Open Access Research

Solvability of right focal boundary value problems with superlinear growth conditions

Minghe Pei1, Sung Kag Chang2* and Young Sun Oh3

Author Affiliations

1 Department of Mathematics, Beihua University, JiLin City, 132013, P.R. China

2 Department of Mathematics, Yeungnam University, Kyongsan, 712-749, Korea

3 Department of Mathematics Education, Daegu University, Kyongsan, 712-714, Korea

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Boundary Value Problems 2012, 2012:60  doi:10.1186/1687-2770-2012-60


The electronic version of this article is the complete one and can be found online at: http://www.boundaryvalueproblems.com/content/2012/1/60


Received:5 March 2012
Accepted:2 May 2012
Published:22 June 2012

© 2012 Pei et al.; licensee Springer

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

Abstract

In this paper, we consider nth-order two-point right focal boundary value problems

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/60/mathml/M2','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/60/mathml/M2">View MathML</a> is a <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/60/mathml/M3','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/60/mathml/M3">View MathML</a>-Carathéodory (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/60/mathml/M4','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/60/mathml/M4">View MathML</a>) function and satisfies superlinear growth conditions. The existence and uniqueness of solutions for the above right focal boundary value problems are obtained by Leray-Schauder continuation theorem and analytical technique. Meanwhile, as an application of our results, examples are given.

MSC: 34B15.

Keywords:
right focal boundary value problem; Leray-Schauder continuation theorem; existence; uniqueness

1 Introduction

In this paper, we shall discuss the existence and uniqueness of solutions of right focal boundary value problems for nth-order nonlinear differential equation

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/60/mathml/M5','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/60/mathml/M5">View MathML</a>

(1.1)

subject to the boundary conditions (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/60/mathml/M6','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/60/mathml/M6">View MathML</a>)

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/60/mathml/M7','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/60/mathml/M7">View MathML</a>

(1.2)

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/60/mathml/M8','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/60/mathml/M8">View MathML</a> satisfies the <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/60/mathml/M3','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/60/mathml/M3">View MathML</a>-Carathéodory (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/60/mathml/M4','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/60/mathml/M4">View MathML</a>) conditions, that is,

(i) for each <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/60/mathml/M11','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/60/mathml/M11">View MathML</a>, the function <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/60/mathml/M12','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/60/mathml/M12">View MathML</a> is measurable on <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/60/mathml/M13','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/60/mathml/M13">View MathML</a>;

(ii) for a.e. <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/60/mathml/M14','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/60/mathml/M14">View MathML</a>, the function <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/60/mathml/M15','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/60/mathml/M15">View MathML</a> is continuous on <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/60/mathml/M16','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/60/mathml/M16">View MathML</a>;

(iii) for each <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/60/mathml/M17','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/60/mathml/M17">View MathML</a>, there exists an <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/60/mathml/M18','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/60/mathml/M18">View MathML</a> such that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/60/mathml/M19','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/60/mathml/M19">View MathML</a> for a.e. <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/60/mathml/M14','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/60/mathml/M14">View MathML</a> and all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/60/mathml/M11','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/60/mathml/M11">View MathML</a> with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/60/mathml/M22','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/60/mathml/M22">View MathML</a>.

As it is well known, the right focal boundary value problems have attracted many scholars’ attention. Among a substantial number of works dealing with right focal boundary value problems, we mention [1-16,18-25].

Recently, using the Leray-Schauder continuation theorem, Hopkins and Kosmatov [16] have obtained sufficient conditions for the existence of at least one sign-changing solution for third-order right focal boundary value problems such as

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/60/mathml/M23','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/60/mathml/M23">View MathML</a>

and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/60/mathml/M24','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/60/mathml/M24">View MathML</a>

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/60/mathml/M25','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/60/mathml/M25">View MathML</a> satisfies the <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/60/mathml/M3','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/60/mathml/M3">View MathML</a>-Carathéodory (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/60/mathml/M4','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/60/mathml/M4">View MathML</a>) conditions and the linear growth conditions.

Motivated by [16], in this paper we study the solvability for general nth-order right focal boundary value problems (1.1), (1.2). The existence and uniqueness of sign-changing solutions for the problems are obtained by Leray-Schauder continuation theorem and analytical technique. We note that the nonlinearity of f in our problem allows up to the superlinear growth conditions.

