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The existence of solutions for nonlinear fractional multipoint boundary value problems at resonance

Na Xu, Wenbin Liu* and Lishun Xiao

Author Affiliations

College of Sciences, China University of Mining and Technology, Xuzhou, Jiangsu, 221116, P.R. China

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Boundary Value Problems 2012, 2012:65  doi:10.1186/1687-2770-2012-65


The electronic version of this article is the complete one and can be found online at: http://www.boundaryvalueproblems.com/content/2012/1/65


Received:17 January 2012
Accepted:18 May 2012
Published:28 June 2012

© 2012 Xu 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

A class of nonlinear fractional multipoint boundary value problems at resonance is considered in this article. The existence results are obtained by the method of the coincidence degree theory of Mawhin. An example is given to illustrate the results.

MSC: 34A08.

Keywords:
coincidence degree; fractional differential equation; resonance; multipoint boundary conditions

1 Introduction

The subject of fractional calculus has gained considerable popularity during the past decades, due mainly to its frequent appearance in a variety of different areas such as physics, aerodynamics, polymer rheology, etc. (see [1-3]). Many methods have been introduced for solving fractional differential equations (FDEs for short in the remaining), such as the Laplace transform method, the iteration method, the Fourier transform method, etc. (see [4]).

Recently, there have been many works related to the existence of solutions for multipoint boundary value problems (BVPs for short in the remaining) at nonresonance of FDEs (see [5-11]). Motivated by the above articles and recent studies on FDEs (see [12-19]), we consider the existence of solutions for a nonlinear fractional multipoint BVPs at resonance in this article.

In [16], Zhang and Bai considered the following fractional three-point boundary value problems at resonance:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/65/mathml/M1','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/65/mathml/M1">View MathML</a>

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/65/mathml/M2','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/65/mathml/M2">View MathML</a> is a natural number; <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/65/mathml/M3','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/65/mathml/M3">View MathML</a> is a real number; <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/65/mathml/M4','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/65/mathml/M4">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/65/mathml/M5','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/65/mathml/M5">View MathML</a> are the standard Riemann-Liouville derivative and integral respectively; <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/65/mathml/M6','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/65/mathml/M6">View MathML</a> is continuous; <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/65/mathml/M7','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/65/mathml/M7">View MathML</a>; <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/65/mathml/M8','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/65/mathml/M8">View MathML</a><a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/65/mathml/M9','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/65/mathml/M9">View MathML</a> are given constants such that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/65/mathml/M10','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/65/mathml/M10">View MathML</a>. In their article, they made the operator <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/65/mathml/M11','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/65/mathml/M11">View MathML</a> and got <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/65/mathml/M12','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/65/mathml/M12">View MathML</a>. In [17], Bai discussed fractional m-point boundary value problems at resonance with the case of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/65/mathml/M12','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/65/mathml/M12">View MathML</a>.

In 2010, Bai and Jiang studied the fractional differential equation of boundary value problems at resonance with the case of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/65/mathml/M14','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/65/mathml/M14">View MathML</a> respectively (see [18,19]), and we can see that they obtained the results by the assumption that a specific algebraic expression is not equal to zero; for example,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/65/mathml/M15','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/65/mathml/M15">View MathML</a>

is referred to as a condition in [18]. We will show that the assumption like above <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/65/mathml/M16','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/65/mathml/M16">View MathML</a> is not necessary.

In this article, we will use the coincidence degree theory to study the existence of solutions for a nonlinear FDEs at resonance which is given by

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/65/mathml/M17','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/65/mathml/M17">View MathML</a>

(1.1)

with boundary conditions

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/65/mathml/M18','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/65/mathml/M18">View MathML</a>

(1.2)

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/65/mathml/M19','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/65/mathml/M19">View MathML</a>; <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/65/mathml/M20','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/65/mathml/M20">View MathML</a>; <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/65/mathml/M21','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/65/mathml/M21">View MathML</a>; <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/65/mathml/M22','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/65/mathml/M22">View MathML</a>; <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/65/mathml/M23','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/65/mathml/M23">View MathML</a> with satisfying Carathéodory conditions; <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/65/mathml/M24','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/65/mathml/M24">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/65/mathml/M25','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/65/mathml/M25">View MathML</a> are the standard Riemann-Liouville fractional derivative and fractional integral respectively.

BVPs (1.1)-(1.2) being at resonance means that the associated linear homogeneous equation <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/65/mathml/M26','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/65/mathml/M26">View MathML</a> with boundary conditions (1.2) has <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/65/mathml/M27','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/65/mathml/M27">View MathML</a> as a nontrivial solution, where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/65/mathml/M28','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/65/mathml/M28">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/65/mathml/M29','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/65/mathml/M29">View MathML</a>.

