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This article is part of the series A Tribute to Professor Ivan Kiguradze.

Open Access Research

Multiple solutions of three-point boundary value problems for second-order impulsive differential equation at resonance

Yulin Zhao1*, Haibo Chen2 and Qiming Zhang1

Author Affiliations

1 School of Science, Hunan University of Technology, Zhuzhou, Hunan, 412007, PR China

2 Department of Mathematics, Central South University, Changsha, Hunan, 410075, PR China

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Boundary Value Problems 2014, 2014:103  doi:10.1186/1687-2770-2014-103

The electronic version of this article is the complete one and can be found online at: http://www.boundaryvalueproblems.com/content/2014/1/103


Received:20 December 2013
Accepted:8 April 2014
Published:7 May 2014

© 2014 Zhao 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 credited.

Abstract

In this paper, by using the coincidence degree theory and the upper and lower solutions method, we deal with the existence of multiple solutions to three-point boundary value problems for second-order differential equation with impulses at resonance. An example is given to show the validity of our results.

Keywords:
resonance; coincidence degree; upper and lower solutions; impulsive; three-point boundary value problems

1 Introduction

The purpose of the present paper is to investigate the following second-order impulsive differential equations:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/103/mathml/M1','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/103/mathml/M1">View MathML</a>

(1.1)

together with the boundary conditions:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/103/mathml/M2','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/103/mathml/M2">View MathML</a>

(1.2)

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/103/mathml/M3','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/103/mathml/M3">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/103/mathml/M4','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/103/mathml/M4">View MathML</a> is a continuous differentiable function, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/103/mathml/M5','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/103/mathml/M5">View MathML</a> is continuous, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/103/mathml/M6','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/103/mathml/M6">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/103/mathml/M7','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/103/mathml/M7">View MathML</a> for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/103/mathml/M8','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/103/mathml/M8">View MathML</a>, m is a fixed positive integer, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/103/mathml/M9','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/103/mathml/M9">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/103/mathml/M10','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/103/mathml/M10">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/103/mathml/M11','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/103/mathml/M11">View MathML</a> denotes the jump of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/103/mathml/M12','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/103/mathml/M12">View MathML</a> at <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/103/mathml/M13','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/103/mathml/M13">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/103/mathml/M14','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/103/mathml/M14">View MathML</a>. <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/103/mathml/M15','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/103/mathml/M15">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/103/mathml/M16','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/103/mathml/M16">View MathML</a> (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/103/mathml/M17','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/103/mathml/M17">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/103/mathml/M18','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/103/mathml/M18">View MathML</a>) represent the right limit (left limit) of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/103/mathml/M19','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/103/mathml/M19">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/103/mathml/M12','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/103/mathml/M12">View MathML</a> at <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/103/mathml/M21','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/103/mathml/M21">View MathML</a>, respectively.

Impulsive differential equations describe processes which experience a sudden change of their state at certain moments. The theory of impulse differential equations has been a significant development in recent years and played a very important role in modern applied mathematical models of real processes rising in phenomena studied in physics, population dynamics, chemical technology, biotechnology, and economics; see [1-10] and the references therein.

Recently, several authors (see [6,11-14] and the references therein) have studied the existence of nontrivial or positive solutions for second-order three-point boundary value problem of the type

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/103/mathml/M22','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/103/mathml/M22">View MathML</a>

(1.3)

Note that the nonlinear term f depends on u and its derivative <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/103/mathml/M23','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/103/mathml/M23">View MathML</a>, then the relative problem becomes more complicated. A general method to deal with this difficulty is to add some conditions to restrict the growth of the <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/103/mathml/M23','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/103/mathml/M23">View MathML</a> term. One condition is the Caratheodory nonlinearity, the other usual condition is Nagumo condition or Nagumo-Winter condition (see [2,9,12,15-20]). When <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/103/mathml/M25','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/103/mathml/M25">View MathML</a>, the linear operator <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/103/mathml/M26','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/103/mathml/M26">View MathML</a> is invertible, this is the so-called non-resonance case. Gupta et al. made use of the Leray-Schauder continuation theorem to get the results on the existence of the solution for the problems (1.3) when <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/103/mathml/M25','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/103/mathml/M25">View MathML</a> in [14]. By using the Leray-Schauder continuation theorem and in the presence of two pairs of upper and lower solutions, Khan and Webb [12] established the existence of at least three solutions for the problem (1.3) when <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/103/mathml/M25','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/103/mathml/M25">View MathML</a>. The linear operator <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/103/mathml/M26','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/103/mathml/M26">View MathML</a> is non-invertible when <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/103/mathml/M30','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/103/mathml/M30">View MathML</a>, this is the so-called resonance case, and the Leray-Schauder continuation theorem cannot be applied. In [11], by using the coincidence degree theory of Mawhin [21] and some linear or non-linear growth assumptions on f, Feng and Webb obtained the existence of the solution of the problem (1.3) when <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/103/mathml/M30','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/103/mathml/M30">View MathML</a>. By applying the nonlinear alternative of Leray-Schauder, Ma [13] have showed the existence of at least one solution for the problem (1.3) when <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/103/mathml/M32','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/103/mathml/M32">View MathML</a>.