The rest of this paper is organized as follows. In Section 2, we give some lemmas which help to simplify the proofs of our main results. In Section 3, we discuss the existence and uniqueness of sign-changing solutions for nth-order right focal boundary value problems (1.1), (1.2) by Leray-Schauder continuation theorem and analytical technique, and give two examples to demonstrate our results. Our results improve and generalize the corresponding results in [16].

2 Preliminary

In this section, we give some lemmas which help to simplify the presentation of our main results.

Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/60/mathml/M28','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/60/mathml/M28">View MathML</a> denote the space of absolutely continuous functions on <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/60/mathml/M13','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/60/mathml/M13">View MathML</a>, and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/60/mathml/M30','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/60/mathml/M30">View MathML</a> denote the Banach space of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/60/mathml/M31','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/60/mathml/M31">View MathML</a> times continuously differentiable functions defined on <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/60/mathml/M13','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/60/mathml/M13">View MathML</a> with the norm <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/60/mathml/M33','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/60/mathml/M33">View MathML</a>, where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/60/mathml/M34','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/60/mathml/M34">View MathML</a>. Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/60/mathml/M35','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/60/mathml/M35">View MathML</a> be the usual Lebesgue space on <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/60/mathml/M13','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/60/mathml/M13">View MathML</a> with norm <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/60/mathml/M37','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/60/mathml/M37">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/60/mathml/M4','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/60/mathml/M4">View MathML</a>.

For <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/60/mathml/M4','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/60/mathml/M4">View MathML</a>, we introduce the Sobolev space

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/60/mathml/M40','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/60/mathml/M40">View MathML</a>

with the norm <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/60/mathml/M41','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/60/mathml/M41">View MathML</a> . Let us consider a special subspace

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/60/mathml/M42','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/60/mathml/M42">View MathML</a>

Then it is clear that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/60/mathml/M43','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/60/mathml/M43">View MathML</a> is closed in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/60/mathml/M44','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/60/mathml/M44">View MathML</a> and hence is itself a Banach space with the norm <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/60/mathml/M41','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/60/mathml/M41">View MathML</a>.

Lemma 2.1 ([21])

Let<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/60/mathml/M46','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/60/mathml/M46">View MathML</a>be the Green’s function of the differential equation<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/60/mathml/M47','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/60/mathml/M47">View MathML</a>subject to the boundary conditions (1.2). Then

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/60/mathml/M48','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/60/mathml/M48">View MathML</a>

and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/60/mathml/M49','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/60/mathml/M49">View MathML</a>

Lemma 2.2Let<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/60/mathml/M50','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/60/mathml/M50">View MathML</a>. Then the solution of the differential equation

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/60/mathml/M51','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/60/mathml/M51">View MathML</a>

subject to the boundary conditions (1.2) satisfies

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/60/mathml/M52','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/60/mathml/M52">View MathML</a>

(2.1)

where for<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/60/mathml/M53','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/60/mathml/M53">View MathML</a> (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/60/mathml/M54','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/60/mathml/M54">View MathML</a>),

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/60/mathml/M55','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/60/mathml/M55">View MathML</a>

(2.2)

and for<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/60/mathml/M56','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/60/mathml/M56">View MathML</a>,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/60/mathml/M57','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/60/mathml/M57">View MathML</a>

(2.3)

Proof Firstly, let us show the lemma for case <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/60/mathml/M53','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/60/mathml/M53">View MathML</a>. Since

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/60/mathml/M59','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/60/mathml/M59">View MathML</a>

we have that for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/60/mathml/M60','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/60/mathml/M60">View MathML</a>,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/60/mathml/M61','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/60/mathml/M61">View MathML</a>

where, for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/60/mathml/M62','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/60/mathml/M62">View MathML</a>,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/60/mathml/M63','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/60/mathml/M63">View MathML</a>

and for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/60/mathml/M64','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/60/mathml/M64">View MathML</a>,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/60/mathml/M65','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/60/mathml/M65">View MathML</a>

