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Existence of the solutions for a class of nonlinear fractional order three-point boundary value problems with resonance

Zigen Ouyang* and Gangzhao Li

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

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Citation and License

Boundary Value Problems 2012, 2012:68  doi:10.1186/1687-2770-2012-68


The electronic version of this article is the complete one and can be found online at: http://www.boundaryvalueproblems.com/content/2012/1/68


Received:4 December 2011
Accepted:9 May 2012
Published:1 July 2012

© 2012 Ouyang and Li; 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 nonlinear fractional order differential equation

is investigated in this paper, where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/68/mathml/M2','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/68/mathml/M2">View MathML</a> is the standard Riemann-Liouville fractional derivative of order <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/68/mathml/M3','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/68/mathml/M3">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/68/mathml/M4','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/68/mathml/M4">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/68/mathml/M5','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/68/mathml/M5">View MathML</a>. Using intermediate value theorem, we obtain a sufficient condition for the existence of the solutions for the above fractional order differential equations.

1 Introduction

Consider the following boundary value problem

(1.1)

(1.2)

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/68/mathml/M2','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/68/mathml/M2">View MathML</a> is the standard Riemann-Liouville fractional derivative of order <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/68/mathml/M3','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/68/mathml/M3">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/68/mathml/M4','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/68/mathml/M4">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/68/mathml/M5','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/68/mathml/M5">View MathML</a>.

In the last few decades, many authors have investigated fractional differential equations which have been applied in many fields such as physics, mechanics, chemistry, engineering etc. (see references [1,6,10,21-23]). Especially, many works have been devoted to the study of initial value problems and bounded value problems for fractional order differential equations [12,13,15,24].

Recently, the existence of positive solutions of fractional differential equations has attracted many authors’ attention [2-5,8,9,12,14,17-20,25,26]. Using some fixed point theorems, they obtained some nice existence conditions for positive solutions.

In more recent works, Jiang and Yuan [7] considered the following boundary value problem of fractional differential equations

(1.3)

(1.4)

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/68/mathml/M2','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/68/mathml/M2">View MathML</a> is the standard Riemann-Liouville fractional derivative of order <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/68/mathml/M15','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/68/mathml/M15">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/68/mathml/M16','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/68/mathml/M16">View MathML</a> is continuous. Using some properties of the Green function <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/68/mathml/M17','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/68/mathml/M17">View MathML</a>, they obtain some new sufficient conditions for the existence of positive solutions for the above problem.

Further, Li, Luo, and Zhou [4] investigated the following fractional order three-point boundary value problems

(1.5)

(1.6)

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/68/mathml/M2','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/68/mathml/M2">View MathML</a> is the standard Riemann-Liouville fractional derivative of order <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/68/mathml/M3','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/68/mathml/M3">View MathML</a><a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/68/mathml/M22','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/68/mathml/M22">View MathML</a><a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/68/mathml/M23','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/68/mathml/M23">View MathML</a><a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/68/mathml/M24','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/68/mathml/M24">View MathML</a><a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/68/mathml/M25','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/68/mathml/M25">View MathML</a><a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/68/mathml/M26','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/68/mathml/M26">View MathML</a>, and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/68/mathml/M16','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/68/mathml/M16">View MathML</a> is continuous.

In this paper, we discuss the boundary value problem (1.1)-(1.2). Using some properties of the Green function <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/68/mathml/M17','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/68/mathml/M17">View MathML</a> and intermediate value theorem, we establish some sufficient conditions for the existence of the positive solutions of the problem (1.1)-(1.2).

The paper is arranged as follows: In Section 2, we introduce some definitions for fractional order differential equations and give our main results for the boundary value problem (1.1)-(1.2). We give some lemmas for our results in Section 3. In Section 4, we prove our main result; and finally, we give an example to illustrate our results.

2 Main results

In this section, we introduce some definitions and preliminary facts which are used in this paper.

Definition 2.1 ([1,10])

The fractional integral of order α with the lower limit <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/68/mathml/M29','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/68/mathml/M29">View MathML</a> for a function f is defined as

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/68/mathml/M30','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/68/mathml/M30">View MathML</a>

provided that the integral on the right-hand side is point-wise defined on <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/68/mathml/M31','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/68/mathml/M31">View MathML</a>, where Γ is the Gamma function.