We will always suppose that the following conditions hold:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/65/mathml/M30','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/65/mathml/M30">View MathML</a>

(C)

The rest of this article is organized as follows: In Section 2, we give some definitions, lemmas and notations. In Section 3, we establish theorems of existence result for BVPs (1.1)-(1.2). In Section 4, we give an example to illustrate our result.

2 Preliminaries

We present here some necessary basic knowledge and definitions of the fractional calculus theory, which can be found in [1-3].

Definition 2.1 The Riemann-Liouville fractional integral of order <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/65/mathml/M31','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/65/mathml/M31">View MathML</a> of a function <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/65/mathml/M32','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/65/mathml/M32">View MathML</a> is given by

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/65/mathml/M33','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/65/mathml/M33">View MathML</a>

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/65/mathml/M34','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/65/mathml/M34">View MathML</a> is the Gamma function, provided the right side is pointwise defined on <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/65/mathml/M35','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/65/mathml/M35">View MathML</a>.

Definition 2.2 The Riemann-Liouville fractional derivative of order <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/65/mathml/M31','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/65/mathml/M31">View MathML</a> of a function <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/65/mathml/M32','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/65/mathml/M32">View MathML</a> is given by

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/65/mathml/M38','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/65/mathml/M38">View MathML</a>

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/65/mathml/M39','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/65/mathml/M39">View MathML</a>, provided the right side is pointwise defined on <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/65/mathml/M35','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/65/mathml/M35">View MathML</a>.

Definition 2.3 ([18])

We say that the map <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/65/mathml/M41','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/65/mathml/M41">View MathML</a> satisfies Carathéodory conditions with respect to <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/65/mathml/M42','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/65/mathml/M42">View MathML</a> if the following conditions are satisfied:

(i) for each <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/65/mathml/M43','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/65/mathml/M43">View MathML</a>, the mapping <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/65/mathml/M44','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/65/mathml/M44">View MathML</a> is Lebesgue measurable;

(ii) for almost every <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/65/mathml/M45','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/65/mathml/M45">View MathML</a>, the mapping <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/65/mathml/M44','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/65/mathml/M44">View MathML</a> is continuous on <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/65/mathml/M47','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/65/mathml/M47">View MathML</a>;

(iii) for each <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/65/mathml/M48','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/65/mathml/M48">View MathML</a>, there exists a <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/65/mathml/M49','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/65/mathml/M49">View MathML</a> such that, for a.e. <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/65/mathml/M45','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/65/mathml/M45">View MathML</a> and every <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/65/mathml/M51','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/65/mathml/M51">View MathML</a>, we have <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/65/mathml/M52','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/65/mathml/M52">View MathML</a>.

Lemma 2.4 ([2])

Assume<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/65/mathml/M53','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/65/mathml/M53">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/65/mathml/M54','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/65/mathml/M54">View MathML</a>, then<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/65/mathml/M55','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/65/mathml/M55">View MathML</a>. And, for all<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/65/mathml/M56','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/65/mathml/M56">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/65/mathml/M57','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/65/mathml/M57">View MathML</a>, we have that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/65/mathml/M58','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/65/mathml/M58">View MathML</a>

Lemma 2.5 ([2])

Let<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/65/mathml/M31','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/65/mathml/M31">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/65/mathml/M39','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/65/mathml/M39">View MathML</a>and assume that<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/65/mathml/M61','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/65/mathml/M61">View MathML</a>, then the following equality holds almost everywhere on<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/65/mathml/M62','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/65/mathml/M62">View MathML</a>,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/65/mathml/M63','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/65/mathml/M63">View MathML</a>

Now, we briefly recall some notations and an abstract existence result, which can be found in [20]. Let YZ be real Banach spaces, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/65/mathml/M64','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/65/mathml/M64">View MathML</a> be a Fredholm map of index zero, and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/65/mathml/M65','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/65/mathml/M65">View MathML</a><a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/65/mathml/M66','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/65/mathml/M66">View MathML</a> be continuous projectors such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/65/mathml/M67','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/65/mathml/M67">View MathML</a>

It follows that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/65/mathml/M68','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/65/mathml/M68">View MathML</a> is invertible. We denote the inverse by <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/65/mathml/M69','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/65/mathml/M69">View MathML</a>. If Ω is an open bounded subset of Y such that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/65/mathml/M70','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/65/mathml/M70">View MathML</a>, the map <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/65/mathml/M71','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/65/mathml/M71">View MathML</a> will be called L-compact on Ω if <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/65/mathml/M72','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/65/mathml/M72">View MathML</a> is bounded and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/65/mathml/M73','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/65/mathml/M73">View MathML</a> is compact.

