/197/mathml/M61','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/197/mathml/M61">View MathML</a>, and for any<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/197/mathml/M62','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/197/mathml/M62">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/197/mathml/M63','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/197/mathml/M63">View MathML</a>, and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/197/mathml/M64','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/197/mathml/M64">View MathML</a>for<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/197/mathml/M65','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/197/mathml/M65">View MathML</a>. That is, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/197/mathml/M66','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/197/mathml/M66">View MathML</a>, where

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/197/mathml/M67','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/197/mathml/M67">View MathML</a>

(2.10)

and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/197/mathml/M68','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/197/mathml/M68">View MathML</a>

(2.11)

Let

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/197/mathml/M69','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/197/mathml/M69">View MathML</a>

(2.12)

Then <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/197/mathml/M70','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/197/mathml/M70">View MathML</a>, and equation (2.8) gives

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/197/mathml/M71','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/197/mathml/M71">View MathML</a>

(2.13)

Now we let

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/197/mathml/M72','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/197/mathml/M72">View MathML</a>

(2.14)

Then <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/197/mathml/M73','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/197/mathml/M73">View MathML</a>, and equation (2.13) gives

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/197/mathml/M74','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/197/mathml/M74">View MathML</a>

(2.15)

We replace <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/197/mathml/M75','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/197/mathml/M75">View MathML</a> by any real number μ, then (2.15) can be rewritten as

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/197/mathml/M76','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/197/mathml/M76">View MathML</a>

(2.16)

To present our result, we assume that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/197/mathml/M77','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/197/mathml/M77">View MathML</a> satisfies the following:

(H) <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/197/mathml/M78','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/197/mathml/M78">View MathML</a>and there exist two positive continuous functions<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/197/mathml/M79','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/197/mathml/M79">View MathML</a>such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/197/mathml/M80','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/197/mathml/M80">View MathML</a>

(2.17)

where<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/197/mathml/M81','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/197/mathml/M81">View MathML</a>. Furthermore,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/197/mathml/M82','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/197/mathml/M82">View MathML</a>

(2.18)

for any<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/197/mathml/M83','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/197/mathml/M83">View MathML</a>.

Our results are the following theorems.

Theorem 2.1Assume that (H) holds. If

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/197/mathml/M84','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/197/mathml/M84">View MathML</a>

(2.19)

then problem (1.1)-(1.2) has at least one solution, where

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/197/mathml/M85','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/197/mathml/M85">View MathML</a>

(2.20)

We define an operator T on the set Ω as follows:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/197/mathml/M86','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/197/mathml/M86">View MathML</a>

(2.21)

Lemma 2.6Assume that<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/197/mathml/M5','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/197/mathml/M5">View MathML</a>and (2.19) hold. Then the operatorTis completely continuous in Ω.

Proof It is not difficult to check that T maps Ω into itself. Next, we divide the proof into three steps.

Step 1. <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/197/mathml/M88','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/197/mathml/M88">View MathML</a> is continuous with respect to <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/197/mathml/M89','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/197/mathml/M89">View MathML</a>.

Suppose that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/197/mathml/M90','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/197/mathml/M90">View MathML</a> is a sequence in Ω, and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/197/mathml/M90','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/197/mathml/M90">View MathML</a> converges to <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/197/mathml/M92','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/197/mathml/M92">View MathML</a>. Because of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/197/mathml/M93','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/197/mathml/M93">View MathML</a> being continuous with respect to <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/197/mathml/M94','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/197/mathml/M94">View MathML</a> and from Lemma 2.4, it is obvious that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/197/mathml/M53','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/197/mathml/M53">View MathML</a> is uniformly continuous with respect to <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/197/mathml/M41','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/197/mathml/M41">View MathML</a>. Then, for any positive number ε, there exists an integer N. When <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/197/mathml/M97','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/197/mathml/M97">View MathML</a>, we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/197/mathml/M98','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/197/mathml/M98">View MathML</a>

(2.22)

It follows from (2.21) and (2.22) that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/197/mathml/M99','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/197/mathml/M99">View MathML</a>

Thus the operator T is continuous in Ω.

Step 2. T maps a bounded set in Ω into a bounded set.

Assume that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/197/mathml/M100','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/197/mathml/M100">View MathML</a> is a bounded set with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/197/mathml/M101','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/197/mathml/M101">View MathML</a> for any <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/197/mathml/M102','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/197/mathml/M102">View MathML</a>. Then we have from (2.17) and (2.21) that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/197/mathml/M103','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/197/mathml/M103">View MathML</a>

(2.23)

This implies that the operator T maps a bounded set into a bounded set in Ω.

Step 3. T is equicontinuous in Ω.

It suffices to show that for any <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/197/mathml/M104','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/197/mathml/M104">View MathML</a> and any <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/197/mathml/M105','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/197/mathml/M105">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/197/mathml/M106','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/197/mathml/M106">View MathML</a> as <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/197/mathml/M107','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/197/mathml/M107">View MathML</a>. There are the following three possible cases:

Case (i) <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/197/mathml/M108','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/197/mathml/M108">View MathML</a>;

Case (ii) <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/197/mathml/M109','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/197/mathml/M109">View MathML</a>;

Case (iii) <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/197/mathml/M110','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/197/mathml/M110">View MathML</a>.