Definition 2.2 ([1,10])

Riemann-Liouville derivative of order α with the lower limit <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/68/mathml/M29','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/68/mathml/M29">View MathML</a> for a function <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/68/mathml/M33','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/68/mathml/M33">View MathML</a> can be written as

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/68/mathml/M34','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/68/mathml/M34">View MathML</a>

where n is a positive integer.

We call the function <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/68/mathml/M35','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/68/mathml/M35">View MathML</a> a solution of (1.1)-(1.2) if <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/68/mathml/M36','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/68/mathml/M36">View MathML</a> with a fractional derivative of order α belongs to <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/68/mathml/M37','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/68/mathml/M37">View MathML</a> and satisfies Equation (1.1) and the boundary condition (1.2).

We also need to introduce some lemmas as follows, which will be used in the proof of our main theorems.

Lemma 2.1 ([26])

Assume that<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/68/mathml/M38','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/68/mathml/M38">View MathML</a>with a fractional derivative of order<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/68/mathml/M39','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/68/mathml/M39">View MathML</a>belongs to<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/68/mathml/M40','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/68/mathml/M40">View MathML</a>. Then, the fractional equation

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/68/mathml/M41','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/68/mathml/M41">View MathML</a>

(2.1)
has solutions

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/68/mathml/M42','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/68/mathml/M42">View MathML</a>

(2.2)

Lemma 2.2 ([26])

Assume that<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/68/mathml/M38','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/68/mathml/M38">View MathML</a>with a fractional derivative of order<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/68/mathml/M39','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/68/mathml/M39">View MathML</a>belongs to<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/68/mathml/M40','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/68/mathml/M40">View MathML</a>. Then

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/68/mathml/M46','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/68/mathml/M46">View MathML</a>

(2.3)

for some<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/68/mathml/M47','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/68/mathml/M47">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/68/mathml/M48','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/68/mathml/M48">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/68/mathml/M49','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/68/mathml/M49">View MathML</a>.

Lemma 2.3 ([16])

Suppose thatXbe a Banach space, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/68/mathml/M50','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/68/mathml/M50">View MathML</a>is closed and convex. Assume thatUis a relatively open subset ofCwith<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/68/mathml/M51','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/68/mathml/M51">View MathML</a>, and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/68/mathml/M52','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/68/mathml/M52">View MathML</a>is a completely continuous operator. Then, either

(i) Thas a fixed point in<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/68/mathml/M53','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/68/mathml/M53">View MathML</a>, or

(ii) there exist<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/68/mathml/M54','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/68/mathml/M54">View MathML</a>and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/68/mathml/M55','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/68/mathml/M55">View MathML</a>with<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/68/mathml/M56','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/68/mathml/M56">View MathML</a>.

Throughout this paper, we assume that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/68/mathml/M57','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/68/mathml/M57">View MathML</a> satisfies the following:

(H)<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/68/mathml/M58','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/68/mathml/M58">View MathML</a>, and there exist two positive functions<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/68/mathml/M59','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/68/mathml/M59">View MathML</a>and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/68/mathml/M60','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/68/mathml/M60">View MathML</a>such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/68/mathml/M61','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/68/mathml/M61">View MathML</a>

(2.4)

where<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/68/mathml/M62','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/68/mathml/M62">View MathML</a>. Furthermore,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/68/mathml/M63','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/68/mathml/M63">View MathML</a>

(2.5)

for any<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/68/mathml/M64','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/68/mathml/M64">View MathML</a>.

We have our main results:

Theorem 2.1Suppose that (H) holds. If

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/68/mathml/M65','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/68/mathml/M65">View MathML</a>

(2.6)

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

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/68/mathml/M66','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/68/mathml/M66">View MathML</a>

3 Some lemmas

Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/68/mathml/M67','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/68/mathml/M67">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/68/mathml/M68','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/68/mathml/M68">View MathML</a> equipped the norm

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/68/mathml/M69','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/68/mathml/M69">View MathML</a>

(3.1)

then Ω is a Banach space.

We first give some lemmas as follows:

Lemma 3.1Problem (1.1)-(1.2) is equivalent to the following integral equation

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/68/mathml/M70','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/68/mathml/M70">View MathML</a>

(3.2)
where

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/68/mathml/M71','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/68/mathml/M71">View MathML</a>

(3.3)

Proof The sufficiency is obvious, we only need to prove the necessity.