Recently, using the coincidence degree theory and the concept of autonomous curvature bound set, Liu and Yu [6] have studied the existence of at least one solution for the problem (1.1)-(1.2) when <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/103/mathml/M33','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/103/mathml/M33">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/103/mathml/M34','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/103/mathml/M34">View MathML</a>.

In the present paper, we assume that there exist n (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/103/mathml/M35','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/103/mathml/M35">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/103/mathml/M36','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/103/mathml/M36">View MathML</a>) pairs of upper and lower solutions for problem (1.1)-(1.2) and the nonlinear f satisfies a Nagumo-like growth condition with respect to <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/103/mathml/M23','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/103/mathml/M23">View MathML</a>. By considering a suitably modified nonlinearity and applying the coincidence degree method of Mawhin [21], the existence of multiple solutions for the problem (1.1)-(1.2) is given.

2 Preliminaries

Let

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/103/mathml/M38','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/103/mathml/M38">View MathML</a>

where

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/103/mathml/M39','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/103/mathml/M39">View MathML</a>

Obviously, X is a Banach space with the following norm:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/103/mathml/M40','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/103/mathml/M40">View MathML</a>

In the following, we recall the concept of strict upper and lower solutions for problem (1.1)-(1.2).

Definition 2.1 A function <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/103/mathml/M41','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/103/mathml/M41">View MathML</a> is said to be a strict lower solution of the problem (1.1)-(1.2) if

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/103/mathml/M42','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/103/mathml/M42">View MathML</a>

(2.1)

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/103/mathml/M43','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/103/mathml/M43">View MathML</a>

(2.2)

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/103/mathml/M44','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/103/mathml/M44">View MathML</a>

(2.3)

Similarly, a function <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/103/mathml/M45','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/103/mathml/M45">View MathML</a> is said to be a strict upper solution of the problem (1.1)-(1.2) if

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/103/mathml/M46','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/103/mathml/M46">View MathML</a>

(2.4)

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/103/mathml/M47','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/103/mathml/M47">View MathML</a>

(2.5)

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/103/mathml/M48','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/103/mathml/M48">View MathML</a>

(2.6)

Remark 2.2 Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/103/mathml/M5','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/103/mathml/M5">View MathML</a> be continuous, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/103/mathml/M7','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/103/mathml/M7">View MathML</a>, and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/103/mathml/M51','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/103/mathml/M51">View MathML</a> is a solution of the problem (1.1)-(1.2), if <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/103/mathml/M52','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/103/mathml/M52">View MathML</a> is a strict lower solution (strict upper solution) for the problem (1.1)-(1.2) with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/103/mathml/M53','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/103/mathml/M53">View MathML</a> (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/103/mathml/M54','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/103/mathml/M54">View MathML</a>), then <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/103/mathml/M55','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/103/mathml/M55">View MathML</a> (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/103/mathml/M56','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/103/mathml/M56">View MathML</a>) on <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/103/mathml/M57','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/103/mathml/M57">View MathML</a>.

Definition 2.3 Let α be a strict lower solution and β be a strict upper solution for the problem (1.1)-(1.2) satisfying <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/103/mathml/M58','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/103/mathml/M58">View MathML</a> on J. We say that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/103/mathml/M5','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/103/mathml/M5">View MathML</a> has property (H) relative to α and β, if there exists a function <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/103/mathml/M60','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/103/mathml/M60">View MathML</a> such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/103/mathml/M61','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/103/mathml/M61">View MathML</a>

(2.7)

for all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/103/mathml/M62','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/103/mathml/M62">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/103/mathml/M63','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/103/mathml/M63">View MathML</a>, and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/103/mathml/M64','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/103/mathml/M64">View MathML</a>

(2.8)

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/103/mathml/M65','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/103/mathml/M65">View MathML</a> with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/103/mathml/M66','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/103/mathml/M66">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/103/mathml/M67','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/103/mathml/M67">View MathML</a>.

3 The key lemmas

Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/103/mathml/M68','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/103/mathml/M68">View MathML</a>, and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/103/mathml/M69','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/103/mathml/M69">View MathML</a>

Then the problem (1.1)-(1.2) can be written as

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/103/mathml/M70','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/103/mathml/M70">View MathML</a>

Lemma 3.1Suppose thatLbe defined in the above. ThenLis a Fredholm operator of index zero. Furthermore

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/103/mathml/M71','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/103/mathml/M71">View MathML</a>

(3.1)

and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/103/mathml/M72','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/103/mathml/M72">View MathML</a>

(3.2)

Proof Firstly, it is clear that (3.1) holds. Next, we shall prove that (3.2) holds.