We only need to consider case (i) because the proofs of the other two are similar. Since D is bounded, then there exists <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/197/mathml/M111','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/197/mathml/M111">View MathML</a> such that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/197/mathml/M112','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/197/mathml/M112">View MathML</a>. From (2.21), for any <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/197/mathml/M113','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/197/mathml/M113">View MathML</a>, we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/197/mathml/M114','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/197/mathml/M114">View MathML</a>

Because of Step 1 to Step 3, it follows that the operator T is completely continuous in Ω. The proof is completed. □

Lemma 2.7Assume that<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/197/mathml/M5','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/197/mathml/M5">View MathML</a>and (2.17) and (2.19) hold. Then the integral equation (2.16) has at least one solution for any real numberμ.

Proof We only need to present that the operator T is a priori bounded. Set

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/197/mathml/M116','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/197/mathml/M116">View MathML</a>

(2.24)

and define a set <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/197/mathml/M117','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/197/mathml/M117">View MathML</a> as follows:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/197/mathml/M118','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/197/mathml/M118">View MathML</a>

To use Lemma 2.1 to prove the existence of a fixed point of the operator T, we need to show that the second possibility of Lemma 2.1 should not happen.

In fact, assume that there exists <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/197/mathml/M119','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/197/mathml/M119">View MathML</a> with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/197/mathml/M120','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/197/mathml/M120">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/197/mathml/M23','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/197/mathml/M23">View MathML</a> such that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/197/mathml/M122','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/197/mathml/M122">View MathML</a>. It follows that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/197/mathml/M123','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/197/mathml/M123">View MathML</a>

and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/197/mathml/M124','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/197/mathml/M124">View MathML</a>

(2.25)

Here we use the inequality

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/197/mathml/M125','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/197/mathml/M125">View MathML</a>

Obviously, (2.25) contradicts our assumption that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/197/mathml/M126','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/197/mathml/M126">View MathML</a>. Therefore, by Lemma 2.1, it follows that T has a fixed point <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/197/mathml/M127','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/197/mathml/M127">View MathML</a>. Hence, the integral equation (2.21) has at least a solution <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/197/mathml/M128','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/197/mathml/M128">View MathML</a>. The proof is completed. □

3 The proof of Theorem 2.1

In this section, we prove Theorem 2.1 by using Lemmas 2.5-2.7 and the intermediate value theorem.

Proof of Theorem 2.1 From the right-hand side of (2.21), we know that (2.21) is continuously dependent on the parameter μ. So, we just need to find μ such that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/197/mathml/M129','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/197/mathml/M129">View MathML</a>, which implies that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/197/mathml/M130','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/197/mathml/M130">View MathML</a>.

We rewrite (2.16) for any given real number μ as follows:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/197/mathml/M131','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/197/mathml/M131">View MathML</a>

(3.1)

From (3.1), it suffices to show that there exists μ such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/197/mathml/M132','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/197/mathml/M132">View MathML</a>

(3.2)

Obviously, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/197/mathml/M133','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/197/mathml/M133">View MathML</a> is continuously dependent on the parameter μ. Our aim here is to prove that there exists <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/197/mathml/M134','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/197/mathml/M134">View MathML</a> such that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/197/mathml/M135','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/197/mathml/M135">View MathML</a>, we only need to prove that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/197/mathml/M136','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/197/mathml/M136">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/197/mathml/M137','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/197/mathml/M137">View MathML</a>.

Firstly, we prove that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/197/mathml/M138','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/197/mathml/M138">View MathML</a>. On the contrary, we suppose that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/197/mathml/M139','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/197/mathml/M139">View MathML</a>. Then there exists a sequence <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/197/mathml/M140','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/197/mathml/M140">View MathML</a> with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/197/mathml/M141','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/197/mathml/M141">View MathML</a> such that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/197/mathml/M142','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/197/mathml/M142">View MathML</a>, which implies that the sequence <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/197/mathml/M143','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/197/mathml/M143">View MathML</a> is bounded. Notice that the function <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/197/mathml/M93','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/197/mathml/M93">View MathML</a> is continuous with respect to <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/197/mathml/M61','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/197/mathml/M61">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/197/mathml/M94','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/197/mathml/M94">View MathML</a>. So, it is impossible to have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/197/mathml/M147','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/197/mathml/M147">View MathML</a>

(3.3)

as <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/197/mathml/M148','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/197/mathml/M148">View MathML</a> is large enough. Indeed, assume that (3.3) is true. Then by (3.1) we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/197/mathml/M149','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/197/mathml/M149">View MathML</a>

(3.4)

Thus we get that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/197/mathml/M150','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/197/mathml/M150">View MathML</a>

(3.5)

Since we have from (H) that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/197/mathml/M151','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/197/mathml/M151">View MathML</a>

(3.6)

by (3.2), (3.5) and (3.6), we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/197/mathml/M152','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/197/mathml/M152">View MathML</a>

(3.7)

which contradicts our assumption.