Suppose that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/68/mathml/M35','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/68/mathml/M35">View MathML</a> is a solution of the problem (1.1)-(1.2). Integrating both sides of (1.1) of α order from 0 to t with respect to t, it follows that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/68/mathml/M73','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/68/mathml/M73">View MathML</a>

(3.4)

According to (1.2) and (3.4), we have

(3.5)

Combining (3.4) and (3.5), we obtain

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/68/mathml/M75','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/68/mathml/M75">View MathML</a>

According to (3.3), it is easy to show that (3.2) holds. The proof is completed. □

Lemma 3.2For any<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/68/mathml/M76','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/68/mathml/M76">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/68/mathml/M17','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/68/mathml/M17">View MathML</a>is continuous, and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/68/mathml/M78','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/68/mathml/M78">View MathML</a>for any<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/68/mathml/M79','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/68/mathml/M79">View MathML</a>.

Proof The continuity of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/68/mathml/M17','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/68/mathml/M17">View MathML</a> for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/68/mathml/M76','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/68/mathml/M76">View MathML</a> is obvious.

Let

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/68/mathml/M82','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/68/mathml/M82">View MathML</a>

we only need to show that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/68/mathml/M83','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/68/mathml/M83">View MathML</a> for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/68/mathml/M84','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/68/mathml/M84">View MathML</a>, the rest of the proof is similar or obvious. From the definition of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/68/mathml/M85','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/68/mathml/M85">View MathML</a>, we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/68/mathml/M86','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/68/mathml/M86">View MathML</a>

for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/68/mathml/M84','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/68/mathml/M84">View MathML</a>. The proof is completed. □

Let

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/68/mathml/M88','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/68/mathml/M88">View MathML</a>

then

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/68/mathml/M89','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/68/mathml/M89">View MathML</a>

(3.6)

The new Green’s function <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/68/mathml/M90','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/68/mathml/M90">View MathML</a> has the following properties:

Lemma 3.3<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/68/mathml/M90','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/68/mathml/M90">View MathML</a>is continuous for<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/68/mathml/M79','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/68/mathml/M79">View MathML</a>, and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/68/mathml/M93','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/68/mathml/M93">View MathML</a>

Furthermore, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/68/mathml/M94','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/68/mathml/M94">View MathML</a>for<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/68/mathml/M79','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/68/mathml/M79">View MathML</a>.

Lemma 3.4For any<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/68/mathml/M96','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/68/mathml/M96">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/68/mathml/M90','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/68/mathml/M90">View MathML</a>is nonincreasing with respect to<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/68/mathml/M98','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/68/mathml/M98">View MathML</a>. Especially, for any<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/68/mathml/M99','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/68/mathml/M99">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/68/mathml/M100','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/68/mathml/M100">View MathML</a>for<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/68/mathml/M101','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/68/mathml/M101">View MathML</a>, and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/68/mathml/M102','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/68/mathml/M102">View MathML</a>for<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/68/mathml/M103','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/68/mathml/M103">View MathML</a>. That is<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/68/mathml/M104','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/68/mathml/M104">View MathML</a>, where

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/68/mathml/M105','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/68/mathml/M105">View MathML</a>

(3.7)
and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/68/mathml/M106','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/68/mathml/M106">View MathML</a>

(3.8)

Let

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/68/mathml/M107','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/68/mathml/M107">View MathML</a>

(3.9)

Then, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/68/mathml/M108','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/68/mathml/M108">View MathML</a>, we have from Lemma 3.1, (3.6) and (3.9) that the integral Equation (3.2) can be rewritten as follows:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/68/mathml/M109','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/68/mathml/M109">View MathML</a>

(3.10)

Let

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/68/mathml/M110','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/68/mathml/M110">View MathML</a>

(3.11)

Then, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/68/mathml/M111','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/68/mathml/M111">View MathML</a> and (3.10) is equivalent to the following

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/68/mathml/M112','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/68/mathml/M112">View MathML</a>

(3.12)

We can divide our proof into the following two steps:

First, we replace <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/68/mathml/M113','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/68/mathml/M113">View MathML</a> by any real number μ, then (3.12) can be rewritten as

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/68/mathml/M114','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/68/mathml/M114">View MathML</a>

(3.13)

It suffices to show that for any given real number μ, (3.13) has a solution <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/68/mathml/M115','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/68/mathml/M115">View MathML</a>, which implies that Equation (1.1) has a solution <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/68/mathml/M35','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/68/mathml/M35">View MathML</a> which satisfies the first boundary value condition <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/68/mathml/M117','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/68/mathml/M117">View MathML</a>.