It follows by Hölder’s inequality that, for each <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/60/mathml/M60','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/60/mathml/M60">View MathML</a>,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/60/mathml/M67','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/60/mathml/M67">View MathML</a>

and consequently, for each <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/60/mathml/M60','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/60/mathml/M60">View MathML</a>,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/60/mathml/M69','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/60/mathml/M69">View MathML</a>

(2.4)

But for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/60/mathml/M64','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/60/mathml/M64">View MathML</a>,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/60/mathml/M71','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/60/mathml/M71">View MathML</a>

It follows by (2.4) that for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/60/mathml/M64','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/60/mathml/M64">View MathML</a>,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/60/mathml/M73','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/60/mathml/M73">View MathML</a>

For <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/60/mathml/M62','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/60/mathml/M62">View MathML</a>, by Lemma 2.1, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/60/mathml/M75','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/60/mathml/M75">View MathML</a> is nondecreasing in t, and thus

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/60/mathml/M76','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/60/mathml/M76">View MathML</a>

Hence, by (2.4) we have for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/60/mathml/M62','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/60/mathml/M62">View MathML</a>,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/60/mathml/M78','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/60/mathml/M78">View MathML</a>

In summary,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/60/mathml/M79','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/60/mathml/M79">View MathML</a>

Next, we show the lemma for the case <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/60/mathml/M56','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/60/mathml/M56">View MathML</a>. It is easy to see that for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/60/mathml/M64','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/60/mathml/M64">View MathML</a>,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/60/mathml/M82','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/60/mathml/M82">View MathML</a>

and thus for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/60/mathml/M64','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/60/mathml/M64">View MathML</a>,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/60/mathml/M84','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/60/mathml/M84">View MathML</a>

Also by Lemma 2.1, we have for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/60/mathml/M85','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/60/mathml/M85">View MathML</a>,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/60/mathml/M86','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/60/mathml/M86">View MathML</a>

so that for each <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/60/mathml/M62','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/60/mathml/M62">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/60/mathml/M75','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/60/mathml/M75">View MathML</a> is nondecreasing in t, it follows that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/60/mathml/M89','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/60/mathml/M89">View MathML</a>

(2.5)

Let

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/60/mathml/M90','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/60/mathml/M90">View MathML</a>

Then

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/60/mathml/M91','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/60/mathml/M91">View MathML</a>

Since

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/60/mathml/M92','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/60/mathml/M92">View MathML</a>

we have for each <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/60/mathml/M93','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/60/mathml/M93">View MathML</a>,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/60/mathml/M94','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/60/mathml/M94">View MathML</a>

in particular

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/60/mathml/M95','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/60/mathml/M95">View MathML</a>

so that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/60/mathml/M96','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/60/mathml/M96">View MathML</a> is nondecreasing on <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/60/mathml/M13','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/60/mathml/M13">View MathML</a>. Hence by (2.5), we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/60/mathml/M98','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/60/mathml/M98">View MathML</a>

Thus for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/60/mathml/M62','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/60/mathml/M62">View MathML</a>,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/60/mathml/M100','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/60/mathml/M100">View MathML</a>

In summary,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/60/mathml/M101','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/60/mathml/M101">View MathML</a>

 □

Lemma 2.3 ([17] Leray-Schauder continuation theorem)

LetXbe a real Banach space and let Ω be a bounded open neighbourhood of 0 inX. Let<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/60/mathml/M102','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/60/mathml/M102">View MathML</a>be a completely continuous operator such that for all<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/60/mathml/M103','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/60/mathml/M103">View MathML</a>, and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/60/mathml/M104','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/60/mathml/M104">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/60/mathml/M105','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/60/mathml/M105">View MathML</a>. Then the operator equation

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/60/mathml/M106','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/60/mathml/M106">View MathML</a>

has a solution<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/60/mathml/M107','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/60/mathml/M107">View MathML</a>.

3 Main results

Now we are ready to establish our existence theorems of solutions for nth-order right focal boundary value problems (1.1), (1.2). The Leray-Schauder continuation theorem plays key roles in the proofs.