The following problem:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/103/mathml/M73','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/103/mathml/M73">View MathML</a>

(3.3)

has a solution <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/103/mathml/M12','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/103/mathml/M12">View MathML</a> satisfying <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/103/mathml/M75','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/103/mathml/M75">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/103/mathml/M76','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/103/mathml/M76">View MathML</a> if and only if

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/103/mathml/M77','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/103/mathml/M77">View MathML</a>

(3.4)

In fact, if (3.3) has a solution <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/103/mathml/M12','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/103/mathml/M12">View MathML</a> satisfying <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/103/mathml/M75','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/103/mathml/M75">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/103/mathml/M76','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/103/mathml/M76">View MathML</a>, then from (3.3) we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/103/mathml/M81','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/103/mathml/M81">View MathML</a>

According to <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/103/mathml/M75','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/103/mathml/M75">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/103/mathml/M76','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/103/mathml/M76">View MathML</a>, we get (3.4).

On the other hand, if (3.4) holds, setting

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/103/mathml/M84','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/103/mathml/M84">View MathML</a>

where c is an arbitrary constant, then <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/103/mathml/M12','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/103/mathml/M12">View MathML</a> is a solution of (3.3) with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/103/mathml/M75','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/103/mathml/M75">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/103/mathml/M76','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/103/mathml/M76">View MathML</a>. Hence (3.2) holds. □

Take the projector <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/103/mathml/M88','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/103/mathml/M88">View MathML</a> as follows:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/103/mathml/M89','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/103/mathml/M89">View MathML</a>

(3.5)

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/103/mathml/M90','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/103/mathml/M90">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/103/mathml/M91','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/103/mathml/M91">View MathML</a>. For every <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/103/mathml/M92','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/103/mathml/M92">View MathML</a>, set

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/103/mathml/M93','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/103/mathml/M93">View MathML</a>

Thus, we obtain

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/103/mathml/M94','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/103/mathml/M94">View MathML</a>

Then <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/103/mathml/M95','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/103/mathml/M95">View MathML</a>. Hence <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/103/mathml/M96','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/103/mathml/M96">View MathML</a>. Since <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/103/mathml/M97','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/103/mathml/M97">View MathML</a>, we have <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/103/mathml/M98','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/103/mathml/M98">View MathML</a>, which implies <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/103/mathml/M99','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/103/mathml/M99">View MathML</a>. Hence L is a Fredholm operator of index zero.

Take <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/103/mathml/M100','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/103/mathml/M100">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/103/mathml/M101','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/103/mathml/M101">View MathML</a>. So the generalized inverse <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/103/mathml/M102','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/103/mathml/M102">View MathML</a> of L can be written as

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/103/mathml/M103','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/103/mathml/M103">View MathML</a>

(3.6)

Set <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/103/mathml/M104','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/103/mathml/M104">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/103/mathml/M105','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/103/mathml/M105">View MathML</a>. Then for the function ψ defined by (2.8), let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/103/mathml/M106','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/103/mathml/M106">View MathML</a> be the solution of the following initial value problem:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/103/mathml/M107','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/103/mathml/M107">View MathML</a>

(3.7)

and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/103/mathml/M108','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/103/mathml/M108">View MathML</a> be the solution of the following initial value problem:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/103/mathml/M109','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/103/mathml/M109">View MathML</a>

(3.8)

Lemma 3.2Suppose that there exists a constant<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/103/mathml/M110','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/103/mathml/M110">View MathML</a>, then<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/103/mathml/M106','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/103/mathml/M106">View MathML</a>is well defined in<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/103/mathml/M112','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/103/mathml/M112">View MathML</a>and positive on this interval, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/103/mathml/M108','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/103/mathml/M108">View MathML</a>is also well defined and positive in<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/103/mathml/M114','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/103/mathml/M114">View MathML</a>. Moreover, if<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/103/mathml/M115','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/103/mathml/M115">View MathML</a>, then<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/103/mathml/M116','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/103/mathml/M116">View MathML</a>for any<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/103/mathml/M117','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/103/mathml/M117">View MathML</a>; <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/103/mathml/M118','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/103/mathml/M118">View MathML</a>for any<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/103/mathml/M119','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/103/mathml/M119">View MathML</a>.