Second, we show that there exists a μ such that the solution <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/68/mathml/M115','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/68/mathml/M115">View MathML</a> of (3.13) satisfies <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/68/mathml/M111','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/68/mathml/M111">View MathML</a>, which implies that the solution <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/68/mathml/M35','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/68/mathml/M35">View MathML</a> of (1.1) also satisfies the boundary value condition <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/68/mathml/M121','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/68/mathml/M121">View MathML</a>.

In this section, we will prove the first step. For convenience sake, we define an operator T on the set Ω as follows:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/68/mathml/M122','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/68/mathml/M122">View MathML</a>

(3.14)

Lemma 3.5Suppose that<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/68/mathml/M5','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/68/mathml/M5">View MathML</a>, and (2.4) hold, then the operatorTis completely continuous in Ω.

Proof It is easy to show that the operator T maps Ω into itself. We divide the proof into the following three steps.

Step 1. <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/68/mathml/M124','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/68/mathml/M124">View MathML</a> is continuous with respect to <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/68/mathml/M125','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/68/mathml/M125">View MathML</a>.

In fact, suppose that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/68/mathml/M126','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/68/mathml/M126">View MathML</a> is a sequence in Ω, and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/68/mathml/M126','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/68/mathml/M126">View MathML</a> converges to <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/68/mathml/M128','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/68/mathml/M128">View MathML</a>. Since <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/68/mathml/M129','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/68/mathml/M129">View MathML</a> is continuous with respect to <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/68/mathml/M130','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/68/mathml/M130">View MathML</a>, and it is obvious that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/68/mathml/M90','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/68/mathml/M90">View MathML</a> is uniformly continuous with respect to <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/68/mathml/M132','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/68/mathml/M132">View MathML</a> from Lemma 3.3, then for any <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/68/mathml/M133','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/68/mathml/M133">View MathML</a>, there exists an integer N, when <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/68/mathml/M134','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/68/mathml/M134">View MathML</a>,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/68/mathml/M135','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/68/mathml/M135">View MathML</a>

(3.15)

which follows from (3.14)-(3.15) that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/68/mathml/M136','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/68/mathml/M136">View MathML</a>

Thus, the operator T is continuous in Ω.

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

Suppose that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/68/mathml/M137','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/68/mathml/M137">View MathML</a> is a bounded set with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/68/mathml/M138','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/68/mathml/M138">View MathML</a> for any <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/68/mathml/M139','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/68/mathml/M139">View MathML</a>. Then, we have from (2.4) and (3.14) that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/68/mathml/M140','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/68/mathml/M140">View MathML</a>

This gives that the operator T maps bounded set into 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/2012/1/68/mathml/M141','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/68/mathml/M141">View MathML</a> and any <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/68/mathml/M142','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/68/mathml/M142">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/68/mathml/M143','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/68/mathml/M143">View MathML</a> as <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/68/mathml/M144','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/68/mathml/M144">View MathML</a>. We consider the following three cases:

(i) <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/68/mathml/M145','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/68/mathml/M145">View MathML</a>;

(ii) <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/68/mathml/M146','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/68/mathml/M146">View MathML</a>;

(iii) <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/68/mathml/M147','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/68/mathml/M147">View MathML</a>.

We only prove the case (i), the rest two cases are similar. Since B is bounded, then there exists a <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/68/mathml/M148','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/68/mathml/M148">View MathML</a> such that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/68/mathml/M149','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/68/mathml/M149">View MathML</a>. According to (3.14), we have

According to Step 1-Step 3, the operator T is completely continuous in Ω. The proof is completed. □

Further, we have

Lemma 3.6Suppose that<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/68/mathml/M5','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/68/mathml/M5">View MathML</a>, and (2.4) and (2.6) holds, then, for any real numberμ, the integral Equation (3.13) has at least one solution.

Proof We only need to show that the operator T is priori bounded. Let

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/68/mathml/M152','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/68/mathml/M152">View MathML</a>

(3.16)

Define a set <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/68/mathml/M153','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/68/mathml/M153">View MathML</a> as follows

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/68/mathml/M154','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/68/mathml/M154">View MathML</a>

(3.17)

To show the existence of a fixed point of T by Lemma 2.3, we need to verify that the second possibility in Lemma 2.3 cannot happen.