Now, for large <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/197/mathml/M148','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/197/mathml/M148">View MathML</a>, we define

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/197/mathml/M154','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/197/mathml/M154">View MathML</a>

Then <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/197/mathml/M155','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/197/mathml/M155">View MathML</a> is not empty.

Secondly, we divide the set <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/197/mathml/M155','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/197/mathml/M155">View MathML</a> into set <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/197/mathml/M157','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/197/mathml/M157">View MathML</a> and set <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/197/mathml/M158','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/197/mathml/M158">View MathML</a> as follows:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/197/mathml/M159','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/197/mathml/M159">View MathML</a>

Obviously, we get that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/197/mathml/M160','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/197/mathml/M160">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/197/mathml/M161','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/197/mathml/M161">View MathML</a>. So, we have from (H) that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/197/mathml/M158','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/197/mathml/M158">View MathML</a> is not empty.

From (H) again, the function <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/197/mathml/M163','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/197/mathml/M163">View MathML</a> is bounded below by a constant for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/197/mathml/M61','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/197/mathml/M61">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/197/mathml/M165','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/197/mathml/M165">View MathML</a>. Thus, there exists a constant M (<0), independent of t and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/197/mathml/M148','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/197/mathml/M148">View MathML</a>, such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/197/mathml/M167','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/197/mathml/M167">View MathML</a>

(3.8)

Let

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/197/mathml/M168','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/197/mathml/M168">View MathML</a>

From the definitions of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/197/mathml/M157','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/197/mathml/M157">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/197/mathml/M158','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/197/mathml/M158">View MathML</a>, we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/197/mathml/M171','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/197/mathml/M171">View MathML</a>

and it follows that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/197/mathml/M172','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/197/mathml/M172">View MathML</a> as <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/197/mathml/M173','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/197/mathml/M173">View MathML</a> (since if <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/197/mathml/M174','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/197/mathml/M174">View MathML</a> is bounded below by a constant as <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/197/mathml/M173','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/197/mathml/M173">View MathML</a>, then (3.7) holds). Therefore, we can choose <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/197/mathml/M176','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/197/mathml/M176">View MathML</a> large enough such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/197/mathml/M177','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/197/mathml/M177">View MathML</a>

(3.9)

for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/197/mathml/M178','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/197/mathml/M178">View MathML</a>. From (H), (3.1), (3.8) and (3.9) and the definitions of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/197/mathml/M157','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/197/mathml/M157">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/197/mathml/M158','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/197/mathml/M158">View MathML</a>, for any <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/197/mathml/M181','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/197/mathml/M181">View MathML</a>, we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/197/mathml/M182','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/197/mathml/M182">View MathML</a>

from which it follows that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/197/mathml/M183','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/197/mathml/M183">View MathML</a>

which implies that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/197/mathml/M184','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/197/mathml/M184">View MathML</a>

This contradicts (3.9). Thus, we have proved that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/197/mathml/M138','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/197/mathml/M138">View MathML</a>. By a similar method, we can also prove that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/197/mathml/M186','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/197/mathml/M186">View MathML</a>.

Notice that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/197/mathml/M187','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/197/mathml/M187">View MathML</a> is continuous with respect to <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/197/mathml/M188','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/197/mathml/M188">View MathML</a>. It follows from the intermediate value theorem [21] that there exists <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/197/mathml/M189','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/197/mathml/M189">View MathML</a> such that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/197/mathml/M190','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/197/mathml/M190">View MathML</a>, that is, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/197/mathml/M191','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/197/mathml/M191">View MathML</a>, which satisfies the second boundary value condition of (1.2). The proof is completed. □

4 Example

In this section, we give an example to illustrate our main result.

Example Consider the boundary value problem

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/197/mathml/M192','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/197/mathml/M192">View MathML</a>

(4.1)

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/197/mathml/M193','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/197/mathml/M193">View MathML</a>

(4.2)

where

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/197/mathml/M194','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/197/mathml/M194">View MathML</a>

So, we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/197/mathml/M195','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/197/mathml/M195">View MathML</a>

and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/197/mathml/M196','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/197/mathml/M196">View MathML</a>

Now we take

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/197/mathml/M197','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/197/mathml/M197">View MathML</a>

It is easy to check that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/197/mathml/M198','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/197/mathml/M198">View MathML</a>

and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/197/mathml/M199','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/197/mathml/M199">View MathML</a>

Thus the conditions of Theorem 2.1 are satisfied. Therefore problem (4.1)-(4.2) has at least a nontrivial solution.

Competing interests

The authors declare that they have no competing interest.

Authors’ contributions

Each of the authors HL and ZO contributes to each part of this study equally and read and approved the final vision of the manuscript.

Acknowledgements

The work was partially supported by the Natural Science Foundation of Hunan Province (No. 13JJ3074), the Foundation of Science and Technology of Hengyang city (No. J1) and the Scientific Research Foundation for Returned Scholars of University of South China (No. 2012XQD43).

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