Theorem 3.1Let<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/60/mathml/M108','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/60/mathml/M108">View MathML</a>satisfy<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/60/mathml/M3','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/60/mathml/M3">View MathML</a>-Carathéodory’s conditions. Suppose that

(i) there exist functions<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/60/mathml/M110','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/60/mathml/M110">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/60/mathml/M60','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/60/mathml/M60">View MathML</a>, and a constant<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/60/mathml/M112','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/60/mathml/M112">View MathML</a>such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/60/mathml/M113','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/60/mathml/M113">View MathML</a>

(3.1)

for a.e. <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/60/mathml/M114','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/60/mathml/M114">View MathML</a>and all<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/60/mathml/M115','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/60/mathml/M115">View MathML</a>;

(ii)

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/60/mathml/M116','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/60/mathml/M116">View MathML</a>

(3.2)

where the constants<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/60/mathml/M117','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/60/mathml/M117">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/60/mathml/M60','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/60/mathml/M60">View MathML</a>are given in Lemma 2.2;

(iii)

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/60/mathml/M119','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/60/mathml/M119">View MathML</a>

(3.3)

where<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/60/mathml/M120','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/60/mathml/M120">View MathML</a>.

Then BVP (1.1), (1.2) has at least one solution in<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/60/mathml/M44','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/60/mathml/M44">View MathML</a>.

Proof We define a linear mapping <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/60/mathml/M122','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/60/mathml/M122">View MathML</a>, by setting for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/60/mathml/M123','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/60/mathml/M123">View MathML</a>,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/60/mathml/M124','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/60/mathml/M124">View MathML</a>

We also define a nonlinear mapping <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/60/mathml/M125','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/60/mathml/M125">View MathML</a> by setting for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/60/mathml/M126','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/60/mathml/M126">View MathML</a>,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/60/mathml/M127','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/60/mathml/M127">View MathML</a>

Then, we note that N is a bounded continuous mapping by Lebesgue’s dominated convergence theorem. It is easy to see that the linear mapping <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/60/mathml/M128','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/60/mathml/M128">View MathML</a> is a one-to-one mapping. Also, let the linear mapping <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/60/mathml/M129','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/60/mathml/M129">View MathML</a> for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/60/mathml/M130','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/60/mathml/M130">View MathML</a> be defined by

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/60/mathml/M131','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/60/mathml/M131">View MathML</a>

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/60/mathml/M46','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/60/mathml/M46">View MathML</a> is the Green’s function of BVP in Lemma 2.1.

Then K satisfies that for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/60/mathml/M130','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/60/mathml/M130">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/60/mathml/M134','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/60/mathml/M134">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/60/mathml/M135','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/60/mathml/M135">View MathML</a>, and also for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/60/mathml/M123','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/60/mathml/M123">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/60/mathml/M137','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/60/mathml/M137">View MathML</a>. Furthermore, it follows easily by using Arzelà-Ascoli theorem that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/60/mathml/M138','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/60/mathml/M138">View MathML</a> is a completely continuous operator.

Here we also note that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/60/mathml/M123','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/60/mathml/M123">View MathML</a> is a solution of BVP (1.1), (1.2) if and only if <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/60/mathml/M140','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/60/mathml/M140">View MathML</a> is a solution of the operator equation

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/60/mathml/M141','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/60/mathml/M141">View MathML</a>

which is equivalent to the operator equation

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/60/mathml/M142','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/60/mathml/M142">View MathML</a>

We now apply the Leray-Schauder continuation theorem to the operator equation <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/60/mathml/M143','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/60/mathml/M143">View MathML</a>. To do this, it is sufficient to verify that the set of all possible solutions of the family of equations

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/60/mathml/M144','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/60/mathml/M144">View MathML</a>

(3.4)

with boundary conditions

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/60/mathml/M145','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/60/mathml/M145">View MathML</a>

(3.5)

is, a priori, bounded in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/60/mathml/M43','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/60/mathml/M43">View MathML</a> by a constant independent of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/60/mathml/M147','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/60/mathml/M147">View MathML</a>.