Proof We only consider the case <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/103/mathml/M120','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/103/mathml/M120">View MathML</a> (in the case <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/103/mathml/M121','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/103/mathml/M121">View MathML</a>, the proof is similar). Assume that there exists a <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/103/mathml/M117','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/103/mathml/M117">View MathML</a> such that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/103/mathml/M123','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/103/mathml/M123">View MathML</a>. Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/103/mathml/M124','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/103/mathml/M124">View MathML</a>. It follows from (3.7) that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/103/mathml/M125','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/103/mathml/M125">View MathML</a>

Integrating the above equation over <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/103/mathml/M126','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/103/mathml/M126">View MathML</a> we get (let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/103/mathml/M127','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/103/mathml/M127">View MathML</a>)

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/103/mathml/M128','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/103/mathml/M128">View MathML</a>

However, the left side of the above equation equals ∞ by (2.8). We reach a contradiction. Hence <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/103/mathml/M129','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/103/mathml/M129">View MathML</a> for every <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/103/mathml/M130','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/103/mathml/M130">View MathML</a>. From <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/103/mathml/M105','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/103/mathml/M105">View MathML</a>, by the continuity of solution of differential equations on the initial values, we obtain <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/103/mathml/M116','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/103/mathml/M116">View MathML</a> for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/103/mathml/M117','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/103/mathml/M117">View MathML</a>. The proof is complete. □

Define the following sets:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/103/mathml/M134','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/103/mathml/M134">View MathML</a>

Define the function <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/103/mathml/M135','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/103/mathml/M135">View MathML</a> as

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/103/mathml/M136','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/103/mathml/M136">View MathML</a>

(3.9)

Lemma 3.3Let Deg denote the coincidence degree. Let the following conditions hold:

(i) <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/103/mathml/M137','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/103/mathml/M137">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/103/mathml/M138','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/103/mathml/M138">View MathML</a> (Mis given in Lemma 3.2);

(ii) <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/103/mathml/M139','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/103/mathml/M139">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/103/mathml/M138','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/103/mathml/M138">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/103/mathml/M141','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/103/mathml/M141">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/103/mathml/M142','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/103/mathml/M142">View MathML</a>;

(iii) <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/103/mathml/M143','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/103/mathml/M143">View MathML</a>, and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/103/mathml/M144','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/103/mathml/M144">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/103/mathml/M145','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/103/mathml/M145">View MathML</a>.

Then

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/103/mathml/M146','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/103/mathml/M146">View MathML</a>

Proof Consider the following family of equations:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/103/mathml/M147','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/103/mathml/M147">View MathML</a>

(3.10)

We will show

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/103/mathml/M148','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/103/mathml/M148">View MathML</a>

(3.11)

If not, then there exist some <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/103/mathml/M149','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/103/mathml/M149">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/103/mathml/M150','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/103/mathml/M150">View MathML</a> such that (3.10) holds. Note that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/103/mathml/M150','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/103/mathml/M150">View MathML</a> if and only if <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/103/mathml/M152','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/103/mathml/M152">View MathML</a> and either <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/103/mathml/M153','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/103/mathml/M153">View MathML</a> for some <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/103/mathml/M154','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/103/mathml/M154">View MathML</a>, or <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/103/mathml/M155','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/103/mathml/M155">View MathML</a> for some <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/103/mathml/M156','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/103/mathml/M156">View MathML</a>. There are two possibilities.

Case (I). If <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/103/mathml/M153','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/103/mathml/M153">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/103/mathml/M158','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/103/mathml/M158">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/103/mathml/M159','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/103/mathml/M159">View MathML</a>. In this case, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/103/mathml/M160','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/103/mathml/M160">View MathML</a>, or <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/103/mathml/M161','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/103/mathml/M161">View MathML</a>.

Subcase (1). Suppose <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/103/mathml/M160','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/103/mathml/M160">View MathML</a>. Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/103/mathml/M163','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/103/mathml/M163">View MathML</a>. Then <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/103/mathml/M164','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/103/mathml/M164">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/103/mathml/M63','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/103/mathml/M63">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/103/mathml/M166','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/103/mathml/M166">View MathML</a>. When <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/103/mathml/M167','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/103/mathml/M167">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/103/mathml/M164','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/103/mathml/M164">View MathML</a>, we get

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/103/mathml/M169','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/103/mathml/M169">View MathML</a>

If <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/103/mathml/M170','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/103/mathml/M170">View MathML</a>, then <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/103/mathml/M171','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/103/mathml/M171">View MathML</a>. From condition (i), we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/103/mathml/M172','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/103/mathml/M172">View MathML</a>

which is a contradiction.

If <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/103/mathml/M173','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/103/mathml/M173">View MathML</a>, then <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/103/mathml/M174','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/103/mathml/M174">View MathML</a> for some <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/103/mathml/M175','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/103/mathml/M175">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/103/mathml/M176','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/103/mathml/M176">View MathML</a>. Thus from (iii), we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/103/mathml/M177','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/103/mathml/M177">View MathML</a>

On the other hand, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/103/mathml/M164','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/103/mathml/M164">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/103/mathml/M63','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/103/mathml/M63">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/103/mathml/M180','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/103/mathml/M180">View MathML</a>, thus

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/103/mathml/M181','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/103/mathml/M181">View MathML</a>

which is a contradiction.