Lemma 2.6 ([20])

LetLbe a Fredholm operator of index zero andNbeL-compact on<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/65/mathml/M74','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/65/mathml/M74">View MathML</a>. The equation<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/65/mathml/M75','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/65/mathml/M75">View MathML</a>has at least one solution in<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/65/mathml/M76','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/65/mathml/M76">View MathML</a>if the following conditions are satisfied:

(i) <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/65/mathml/M77','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/65/mathml/M77">View MathML</a>for each<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/65/mathml/M78','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/65/mathml/M78">View MathML</a>;

(ii) <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/65/mathml/M79','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/65/mathml/M79">View MathML</a>for each<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/65/mathml/M80','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/65/mathml/M80">View MathML</a>;

(iii) <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/65/mathml/M81','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/65/mathml/M81">View MathML</a>,

where<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/65/mathml/M66','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/65/mathml/M66">View MathML</a>is a projection such that<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/65/mathml/M83','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/65/mathml/M83">View MathML</a>and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/65/mathml/M84','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/65/mathml/M84">View MathML</a>is a any isomorphism.

In this article, we use the Banach space <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/65/mathml/M85','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/65/mathml/M85">View MathML</a> with the norm <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/65/mathml/M86','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/65/mathml/M86">View MathML</a>.

Lemma 2.7 ([16])

Given<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/65/mathml/M87','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/65/mathml/M87">View MathML</a>and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/65/mathml/M88','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/65/mathml/M88">View MathML</a>, for any<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/65/mathml/M89','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/65/mathml/M89">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/65/mathml/M90','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/65/mathml/M90">View MathML</a> (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/65/mathml/M91','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/65/mathml/M91">View MathML</a>), we can define a Banach space

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/65/mathml/M92','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/65/mathml/M92">View MathML</a>

with the norm defined by<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/65/mathml/M93','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/65/mathml/M93">View MathML</a>.

Lemma 2.8 ([16])

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/65/mathml/M94','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/65/mathml/M94">View MathML</a>is a sequentially compact set if and only ifEis uniformly bounded and equicontinuous. Here, a uniform bound means that there exists a constant<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/65/mathml/M95','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/65/mathml/M95">View MathML</a>with each<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/65/mathml/M96','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/65/mathml/M96">View MathML</a>, such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/65/mathml/M97','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/65/mathml/M97">View MathML</a>

and equicontinuation means that there exists a<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/65/mathml/M98','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/65/mathml/M98">View MathML</a>with<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/65/mathml/M99','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/65/mathml/M99">View MathML</a>for any<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/65/mathml/M100','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/65/mathml/M100">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/65/mathml/M96','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/65/mathml/M96">View MathML</a>and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/65/mathml/M102','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/65/mathml/M102">View MathML</a>, such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/65/mathml/M103','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/65/mathml/M103">View MathML</a>

In this article, let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/65/mathml/M104','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/65/mathml/M104">View MathML</a> with the norm <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/65/mathml/M105','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/65/mathml/M105">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/65/mathml/M106','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/65/mathml/M106">View MathML</a> with the norm <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/65/mathml/M107','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/65/mathml/M107">View MathML</a>. Define the operator <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/65/mathml/M108','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/65/mathml/M108">View MathML</a> by

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/65/mathml/M109','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/65/mathml/M109">View MathML</a>

(2.1)

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/65/mathml/M110','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/65/mathml/M110">View MathML</a>. Define the operator <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/65/mathml/M71','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/65/mathml/M71">View MathML</a> by

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/65/mathml/M112','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/65/mathml/M112">View MathML</a>

(2.2)

Thus, BVP (1.1) can be written as <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/65/mathml/M113','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/65/mathml/M113">View MathML</a> for each <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/65/mathml/M114','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/65/mathml/M114">View MathML</a>.

3 Main results

First, let us introduce the following notations for convenience, with setting <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/65/mathml/M115','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/65/mathml/M115">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/65/mathml/M116','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/65/mathml/M116">View MathML</a> with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/65/mathml/M117','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/65/mathml/M117">View MathML</a>,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/65/mathml/M118','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/65/mathml/M118">View MathML</a>

Then, let us make some assumptions which will be used throughout the article.

(H1) There exist functions <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/65/mathml/M119','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/65/mathml/M119">View MathML</a> and a constant <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/65/mathml/M120','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/65/mathml/M120">View MathML</a> such that for all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/65/mathml/M121','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/65/mathml/M121">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/65/mathml/M45','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/65/mathml/M45">View MathML</a>,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/65/mathml/M123','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/65/mathml/M123">View MathML</a>

(H2) For any <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/65/mathml/M114','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/65/mathml/M114">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/65/mathml/M45','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/65/mathml/M45">View MathML</a>, there exists a constant <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/65/mathml/M126','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/65/mathml/M126">View MathML</a> such that if <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/65/mathml/M127','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/65/mathml/M127">View MathML</a>, then either

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/65/mathml/M128','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/65/mathml/M128">View MathML</a>

(H3) For any <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/65/mathml/M114','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/65/mathml/M114">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/65/mathml/M45','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/65/mathml/M45">View MathML</a>, there exists a constant <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/65/mathml/M131','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/65/mathml/M131">View MathML</a> such that if <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/65/mathml/M132','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/65/mathml/M132">View MathML</a>, then either

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/65/mathml/M133','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/65/mathml/M133">View MathML</a>

Theorem 3.1If conditions (C), (H1)-(H3) hold, then BVPs (1.1)-(1.2) have at least one solution provided that<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/65/mathml/M134','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/65/mathml/M134">View MathML</a>.