Suppose <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/60/mathml/M148','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/60/mathml/M148">View MathML</a> is a solution of BVP (3.4), (3.5) for some <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/60/mathml/M147','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/60/mathml/M147">View MathML</a>. Then from (3.4), (3.1) and (2.2) in Lemma 2.2, we obtain

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/60/mathml/M150','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/60/mathml/M150">View MathML</a>

Consequently we obtain

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/60/mathml/M151','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/60/mathml/M151">View MathML</a>

(3.6)

Now we have two cases to consider:

Case 1. <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/60/mathml/M152','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/60/mathml/M152">View MathML</a>. In this case (3.6) becomes <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/60/mathml/M153','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/60/mathml/M153">View MathML</a>, i.e. <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/60/mathml/M154','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/60/mathml/M154">View MathML</a>. Thus from (2.1) in Lemma 2.2, we have that there exists a constant <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/60/mathml/M155','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/60/mathml/M155">View MathML</a> which is independent of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/60/mathml/M147','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/60/mathml/M147">View MathML</a> such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/60/mathml/M157','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/60/mathml/M157">View MathML</a>

(3.7)

Now, let

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/60/mathml/M158','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/60/mathml/M158">View MathML</a>

Then estimate (3.7) show that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/60/mathml/M159','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/60/mathml/M159">View MathML</a> has no fixed point on Ω. Hence KN has a fixed point in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/60/mathml/M160','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/60/mathml/M160">View MathML</a> by the Leray-Schauder continuation theorem.

Case 2. <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/60/mathml/M161','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/60/mathml/M161">View MathML</a>. When <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/60/mathml/M162','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/60/mathml/M162">View MathML</a> in (3.1), it is easy to see that BVP (1.1), (1.2) has the trivial solution <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/60/mathml/M163','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/60/mathml/M163">View MathML</a>. Thus assume <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/60/mathml/M164','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/60/mathml/M164">View MathML</a> and let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/60/mathml/M165','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/60/mathml/M165">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/60/mathml/M166','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/60/mathml/M166">View MathML</a>. Then from (3.6), <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/60/mathml/M167','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/60/mathml/M167">View MathML</a>. It is easy to see that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/60/mathml/M168','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/60/mathml/M168">View MathML</a> has a unique positive solution <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/60/mathml/M169','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/60/mathml/M169">View MathML</a>, say <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/60/mathml/M170','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/60/mathml/M170">View MathML</a>. By (3.3), we have <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/60/mathml/M171','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/60/mathml/M171">View MathML</a> and thus <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/60/mathml/M172','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/60/mathml/M172">View MathML</a> has a minimum positive solution, say <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/60/mathml/M173','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/60/mathml/M173">View MathML</a> which is less than <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/60/mathml/M170','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/60/mathml/M170">View MathML</a> and independent of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/60/mathml/M147','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/60/mathml/M147">View MathML</a>. Hence it follows that if <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/60/mathml/M176','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/60/mathml/M176">View MathML</a>, then

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/60/mathml/M177','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/60/mathml/M177">View MathML</a>

(3.8)

From (2.1) in Lemma 2.2, we get

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/60/mathml/M178','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/60/mathml/M178">View MathML</a>

(3.9)

Now, we let

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/60/mathml/M179','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/60/mathml/M179">View MathML</a>

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/60/mathml/M180','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/60/mathml/M180">View MathML</a>. Then estimates (3.8) and (3.9) show that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/60/mathml/M159','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/60/mathml/M159">View MathML</a> has no fixed point on Ω. Consequently, KN has a fixed point in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/60/mathml/M160','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/60/mathml/M160">View MathML</a> by the Leray-Schauder continuation theorem. This completes the proof of the theorem. □

Corollary 3.1Let conditions (i) and (ii) of Theorem 3.1 hold. If<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/60/mathml/M152','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/60/mathml/M152">View MathML</a>or<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/60/mathml/M161','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/60/mathml/M161">View MathML</a>is small enough, then BVP (1.1), (1.2) has at least one solution in<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/60/mathml/M44','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/60/mathml/M44">View MathML</a>.

Corollary 3.2Let conditions (i) and (ii) of Theorem 3.1 hold. If<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/60/mathml/M186','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/60/mathml/M186">View MathML</a>is small enough, then BVP (1.1), (1.2) has at least one solution in<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/60/mathml/M44','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/60/mathml/M44">View MathML</a>.