If <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/103/mathml/M182','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/103/mathml/M182">View MathML</a>, it is easy to see that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/103/mathml/M183','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/103/mathml/M183">View MathML</a>. Since <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/103/mathml/M75','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/103/mathml/M75">View MathML</a>, we have <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/103/mathml/M185','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/103/mathml/M185">View MathML</a>, thus we can obtain <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/103/mathml/M186','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/103/mathml/M186">View MathML</a>. However, from condition (i) we know

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/103/mathml/M187','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/103/mathml/M187">View MathML</a>

which is a contradiction.

If <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/103/mathml/M188','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/103/mathml/M188">View MathML</a>, then <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/103/mathml/M189','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/103/mathml/M189">View MathML</a>. This means that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/103/mathml/M190','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/103/mathml/M190">View MathML</a>. Since <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/103/mathml/M191','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/103/mathml/M191">View MathML</a>, we have <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/103/mathml/M192','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/103/mathml/M192">View MathML</a>. However, according to the above arguments, we know <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/103/mathml/M193','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/103/mathml/M193">View MathML</a>, which is a contradiction.

Subcase (2). Suppose <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/103/mathml/M161','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/103/mathml/M161">View MathML</a> for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/103/mathml/M195','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/103/mathml/M195">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/103/mathml/M196','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/103/mathml/M196">View MathML</a>. Obviously, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/103/mathml/M197','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/103/mathml/M197">View MathML</a>. Since <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/103/mathml/M198','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/103/mathml/M198">View MathML</a>. Thus we get

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/103/mathml/M199','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/103/mathml/M199">View MathML</a>

(3.12)

Let

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/103/mathml/M200','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/103/mathml/M200">View MathML</a>

Without loss of generality, we suppose that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/103/mathml/M201','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/103/mathml/M201">View MathML</a> for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/103/mathml/M195','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/103/mathml/M195">View MathML</a>. Then we have three possibilities.

(i) If <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/103/mathml/M203','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/103/mathml/M203">View MathML</a>, then there exists a sufficiently small neighborhood of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/103/mathml/M204','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/103/mathml/M204">View MathML</a> such that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/103/mathml/M205','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/103/mathml/M205">View MathML</a>. Thus <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/103/mathml/M206','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/103/mathml/M206">View MathML</a>, and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/103/mathml/M207','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/103/mathml/M207">View MathML</a> has a local maximum value on <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/103/mathml/M204','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/103/mathml/M204">View MathML</a>, which implies <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/103/mathml/M209','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/103/mathml/M209">View MathML</a>. But in this case, from (3.7), (3.12), and (ii), we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/103/mathml/M210','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/103/mathml/M210">View MathML</a>

(3.13)

which is a contradiction.

(ii) If <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/103/mathml/M211','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/103/mathml/M211">View MathML</a>, then there exists a sufficiently small neighborhood of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/103/mathml/M204','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/103/mathml/M204">View MathML</a> such that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/103/mathml/M213','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/103/mathml/M213">View MathML</a>. Thus <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/103/mathml/M214','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/103/mathml/M214">View MathML</a>, and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/103/mathml/M207','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/103/mathml/M207">View MathML</a> has a local maximum value on <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/103/mathml/M204','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/103/mathml/M204">View MathML</a>, which implies <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/103/mathml/M217','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/103/mathml/M217">View MathML</a>. But in this case, from (3.8), (3.12), and (ii), we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/103/mathml/M218','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/103/mathml/M218">View MathML</a>

which is a contradiction.

(iii) If <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/103/mathml/M219','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/103/mathml/M219">View MathML</a>, then there exists a sufficiently small neighborhood of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/103/mathml/M204','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/103/mathml/M204">View MathML</a> such that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/103/mathml/M205','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/103/mathml/M205">View MathML</a>. Thus <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/103/mathml/M206','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/103/mathml/M206">View MathML</a>. By the same argument as in (3.13), we reach a contradiction.

Case (II). <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/103/mathml/M223','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/103/mathml/M223">View MathML</a> for some <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/103/mathml/M156','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/103/mathml/M156">View MathML</a>. In this case, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/103/mathml/M225','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/103/mathml/M225">View MathML</a>, or <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/103/mathml/M226','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/103/mathml/M226">View MathML</a>.