In fact, assume that there exists <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/68/mathml/M155','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/68/mathml/M155">View MathML</a> with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/68/mathml/M156','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/68/mathml/M156">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/68/mathml/M55','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/68/mathml/M55">View MathML</a> such that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/68/mathml/M158','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/68/mathml/M158">View MathML</a>. It follows that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/68/mathml/M159','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/68/mathml/M159">View MathML</a>

and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/68/mathml/M160','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/68/mathml/M160">View MathML</a>

(3.18)

Here we have the use of the inequality

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/68/mathml/M161','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/68/mathml/M161">View MathML</a>

It is obvious that (3.18) contradicts our assumption that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/68/mathml/M156','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/68/mathml/M156">View MathML</a>. Therefore, by Lemma 2.3, it follows that T has a fixed point <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/68/mathml/M163','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/68/mathml/M163">View MathML</a>. Hence, the integral Equation (3.14) has at least a solution <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/68/mathml/M115','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/68/mathml/M115">View MathML</a>. The proof is completed. □

4 The proof of the main results

Now, we prove Theorem 2.1 by Lemma 3.4-3.5 and the intermediate value theorem.

Proof of Theorem 2.1 It is obvious that the right-hand side of (3.14) is continuously dependent on the parameter μ, so we need to find a μ such that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/68/mathml/M111','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/68/mathml/M111">View MathML</a>, which implies that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/68/mathml/M166','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/68/mathml/M166">View MathML</a>.

For any given real number μ, we rewrite (3.13) as follows:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/68/mathml/M167','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/68/mathml/M167">View MathML</a>

(4.1)

From (4.1), it suffices to show that there exists a μ such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/68/mathml/M168','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/68/mathml/M168">View MathML</a>

(4.2)

It is obvious that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/68/mathml/M169','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/68/mathml/M169">View MathML</a> is continuously dependent on the parameter μ. In order to prove that there exists a <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/68/mathml/M170','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/68/mathml/M170">View MathML</a> such that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/68/mathml/M171','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/68/mathml/M171">View MathML</a>, we only need to show that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/68/mathml/M172','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/68/mathml/M172">View MathML</a>, and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/68/mathml/M173','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/68/mathml/M173">View MathML</a>.

Now, we show that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/68/mathml/M172','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/68/mathml/M172">View MathML</a>. On the contrary, we assume that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/68/mathml/M175','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/68/mathml/M175">View MathML</a>. Then, there exists a sequence <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/68/mathml/M176','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/68/mathml/M176">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/68/mathml/M177','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/68/mathml/M177">View MathML</a> such that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/68/mathml/M178','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/68/mathml/M178">View MathML</a>, which implies that the sequence <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/68/mathml/M179','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/68/mathml/M179">View MathML</a> is bounded from above. Notice that the function <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/68/mathml/M129','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/68/mathml/M129">View MathML</a> is continuous with respect to <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/68/mathml/M98','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/68/mathml/M98">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/68/mathml/M130','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/68/mathml/M130">View MathML</a>. We first claim that it is impossible to have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/68/mathml/M183','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/68/mathml/M183">View MathML</a>

(4.3)

as <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/68/mathml/M184','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/68/mathml/M184">View MathML</a> is large enough. Indeed, assume that (4.3) is true. Then, by (4.1), we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/68/mathml/M185','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/68/mathml/M185">View MathML</a>

(4.4)

for all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/68/mathml/M98','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/68/mathml/M98">View MathML</a>. Thus, we get

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/68/mathml/M187','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/68/mathml/M187">View MathML</a>

(4.5)

for all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/68/mathml/M98','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/68/mathml/M98">View MathML</a>. Since we have assumed in (H) that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/68/mathml/M189','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/68/mathml/M189">View MathML</a>

(4.6)

by (4.2), (4.5)-(4.6), we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/68/mathml/M190','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/68/mathml/M190">View MathML</a>

(4.7)

which contradicts our assumption.

Now, for large <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/68/mathml/M184','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/68/mathml/M184">View MathML</a>, we define

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/68/mathml/M192','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/68/mathml/M192">View MathML</a>

Then, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/68/mathml/M193','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/68/mathml/M193">View MathML</a> is not empty.