In order to obtain our main result, we first present and prove Lemmas 3.2-3.8. Now, let us define operators <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/65/mathml/M135','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/65/mathml/M135">View MathML</a> (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/65/mathml/M136','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/65/mathml/M136">View MathML</a>) as follows:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/65/mathml/M137','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/65/mathml/M137">View MathML</a>

Lemma 3.2If condition (C) holds andLis defined by (2.1), then

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/65/mathml/M138','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/65/mathml/M138">View MathML</a>

Proof By (2.1) and Lemma 2.5, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/65/mathml/M26','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/65/mathml/M26">View MathML</a> has a solution

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/65/mathml/M140','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/65/mathml/M140">View MathML</a>

Combining with the condition (1.2), we get <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/65/mathml/M141','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/65/mathml/M141">View MathML</a>.

Suppose <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/65/mathml/M142','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/65/mathml/M142">View MathML</a>, then there exists <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/65/mathml/M114','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/65/mathml/M114">View MathML</a> such that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/65/mathml/M144','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/65/mathml/M144">View MathML</a>, i.e., <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/65/mathml/M145','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/65/mathml/M145">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/65/mathml/M146','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/65/mathml/M146">View MathML</a>. By Lemma 2.5, we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/65/mathml/M147','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/65/mathml/M147">View MathML</a>

Then in view of condition (C), (1.2) and Lemma 2.4, x satisfies

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/65/mathml/M148','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/65/mathml/M148">View MathML</a>

(3.1)

On the other hand, suppose <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/65/mathml/M149','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/65/mathml/M149">View MathML</a> and it satisfies (3.1), let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/65/mathml/M150','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/65/mathml/M150">View MathML</a>, then <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/65/mathml/M114','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/65/mathml/M114">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/65/mathml/M152','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/65/mathml/M152">View MathML</a>, i.e., <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/65/mathml/M153','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/65/mathml/M153">View MathML</a>. Therefore, we obtain that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/65/mathml/M154','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/65/mathml/M154">View MathML</a>

 □

Lemma 3.3If condition (C) holds, then there exist two constants<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/65/mathml/M116','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/65/mathml/M116">View MathML</a>and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/65/mathml/M115','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/65/mathml/M115">View MathML</a>with<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/65/mathml/M117','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/65/mathml/M117">View MathML</a>such that<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/65/mathml/M158','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/65/mathml/M158">View MathML</a>.

Proof From <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/65/mathml/M159','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/65/mathml/M159">View MathML</a>, we obtain that for any nonnegative integer l, there exists <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/65/mathml/M160','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/65/mathml/M160">View MathML</a> such that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/65/mathml/M161','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/65/mathml/M161">View MathML</a>. If else, we obtain that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/65/mathml/M162','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/65/mathml/M162">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/65/mathml/M163','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/65/mathml/M163">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/65/mathml/M164','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/65/mathml/M164">View MathML</a>.

If <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/65/mathml/M165','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/65/mathml/M165">View MathML</a>, we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/65/mathml/M166','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/65/mathml/M166">View MathML</a>

It is equal to

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/65/mathml/M167','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/65/mathml/M167">View MathML</a>

Since the determinant of coefficients is not equal to zero, we have that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/65/mathml/M168','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/65/mathml/M168">View MathML</a> (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/65/mathml/M169','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/65/mathml/M169">View MathML</a>), which is a contradiction to condition (C).

If <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/65/mathml/M170','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/65/mathml/M170">View MathML</a>, we get

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/65/mathml/M171','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/65/mathml/M171">View MathML</a>

Similarly, we can deduce that the determinant of coefficients is not equal to zero, so we have that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/65/mathml/M168','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/65/mathml/M168">View MathML</a> (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/65/mathml/M169','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/65/mathml/M169">View MathML</a>), which is a contradiction to condition (C). Thus, there exists <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/65/mathml/M174','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/65/mathml/M174">View MathML</a> such that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/65/mathml/M161','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/65/mathml/M161">View MathML</a>.