Subcase (3). Suppose <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/103/mathml/M225','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/103/mathml/M225">View MathML</a> for some <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/103/mathml/M228','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/103/mathml/M228">View MathML</a>. Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/103/mathml/M229','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/103/mathml/M229">View MathML</a> be defined in the above. Obviously, we have <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/103/mathml/M164','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/103/mathml/M164">View MathML</a> for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/103/mathml/M63','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/103/mathml/M63">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/103/mathml/M180','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/103/mathml/M180">View MathML</a>. It follows from (iii) that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/103/mathml/M233','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/103/mathml/M233">View MathML</a>

If <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/103/mathml/M234','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/103/mathml/M234">View MathML</a>, then <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/103/mathml/M235','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/103/mathml/M235">View MathML</a>. Form (3.11) and (i), we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/103/mathml/M236','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/103/mathml/M236">View MathML</a>

which is a contradiction.

If <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/103/mathml/M237','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/103/mathml/M237">View MathML</a>, together with (iii), we get <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/103/mathml/M238','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/103/mathml/M238">View MathML</a>. But <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/103/mathml/M239','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/103/mathml/M239">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/103/mathml/M164','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/103/mathml/M164">View MathML</a> for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/103/mathml/M67','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/103/mathml/M67">View MathML</a>, which yields <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/103/mathml/M176','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/103/mathml/M176">View MathML</a>, and we reach a contradiction.

Similar to subcase (2), we can show that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/103/mathml/M226','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/103/mathml/M226">View MathML</a> is also impossible. Combining the results of case (I) and case (II) we obtain (3.11).

On the other hand, for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/103/mathml/M149','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/103/mathml/M149">View MathML</a>, it follows from [21] that (3.10) is equivalent to the following family of operator equations:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/103/mathml/M245','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/103/mathml/M245">View MathML</a>

(3.14)

where E is the identity mapping.

Note that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/103/mathml/M246','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/103/mathml/M246">View MathML</a>, and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/103/mathml/M247','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/103/mathml/M247">View MathML</a>

From (3.5) and (3.6), we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/103/mathml/M248','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/103/mathml/M248">View MathML</a>

and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/103/mathml/M249','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/103/mathml/M249">View MathML</a>

By the Ascoli-Arzela theorem, it is easy to show that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/103/mathml/M250','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/103/mathml/M250">View MathML</a> is bounded and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/103/mathml/M251','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/103/mathml/M251">View MathML</a> is compact. Thus N is L-compact on <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/103/mathml/M252','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/103/mathml/M252">View MathML</a>.

Define a mapping <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/103/mathml/M253','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/103/mathml/M253">View MathML</a>,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/103/mathml/M254','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/103/mathml/M254">View MathML</a>

Then it is easy to prove that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/103/mathml/M255','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/103/mathml/M255">View MathML</a> is completely continuous and we claim that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/103/mathml/M256','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/103/mathml/M256">View MathML</a>

(3.15)

In fact, for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/103/mathml/M257','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/103/mathml/M257">View MathML</a>, it follows from (3.10) and (3.14) that (3.15) holds. For <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/103/mathml/M258','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/103/mathml/M258">View MathML</a>, if there exists a <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/103/mathml/M259','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/103/mathml/M259">View MathML</a> such that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/103/mathml/M260','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/103/mathml/M260">View MathML</a>, that is <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/103/mathml/M261','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/103/mathml/M261">View MathML</a>, in this case, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/103/mathml/M262','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/103/mathml/M262">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/103/mathml/M263','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/103/mathml/M263">View MathML</a>, hence <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/103/mathml/M264','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/103/mathml/M264">View MathML</a> or <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/103/mathml/M265','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/103/mathml/M265">View MathML</a>. However, it follows from (iii) that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/103/mathml/M266','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/103/mathml/M266">View MathML</a>

Thus we get <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/103/mathml/M267','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/103/mathml/M267">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/103/mathml/M268','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/103/mathml/M268">View MathML</a>, which contradicts <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/103/mathml/M269','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/103/mathml/M269">View MathML</a> for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/103/mathml/M270','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/103/mathml/M270">View MathML</a>. Therefore (3.15) holds. As follows from [21] and by using the invariance of Leray-Schauder degree under homotopy, we obtain

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/103/mathml/M271','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/103/mathml/M271">View MathML</a>

Since KerL is one dimensional and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/103/mathml/M267','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/103/mathml/M267">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/103/mathml/M268','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/103/mathml/M268">View MathML</a>, we get

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/103/mathml/M274','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/103/mathml/M274">View MathML</a>

From the property of coincidence degree we proved Lemma 3.3. □

Lemma 3.4Assume that

(c1) there exist lower and upper solutions<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/103/mathml/M275','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/103/mathml/M275">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/103/mathml/M276','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/103/mathml/M276">View MathML</a>of the problem (1.1)-(1.2), respectively, with<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/103/mathml/M58','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/103/mathml/M58">View MathML</a>;

(c2) <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/103/mathml/M5','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/103/mathml/M5">View MathML</a>is continuous and has property (H) relative toα, β;

(c3) <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/103/mathml/M279','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/103/mathml/M279">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/103/mathml/M280','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/103/mathml/M280">View MathML</a>are continuous for each<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/103/mathml/M281','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/103/mathml/M281">View MathML</a>, and satisfy

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/103/mathml/M282','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/103/mathml/M282">View MathML</a>

Then<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/103/mathml/M283','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/103/mathml/M283">View MathML</a>.