Further, we divide the set <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/68/mathml/M193','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/68/mathml/M193">View MathML</a> into two sets <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/68/mathml/M195','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/68/mathml/M195">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/68/mathml/M196','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/68/mathml/M196">View MathML</a> as follows:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/68/mathml/M197','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/68/mathml/M197">View MathML</a>

It is easy to know that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/68/mathml/M198','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/68/mathml/M198">View MathML</a>, and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/68/mathml/M199','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/68/mathml/M199">View MathML</a>, and we have from (H) that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/68/mathml/M196','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/68/mathml/M196">View MathML</a> is not empty.

From (H) again, the function <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/68/mathml/M201','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/68/mathml/M201">View MathML</a> is bounded below by a constant for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/68/mathml/M98','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/68/mathml/M98">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/68/mathml/M203','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/68/mathml/M203">View MathML</a>. Thus, there exists a constant M (<0), independent of t and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/68/mathml/M184','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/68/mathml/M184">View MathML</a>, such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/68/mathml/M205','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/68/mathml/M205">View MathML</a>

(4.8)

Let

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/68/mathml/M206','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/68/mathml/M206">View MathML</a>

From the definitions of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/68/mathml/M195','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/68/mathml/M195">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/68/mathml/M196','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/68/mathml/M196">View MathML</a>, we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/68/mathml/M209','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/68/mathml/M209">View MathML</a>

and it follows that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/68/mathml/M210','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/68/mathml/M210">View MathML</a> as <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/68/mathml/M211','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/68/mathml/M211">View MathML</a> (since if <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/68/mathml/M212','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/68/mathml/M212">View MathML</a> is bounded below by a constant as <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/68/mathml/M211','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/68/mathml/M211">View MathML</a>, then (4.7) holds). Therefore, we can choose <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/68/mathml/M214','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/68/mathml/M214">View MathML</a> large enough so that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/68/mathml/M215','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/68/mathml/M215">View MathML</a>

(4.9)

for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/68/mathml/M216','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/68/mathml/M216">View MathML</a>. From (H), (4.1), (4.8)-(4.9), and the definitions of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/68/mathml/M195','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/68/mathml/M195">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/68/mathml/M196','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/68/mathml/M196">View MathML</a>, for any <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/68/mathml/M219','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/68/mathml/M219">View MathML</a>, we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/68/mathml/M220','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/68/mathml/M220">View MathML</a>

from which it follows that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/68/mathml/M221','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/68/mathml/M221">View MathML</a>

which implies that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/68/mathml/M222','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/68/mathml/M222">View MathML</a>

This contradicts (4.9).

Now, we have proved that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/68/mathml/M172','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/68/mathml/M172">View MathML</a>. By a similar method, we can prove that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/68/mathml/M173','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/68/mathml/M173">View MathML</a>. The detail is omitted.

Notice that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/68/mathml/M225','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/68/mathml/M225">View MathML</a> is continuous with respect to <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/68/mathml/M226','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/68/mathml/M226">View MathML</a>. It follows from the intermediate value theorem [11] that there exists a <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/68/mathml/M227','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/68/mathml/M227">View MathML</a> such that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/68/mathml/M228','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/68/mathml/M228">View MathML</a>, that is <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/68/mathml/M229','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/68/mathml/M229">View MathML</a>, which satisfies the second boundary value condition of (1.2). The proof is completed. □

5 Examples

Example 5.1 Consider the following boundary value problem

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/68/mathml/M230','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/68/mathml/M230">View MathML</a>

(5.1)

where

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/68/mathml/M231','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/68/mathml/M231">View MathML</a>

and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/68/mathml/M232','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/68/mathml/M232">View MathML</a>

It is easy to show that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/68/mathml/M233','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/68/mathml/M233">View MathML</a>

and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/68/mathml/M234','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/68/mathml/M234">View MathML</a>

Thus, the conditions of Theorem 2.1 are satisfied. Therefore, the problem (5.1) has at least a nontrivial solution.

Competing interests

The authors declare that they have no competing interests.

Author’s contributions

Each of the authors, ZO and GL contributed to each part of this study equally and read and approved the final version of the mnanuscript.

Acknowledgements

Supported partially by China Postdoctoral Science Foundation under Grant No.20110491280 and the Subject Lead Foundation of University of South China No. 2007XQD13.

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