Similarly, from <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/65/mathml/M176','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/65/mathml/M176">View MathML</a>, we have that there exists a constant <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/65/mathml/M177','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/65/mathml/M177">View MathML</a> such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/65/mathml/M178','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/65/mathml/M178">View MathML</a>

(3.2)

Let

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/65/mathml/M179','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/65/mathml/M179">View MathML</a>

we shall prove that S is a finite set. If else, there exists a strict increasing sequence <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/65/mathml/M180','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/65/mathml/M180">View MathML</a> such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/65/mathml/M181','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/65/mathml/M181">View MathML</a>

Since <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/65/mathml/M182','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/65/mathml/M182">View MathML</a>, we have <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/65/mathml/M183','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/65/mathml/M183">View MathML</a>. Thus,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/65/mathml/M184','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/65/mathml/M184">View MathML</a>

which is a contradiction to (3.2). Therefore, there exists two constants <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/65/mathml/M115','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/65/mathml/M115">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/65/mathml/M116','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/65/mathml/M116">View MathML</a> with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/65/mathml/M117','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/65/mathml/M117">View MathML</a> such that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/65/mathml/M158','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/65/mathml/M158">View MathML</a>. □

Lemma 3.4If the condition (C) holds andLis defined by (2.1), thenLis a Fredholm operator of index zero. Define the linear operator<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/65/mathml/M189','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/65/mathml/M189">View MathML</a>with<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/65/mathml/M190','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/65/mathml/M190">View MathML</a>, then it is the inverse ofL. Furthermore, we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/65/mathml/M191','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/65/mathml/M191">View MathML</a>

Proof For each <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/65/mathml/M115','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/65/mathml/M115">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/65/mathml/M116','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/65/mathml/M116">View MathML</a> with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/65/mathml/M117','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/65/mathml/M117">View MathML</a>, define operator <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/65/mathml/M66','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/65/mathml/M66">View MathML</a> by

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/65/mathml/M196','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/65/mathml/M196">View MathML</a>

(3.3)

where

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/65/mathml/M197','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/65/mathml/M197">View MathML</a>

(3.4)

It is clear that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/65/mathml/M198','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/65/mathml/M198">View MathML</a>. It follows from (3.4), the definition of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/65/mathml/M199','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/65/mathml/M199">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/65/mathml/M200','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/65/mathml/M200">View MathML</a> that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/65/mathml/M201','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/65/mathml/M201">View MathML</a>

(3.5)

similarly, we can derive that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/65/mathml/M202','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/65/mathml/M202">View MathML</a>

(3.6)

Hence, for each <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/65/mathml/M203','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/65/mathml/M203">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/65/mathml/M45','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/65/mathml/M45">View MathML</a>, it follows from the (3.3)-(3.6) that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/65/mathml/M205','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/65/mathml/M205">View MathML</a>

Furthermore, Q is a continuous linear projector.

For each <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/65/mathml/M153','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/65/mathml/M153">View MathML</a>, we have <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/65/mathml/M207','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/65/mathml/M207">View MathML</a>, i.e., <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/65/mathml/M208','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/65/mathml/M208">View MathML</a>. On the other hand, for each <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/65/mathml/M208','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/65/mathml/M208">View MathML</a>, we have that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/65/mathml/M210','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/65/mathml/M210">View MathML</a>

However, the determinant of coefficients is as follows

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/65/mathml/M211','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/65/mathml/M211">View MathML</a>

then we have <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/65/mathml/M212','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/65/mathml/M212">View MathML</a> (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/65/mathml/M213','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/65/mathml/M213">View MathML</a>), i.e., <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/65/mathml/M153','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/65/mathml/M153">View MathML</a>. Thus, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/65/mathml/M83','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/65/mathml/M83">View MathML</a>.

Take any <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/65/mathml/M203','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/65/mathml/M203">View MathML</a> in the type <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/65/mathml/M217','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/65/mathml/M217">View MathML</a>, obviously, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/65/mathml/M218','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/65/mathml/M218">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/65/mathml/M219','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/65/mathml/M219">View MathML</a>, so <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/65/mathml/M220','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/65/mathml/M220">View MathML</a>. For any <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/65/mathml/M221','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/65/mathml/M221">View MathML</a> with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/65/mathml/M222','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/65/mathml/M222">View MathML</a>, by Lemma 3.2, we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/65/mathml/M223','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/65/mathml/M223">View MathML</a>

That is,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/65/mathml/M224','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/65/mathml/M224">View MathML</a>

but the determinant of coefficients is as follows

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/65/mathml/M225','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/65/mathml/M225">View MathML</a>

we can deduce that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/65/mathml/M226','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/65/mathml/M226">View MathML</a>. Hence, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/65/mathml/M227','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/65/mathml/M227">View MathML</a>. Furthermore, we get <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/65/mathml/M228','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/65/mathml/M228">View MathML</a>. Therefore, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/65/mathml/M229','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/65/mathml/M229">View MathML</a>, which means that L is a Fredholm operator of index zero.

Let operator <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/65/mathml/M65','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/65/mathml/M65">View MathML</a> and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/65/mathml/M231','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/65/mathml/M231">View MathML</a>

(3.7)

It is easy to calculate that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/65/mathml/M232','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/65/mathml/M232">View MathML</a>; furthermore, P is a continuous linear projector. Obviously

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/65/mathml/M233','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/65/mathml/M233">View MathML</a>

It is clear that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/65/mathml/M234','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/65/mathml/M234">View MathML</a>.