Proof Choose <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/103/mathml/M110','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/103/mathml/M110">View MathML</a> large enough for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/103/mathml/M285','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/103/mathml/M285">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/103/mathml/M286','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/103/mathml/M286">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/103/mathml/M63','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/103/mathml/M63">View MathML</a>, such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/103/mathml/M288','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/103/mathml/M288">View MathML</a>

and let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/103/mathml/M289','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/103/mathml/M289">View MathML</a> (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/103/mathml/M290','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/103/mathml/M290">View MathML</a>) be defined in (3.7) and (3.8), then it follows from Lemma 3.2 that we can choose <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/103/mathml/M291','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/103/mathml/M291">View MathML</a> large enough such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/103/mathml/M292','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/103/mathml/M292">View MathML</a>

Define a set <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/103/mathml/M293','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/103/mathml/M293">View MathML</a> as

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/103/mathml/M294','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/103/mathml/M294">View MathML</a>

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/103/mathml/M135','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/103/mathml/M135">View MathML</a> is given in (3.9).

Define the auxiliary functions F and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/103/mathml/M296','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/103/mathml/M296">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/103/mathml/M297','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/103/mathml/M297">View MathML</a> as follows:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/103/mathml/M298','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/103/mathml/M298">View MathML</a>

where

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/103/mathml/M299','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/103/mathml/M299">View MathML</a>

We then generalize F to <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/103/mathml/M300','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/103/mathml/M300">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/103/mathml/M301','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/103/mathml/M301">View MathML</a>. It is easy to see that F, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/103/mathml/M296','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/103/mathml/M296">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/103/mathml/M297','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/103/mathml/M297">View MathML</a> are continuous and satisfy

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/103/mathml/M304','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/103/mathml/M304">View MathML</a>

Moreover, when <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/103/mathml/M141','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/103/mathml/M141">View MathML</a>, F, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/103/mathml/M296','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/103/mathml/M296">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/103/mathml/M297','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/103/mathml/M297">View MathML</a> are bounded. It follows from Lemma 3.3 that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/103/mathml/M308','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/103/mathml/M308">View MathML</a>

where

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/103/mathml/M309','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/103/mathml/M309">View MathML</a>

Next we show

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/103/mathml/M310','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/103/mathml/M310">View MathML</a>

(3.16)

It suffices to show that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/103/mathml/M311','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/103/mathml/M311">View MathML</a>

(3.17)

In fact, let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/103/mathml/M312','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/103/mathml/M312">View MathML</a> such that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/103/mathml/M313','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/103/mathml/M313">View MathML</a> and assume that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/103/mathml/M314','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/103/mathml/M314">View MathML</a>

Case (1). If there exists <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/103/mathml/M315','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/103/mathml/M315">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/103/mathml/M316','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/103/mathml/M316">View MathML</a> such that the function <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/103/mathml/M317','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/103/mathml/M317">View MathML</a> attains its maximum value <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/103/mathml/M318','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/103/mathml/M318">View MathML</a>, which implies <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/103/mathml/M319','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/103/mathml/M319">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/103/mathml/M320','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/103/mathml/M320">View MathML</a>. But on the other hand,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/103/mathml/M321','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/103/mathml/M321">View MathML</a>

(3.18)

which implies <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/103/mathml/M322','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/103/mathml/M322">View MathML</a>. We reach a contradiction.

If <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/103/mathml/M323','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/103/mathml/M323">View MathML</a>, then <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/103/mathml/M324','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/103/mathml/M324">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/103/mathml/M325','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/103/mathml/M325">View MathML</a>. On the other hand, since <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/103/mathml/M75','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/103/mathml/M75">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/103/mathml/M276','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/103/mathml/M276">View MathML</a> is a strict upper solution of problem (1.1)-(1.2), we see that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/103/mathml/M328','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/103/mathml/M328">View MathML</a>. Thus <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/103/mathml/M329','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/103/mathml/M329">View MathML</a>. Note that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/103/mathml/M330','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/103/mathml/M330">View MathML</a> assumes the maximum value at <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/103/mathml/M331','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/103/mathml/M331">View MathML</a>, there exists <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/103/mathml/M332','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/103/mathml/M332">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/103/mathml/M333','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/103/mathml/M333">View MathML</a> such that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/103/mathml/M334','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/103/mathml/M334">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/103/mathml/M335','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/103/mathml/M335">View MathML</a>. By the same argument as in (3.18) where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/103/mathml/M336','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/103/mathml/M336">View MathML</a>, we reach a contradiction.