Remark 3.1 Theorem 3.1-3.4 in [16] are special cases of above Theorem 3.1.

Next, we give some results on the uniqueness of solutions for BVP (1.1), (1.2).

Theorem 3.2Let<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/60/mathml/M108','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/60/mathml/M108">View MathML</a>satisfy<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/60/mathml/M3','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/60/mathml/M3">View MathML</a>-Carathéodory’s conditions. Suppose that

(i) there exist functions<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/60/mathml/M190','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/60/mathml/M190">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/60/mathml/M60','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/60/mathml/M60">View MathML</a>, and a constant<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/60/mathml/M192','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/60/mathml/M192">View MathML</a>such that

(3.10)

for a.e. <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/60/mathml/M194','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/60/mathml/M194">View MathML</a>and all<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/60/mathml/M195','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/60/mathml/M195">View MathML</a>;

(ii)

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/60/mathml/M196','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/60/mathml/M196">View MathML</a>

(3.11)

where the constants<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/60/mathml/M117','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/60/mathml/M117">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/60/mathml/M60','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/60/mathml/M60">View MathML</a>are given in Lemma 2.2;

(iii)

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/60/mathml/M199','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/60/mathml/M199">View MathML</a>

(3.12)

where<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/60/mathml/M120','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/60/mathml/M120">View MathML</a>.

Then BVP (1.1), (1.2) has at least one solution<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/60/mathml/M201','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/60/mathml/M201">View MathML</a>and in particular has at most one solution<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/60/mathml/M201','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/60/mathml/M201">View MathML</a>with<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/60/mathml/M203','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/60/mathml/M203">View MathML</a>.

Proof We note that assumption (3.10) implies

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/60/mathml/M204','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/60/mathml/M204">View MathML</a>

for a.e. <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/60/mathml/M205','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/60/mathml/M205">View MathML</a> and all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/60/mathml/M115','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/60/mathml/M115">View MathML</a>. Accordingly from Theorem 3.1, BVP (1.1), (1.2) has at least one solution in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/60/mathml/M44','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/60/mathml/M44">View MathML</a>.

Now, suppose that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/60/mathml/M208','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/60/mathml/M208">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/60/mathml/M209','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/60/mathml/M209">View MathML</a> are two solutions of BVP (1.1), (1.2) with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/60/mathml/M210','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/60/mathml/M210">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/60/mathml/M211','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/60/mathml/M211">View MathML</a>. Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/60/mathml/M212','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/60/mathml/M212">View MathML</a>. Then <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/60/mathml/M213','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/60/mathml/M213">View MathML</a> satisfies the boundary condition (1.2) and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/60/mathml/M214','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/60/mathml/M214">View MathML</a>

Similarly to the proof of Theorem 3.1, we can show easily that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/60/mathml/M215','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/60/mathml/M215">View MathML</a>

which gives

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/60/mathml/M216','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/60/mathml/M216">View MathML</a>

(3.13)

Now consider two cases. If <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/60/mathml/M152','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/60/mathml/M152">View MathML</a>, then <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/60/mathml/M218','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/60/mathml/M218">View MathML</a> from (3.13). Since <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/60/mathml/M219','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/60/mathml/M219">View MathML</a>, we have <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/60/mathml/M220','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/60/mathml/M220">View MathML</a> on <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/60/mathml/M13','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/60/mathml/M13">View MathML</a>, i.e., <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/60/mathml/M222','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/60/mathml/M222">View MathML</a> on <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/60/mathml/M13','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/60/mathml/M13">View MathML</a>.

If <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/60/mathml/M161','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/60/mathml/M161">View MathML</a>, let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/60/mathml/M225','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/60/mathml/M225">View MathML</a>. Then <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/60/mathml/M226','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/60/mathml/M226">View MathML</a> from (3.13). It follows that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/60/mathml/M227','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/60/mathml/M227">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/60/mathml/M228','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/60/mathml/M228">View MathML</a> on <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/60/mathml/M229','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/60/mathml/M229">View MathML</a>. Since <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/60/mathml/M230','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/60/mathml/M230">View MathML</a>, we get <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/60/mathml/M218','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/60/mathml/M218">View MathML</a>. Consequently, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/60/mathml/M222','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/60/mathml/M222">View MathML</a> on <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/60/mathml/M13','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/60/mathml/M13">View MathML</a>. This completes the proof of the theorem. □

Corollary 3.3Let conditions (i) and (ii) of Theorem 3.2 hold. If<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/60/mathml/M152','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/60/mathml/M152">View MathML</a>, then BVP (1.1), (1.2) has exactly one solution in<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/60/mathml/M44','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/60/mathml/M44">View MathML</a>.