For any <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/65/mathml/M153','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/65/mathml/M153">View MathML</a>, in view of the definition of operators Kp and L, we have <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/65/mathml/M236','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/65/mathml/M236">View MathML</a>. On the other hand, if <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/65/mathml/M237','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/65/mathml/M237">View MathML</a>, we have <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/65/mathml/M238','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/65/mathml/M238">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/65/mathml/M114','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/65/mathml/M114">View MathML</a>. Therefore, by Lemma 2.5 and definitions of operators <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/65/mathml/M69','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/65/mathml/M69">View MathML</a> and L, we know that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/65/mathml/M241','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/65/mathml/M241">View MathML</a>, which implies that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/65/mathml/M242','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/65/mathml/M242">View MathML</a>. By the definition of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/65/mathml/M69','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/65/mathml/M69">View MathML</a>, we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/65/mathml/M244','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/65/mathml/M244">View MathML</a>

It follows from Lemma 2.4 that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/65/mathml/M245','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/65/mathml/M245">View MathML</a>

Then, we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/65/mathml/M246','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/65/mathml/M246">View MathML</a>

By the definition of the norm in space Y, we get <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/65/mathml/M247','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/65/mathml/M247">View MathML</a>. □

Lemma 3.5Assume<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/65/mathml/M248','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/65/mathml/M248">View MathML</a>is an open bounded subset such that<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/65/mathml/M249','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/65/mathml/M249">View MathML</a>, andNis defined by (2.2), thenNisL-compact on<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/65/mathml/M74','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/65/mathml/M74">View MathML</a>.

Proof In order to prove N is L-compact, we only need to prove that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/65/mathml/M72','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/65/mathml/M72">View MathML</a> is bounded and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/65/mathml/M252','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/65/mathml/M252">View MathML</a> is compact. Since the function f satisfies Carathéodory conditions and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/65/mathml/M253','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/65/mathml/M253">View MathML</a>, for each <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/65/mathml/M48','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/65/mathml/M48">View MathML</a>, there exists a <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/65/mathml/M255','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/65/mathml/M255">View MathML</a> such that, for a.e. <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/65/mathml/M45','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/65/mathml/M45">View MathML</a> and every <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/65/mathml/M51','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/65/mathml/M51">View MathML</a>, we have <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/65/mathml/M258','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/65/mathml/M258">View MathML</a>. By the definition of operators Q and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/65/mathml/M69','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/65/mathml/M69">View MathML</a> on the interval <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/65/mathml/M260','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/65/mathml/M260">View MathML</a>, it is easy to get that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/65/mathml/M72','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/65/mathml/M72">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/65/mathml/M262','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/65/mathml/M262">View MathML</a> are bounded. Thus, there exists a constant <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/65/mathml/M263','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/65/mathml/M263">View MathML</a> with each <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/65/mathml/M264','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/65/mathml/M264">View MathML</a>, such that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/65/mathml/M265','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/65/mathml/M265">View MathML</a>.

For all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/65/mathml/M266','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/65/mathml/M266">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/65/mathml/M19','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/65/mathml/M19">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/65/mathml/M268','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/65/mathml/M268">View MathML</a>, we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/65/mathml/M269','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/65/mathml/M269">View MathML</a>

and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/65/mathml/M270','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/65/mathml/M270">View MathML</a>

Since <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/65/mathml/M271','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/65/mathml/M271">View MathML</a> is uniformly continuous on <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/65/mathml/M260','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/65/mathml/M260">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/65/mathml/M255','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/65/mathml/M255">View MathML</a>, so <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/65/mathml/M274','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/65/mathml/M274">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/65/mathml/M275','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/65/mathml/M275">View MathML</a> are equicontinuous. By Lemma 2.8, we get that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/65/mathml/M276','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/65/mathml/M276">View MathML</a> is completely continuous. □

Lemma 3.6Suppose (H1)-(H3) hold, then the set<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/65/mathml/M277','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/65/mathml/M277">View MathML</a>is bounded.

Proof Taking any <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/65/mathml/M278','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/65/mathml/M278">View MathML</a>, then we have <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/65/mathml/M279','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/65/mathml/M279">View MathML</a>, which yields <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/65/mathml/M280','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/65/mathml/M280">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/65/mathml/M281','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/65/mathml/M281">View MathML</a>, i.e., <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/65/mathml/M282','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/65/mathml/M282">View MathML</a> for all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/65/mathml/M45','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/65/mathml/M45">View MathML</a>. It follows from (H2) and (H3) that there exists <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/65/mathml/M284','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/65/mathml/M284">View MathML</a> such that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/65/mathml/M285','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/65/mathml/M285">View MathML</a>. Then we can get that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/65/mathml/M286','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/65/mathml/M286">View MathML</a>

Furthermore, we have that, with setting <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/65/mathml/M287','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/65/mathml/M287">View MathML</a>,

(3.8)

(3.9)

By (3.7)-(3.9) and Lemma 2.4, we have that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/65/mathml/M290','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/65/mathml/M290">View MathML</a>

As before, for any <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/65/mathml/M278','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/65/mathml/M278">View MathML</a>, we have <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/65/mathml/M292','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/65/mathml/M292">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/65/mathml/M293','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/65/mathml/M293">View MathML</a>. From Lemma 3.4 and for each <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/65/mathml/M294','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/65/mathml/M294">View MathML</a>, we can get

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/65/mathml/M295','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/65/mathml/M295">View MathML</a>

Furthermore, we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/65/mathml/M296','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/65/mathml/M296">View MathML</a>

By (H1) and the definition of N, we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/65/mathml/M297','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/65/mathml/M297">View MathML</a>

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/65/mathml/M298','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/65/mathml/M298">View MathML</a>. Since <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/65/mathml/M299','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/65/mathml/M299">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/65/mathml/M134','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/65/mathml/M134">View MathML</a> hold true, we can get that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/65/mathml/M301','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/65/mathml/M301">View MathML</a>

which yield that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/65/mathml/M302','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/65/mathml/M302">View MathML</a>

Furthermore, from the previous inequalities, we know that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/65/mathml/M303','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/65/mathml/M303">View MathML</a>

Since <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/65/mathml/M120','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/65/mathml/M120">View MathML</a>, there exist constants <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/65/mathml/M305','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/65/mathml/M305">View MathML</a> such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/65/mathml/M306','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/65/mathml/M306">View MathML</a>

Therefore, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/65/mathml/M307','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/65/mathml/M307">View MathML</a> is bounded. □

Lemma 3.7Suppose (H2) and (H3) hold, then the set<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/65/mathml/M308','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/65/mathml/M308">View MathML</a>is bounded.

Proof For any <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/65/mathml/M309','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/65/mathml/M309">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/65/mathml/M29','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/65/mathml/M29">View MathML</a>, then <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/65/mathml/M311','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/65/mathml/M311">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/65/mathml/M282','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/65/mathml/M282">View MathML</a>. By (H2), we get that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/65/mathml/M313','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/65/mathml/M313">View MathML</a>, then we have <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/65/mathml/M314','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/65/mathml/M314">View MathML</a>. By (H3), we have that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/65/mathml/M315','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/65/mathml/M315">View MathML</a>, thus <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/65/mathml/M316','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/65/mathml/M316">View MathML</a>. Therefore, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/65/mathml/M317','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/65/mathml/M317">View MathML</a> is bounded. □

Lemma 3.8If the first parts of (H2) and (H3) hold, then the set<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/65/mathml/M318','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/65/mathml/M318">View MathML</a>is bounded.

Proof Taking any <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/65/mathml/M319','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/65/mathml/M319">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/65/mathml/M29','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/65/mathml/M29">View MathML</a>, we have <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/65/mathml/M27','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/65/mathml/M27">View MathML</a>. For all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/65/mathml/M45','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/65/mathml/M45">View MathML</a>, we define the isomorphism <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/65/mathml/M323','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/65/mathml/M323">View MathML</a> by

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/65/mathml/M324','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/65/mathml/M324">View MathML</a>

By the definition of the set <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/65/mathml/M325','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/65/mathml/M325">View MathML</a>, we can get that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/65/mathml/M326','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/65/mathml/M326">View MathML</a>

(3.10)

If <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/65/mathml/M327','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/65/mathml/M327">View MathML</a>, we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/65/mathml/M328','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/65/mathml/M328">View MathML</a>

By the first parts of (H2) and (H3), similar to the proof of Lemma 3.7, then

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/65/mathml/M329','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/65/mathml/M329">View MathML</a>

Therefore, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/65/mathml/M325','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/65/mathml/M325">View MathML</a> is bounded.

If <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/65/mathml/M331','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/65/mathml/M331">View MathML</a>, we have <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/65/mathml/M226','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/65/mathml/M226">View MathML</a>.

If <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/65/mathml/M333','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/65/mathml/M333">View MathML</a>, we get that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/65/mathml/M334','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/65/mathml/M334">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/65/mathml/M335','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/65/mathml/M335">View MathML</a>, similar to the proof of Lemma 3.7, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/65/mathml/M325','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/65/mathml/M325">View MathML</a> is bounded. If else, we have that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/65/mathml/M337','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/65/mathml/M337">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/65/mathml/M338','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/65/mathml/M338">View MathML</a>. It contradicts (3.10), thus <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/65/mathml/M325','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/65/mathml/M325">View MathML</a> is bounded. □

Remark 3.9 If the other parts of (H2) and (H3) hold, then the set <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/65/mathml/M340','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/65/mathml/M340">View MathML</a> is bounded.

Now with Lemmas 3.2-3.8 in hands, we can begin to prove our main result - Theorem 3.1.