Finally, we give two examples to which our results can be applicable.

Example 3.1 Consider the boundary value problem

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/60/mathml/M236','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/60/mathml/M236">View MathML</a>

Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/60/mathml/M237','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/60/mathml/M237">View MathML</a>. Then it is easy to see that f satisfies <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/60/mathml/M238','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/60/mathml/M238">View MathML</a>-Carathéodory’s conditions. By the inequality <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/60/mathml/M239','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/60/mathml/M239">View MathML</a> for any <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/60/mathml/M240','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/60/mathml/M240">View MathML</a> with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/60/mathml/M241','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/60/mathml/M241">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/60/mathml/M54','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/60/mathml/M54">View MathML</a>, we get

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/60/mathml/M243','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/60/mathml/M243">View MathML</a>

Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/60/mathml/M244','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/60/mathml/M244">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/60/mathml/M245','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/60/mathml/M245">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/60/mathml/M246','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/60/mathml/M246">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/60/mathml/M247','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/60/mathml/M247">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/60/mathml/M248','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/60/mathml/M248">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/60/mathml/M249','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/60/mathml/M249">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/60/mathml/M250','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/60/mathml/M250">View MathML</a>. Then we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/60/mathml/M251','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/60/mathml/M251">View MathML</a>

It is easy to compute that

Consequently, we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/60/mathml/M253','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/60/mathml/M253">View MathML</a>

and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/60/mathml/M254','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/60/mathml/M254">View MathML</a>

Thus by Theorem 3.1, the above boundary value problem has at least one solution in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/60/mathml/M255','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/60/mathml/M255">View MathML</a>.

Example 3.2 Consider the boundary value problem

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/60/mathml/M256','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/60/mathml/M256">View MathML</a>

where

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/60/mathml/M257','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/60/mathml/M257">View MathML</a>

Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/60/mathml/M258','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/60/mathml/M258">View MathML</a>. Then it is easy to see that f satisfies <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/60/mathml/M238','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/60/mathml/M238">View MathML</a>-Carathéodory’s conditions and

Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/60/mathml/M261','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/60/mathml/M261">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/60/mathml/M262','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/60/mathml/M262">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/60/mathml/M263','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/60/mathml/M263">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/60/mathml/M247','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/60/mathml/M247">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/60/mathml/M265','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/60/mathml/M265">View MathML</a>. Then it is easy to compute that

Consequently, we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/60/mathml/M267','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/60/mathml/M267">View MathML</a>

Since <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/60/mathml/M268','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/60/mathml/M268">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/60/mathml/M250','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/60/mathml/M250">View MathML</a>, we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/60/mathml/M270','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/60/mathml/M270">View MathML</a>

Thus by Theorem 3.2, the above boundary value problem has at least one solution <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/60/mathml/M271','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/60/mathml/M271">View MathML</a> and in particular has at most one solution <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/60/mathml/M272','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/60/mathml/M272">View MathML</a> with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/60/mathml/M273','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/60/mathml/M273">View MathML</a>.

Also, since from the equation of the boundary value problem we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/60/mathml/M274','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/60/mathml/M274">View MathML</a>

it follows that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/60/mathml/M275','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/60/mathml/M275">View MathML</a>

Hence above boundary value problem has a unique solution <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/60/mathml/M272','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/60/mathml/M272">View MathML</a>.

Competing interests

The authors declare that they have no competing interests.

Authors’ contributions

MP carried out most of calculations and manuscript preparation. SKC carried out literature survey and conceived ideas. YSO participated in discussions and coordination. All authors read and approved the final manuscript.

Acknowledgement

SKC was supported by Yeungnam University Research Grants 2012. YSO was supported by Daegu University Research Grants 2010.

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