Proof of Theorem 3.1 Assume that Ω is a bounded open set of Y with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/65/mathml/M341','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/65/mathml/M341">View MathML</a>. By Lemma 3.5, N is L-compact on <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/65/mathml/M74','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/65/mathml/M74">View MathML</a>. Then by Lemmas 3.6 and 3.7, we have

(i) <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/65/mathml/M343','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/65/mathml/M343">View MathML</a> for every <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/65/mathml/M344','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/65/mathml/M344">View MathML</a>;

(ii) <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/65/mathml/M345','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/65/mathml/M345">View MathML</a> for every <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/65/mathml/M346','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/65/mathml/M346">View MathML</a>.

Finally, we will prove that (iii) of Lemma 2.6 is satisfied. We let I as the identity operator in the Banach space Y and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/65/mathml/M347','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/65/mathml/M347">View MathML</a>, according to Lemma 3.8 (or Remark 3.9) we know that for all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/65/mathml/M348','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/65/mathml/M348">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/65/mathml/M349','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/65/mathml/M349">View MathML</a>. By the homotopic property of degree, we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/65/mathml/M350','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/65/mathml/M350">View MathML</a>

so (iii) of Lemma 2.6 is satisfied.

Consequently, by Lemma 2.6, the equation <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/65/mathml/M113','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/65/mathml/M113">View MathML</a> has at least one solution in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/65/mathml/M76','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/65/mathml/M76">View MathML</a>. Namely, BVPs (1.1)-(1.2) have at least one solution in the space Y. □

According to Theorem 3.1, we have the following corollary.

Corollary 3.10Suppose that (H1) is replaced by the following condition,

(H4) there exist functions<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/65/mathml/M119','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/65/mathml/M119">View MathML</a>and a constant<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/65/mathml/M120','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/65/mathml/M120">View MathML</a>such that for all<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/65/mathml/M121','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/65/mathml/M121">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/65/mathml/M45','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/65/mathml/M45">View MathML</a>,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/65/mathml/M357','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/65/mathml/M357">View MathML</a>

or

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/65/mathml/M358','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/65/mathml/M358">View MathML</a>

and the others in Theorem 3.1 are not changed, then BVPs (1.1)-(1.2) have at least one solution.

4 An example

Example Consider the following boundary value problem for all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/65/mathml/M359','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/65/mathml/M359">View MathML</a>:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/65/mathml/M360','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/65/mathml/M360">View MathML</a>

(4.1)

Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/65/mathml/M361','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/65/mathml/M361">View MathML</a>, and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/65/mathml/M362','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/65/mathml/M362">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/65/mathml/M363','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/65/mathml/M363">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/65/mathml/M364','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/65/mathml/M364">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/65/mathml/M365','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/65/mathml/M365">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/65/mathml/M366','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/65/mathml/M366">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/65/mathml/M367','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/65/mathml/M367">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/65/mathml/M368','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/65/mathml/M368">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/65/mathml/M369','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/65/mathml/M369">View MathML</a>. We can get that the condition (C) holds, i.e., <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/65/mathml/M370','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/65/mathml/M370">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/65/mathml/M371','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/65/mathml/M371">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/65/mathml/M372','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/65/mathml/M372">View MathML</a>. Moreover,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/65/mathml/M373','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/65/mathml/M373">View MathML</a>

Thus, we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/65/mathml/M374','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/65/mathml/M374">View MathML</a>

Taking <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/65/mathml/M375','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/65/mathml/M375">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/65/mathml/M376','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/65/mathml/M376">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/65/mathml/M377','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/65/mathml/M377">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/65/mathml/M378','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/65/mathml/M378">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/65/mathml/M379','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/65/mathml/M379">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/65/mathml/M380','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/65/mathml/M380">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/65/mathml/M381','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/65/mathml/M381">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/65/mathml/M382','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/65/mathml/M382">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/65/mathml/M383','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/65/mathml/M383">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/65/mathml/M384','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/65/mathml/M384">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/65/mathml/M385','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/65/mathml/M385">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/65/mathml/M386','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/65/mathml/M386">View MathML</a>, we can calculate that (H1)-(H3) hold. Furthermore, we can get

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/65/mathml/M387','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/65/mathml/M387">View MathML</a>

By Corollary 3.10, the BVP (4.1) has at least one solution in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/65/mathml/M388','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/65/mathml/M388">View MathML</a>.

Competing interests

The authors declare that they have no competing interests.

Author’s contributions

NX designed all the steps of proof in this research and also wrote the article. WBL suggested many good ideas in this article. LSX helped to draft the first manuscript and gave an example to illustrate our result. All authors read and approved the final manuscript.

Acknowledgement

The authors would like to acknowledge the anonymous referee for many helpful comments and valuable suggestions on this article. This work is sponsored by Fundamental Research Funds for the Central Universities (2012LWB44).

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