If <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/103/mathml/M337','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/103/mathml/M337">View MathML</a>, then <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/103/mathml/M338','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/103/mathml/M338">View MathML</a>. According to (2.6), we get

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/103/mathml/M339','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/103/mathml/M339">View MathML</a>

if <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/103/mathml/M340','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/103/mathml/M340">View MathML</a>, which is a contradiction. Thus <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/103/mathml/M341','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/103/mathml/M341">View MathML</a>, which implies that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/103/mathml/M330','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/103/mathml/M330">View MathML</a> also attains its maximum value at <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/103/mathml/M343','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/103/mathml/M343">View MathML</a>. By the same argument as in (3.18) where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/103/mathml/M344','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/103/mathml/M344">View MathML</a>, we reach a contradiction.

Hence the function <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/103/mathml/M330','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/103/mathml/M330">View MathML</a> cannot have any nonnegative maximum value on the interval <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/103/mathml/M57','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/103/mathml/M57">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/103/mathml/M347','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/103/mathml/M347">View MathML</a> for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/103/mathml/M348','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/103/mathml/M348">View MathML</a>.

Case (2). If there exists a <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/103/mathml/M349','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/103/mathml/M349">View MathML</a> such that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/103/mathml/M350','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/103/mathml/M350">View MathML</a>, from case (1), we get <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/103/mathml/M351','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/103/mathml/M351">View MathML</a> for some <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/103/mathml/M281','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/103/mathml/M281">View MathML</a>. Hence <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/103/mathml/M353','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/103/mathml/M353">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/103/mathml/M354','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/103/mathml/M354">View MathML</a>, and consequently we have <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/103/mathml/M355','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/103/mathml/M355">View MathML</a>, which implies <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/103/mathml/M356','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/103/mathml/M356">View MathML</a>, because

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/103/mathml/M357','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/103/mathml/M357">View MathML</a>

Consequently, we get <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/103/mathml/M358','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/103/mathml/M358">View MathML</a> or <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/103/mathml/M359','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/103/mathml/M359">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/103/mathml/M360','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/103/mathml/M360">View MathML</a>.

By the continuity of f and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/103/mathml/M276','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/103/mathml/M276">View MathML</a> is a strict upper solution of problem (1.1)-(1.2), there exists a sequence <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/103/mathml/M362','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/103/mathml/M362">View MathML</a> with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/103/mathml/M363','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/103/mathml/M363">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/103/mathml/M364','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/103/mathml/M364">View MathML</a>, as <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/103/mathml/M365','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/103/mathml/M365">View MathML</a> such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/103/mathml/M366','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/103/mathml/M366">View MathML</a>

(3.19)

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/103/mathml/M367','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/103/mathml/M367">View MathML</a> are from the mean value theorem. As before, we also get

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/103/mathml/M368','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/103/mathml/M368">View MathML</a>

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/103/mathml/M369','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/103/mathml/M369">View MathML</a>. As a result, we can obtain

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/103/mathml/M370','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/103/mathml/M370">View MathML</a>

which is a contradiction to (3.19).

Case (3). If <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/103/mathml/M371','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/103/mathml/M371">View MathML</a> for all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/103/mathml/M63','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/103/mathml/M63">View MathML</a>, then there must be a <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/103/mathml/M373','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/103/mathml/M373">View MathML</a> (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/103/mathml/M374','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/103/mathml/M374">View MathML</a>) such that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/103/mathml/M375','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/103/mathml/M375">View MathML</a>, and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/103/mathml/M376','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/103/mathml/M376">View MathML</a>, which implies <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/103/mathml/M377','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/103/mathml/M377">View MathML</a>. Namely, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/103/mathml/M378','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/103/mathml/M378">View MathML</a>. However, this is impossible because

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/103/mathml/M379','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/103/mathml/M379">View MathML</a>

Thus we have proved that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/103/mathml/M380','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/103/mathml/M380">View MathML</a> on J. Similarly we can show that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/103/mathml/M381','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/103/mathml/M381">View MathML</a> on J. It then follows that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/103/mathml/M382','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/103/mathml/M382">View MathML</a> on J.

We now shall prove that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/103/mathml/M383','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/103/mathml/M383">View MathML</a>

(3.20)

Assume that (3.20) cannot hold. Thus there are two possibilities:

(a) There exists <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/103/mathml/M384','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/103/mathml/M384">View MathML</a> for some <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/103/mathml/M196','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/103/mathml/M196">View MathML</a> such that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/103/mathml/M386','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/103/mathml/M386">View MathML</a>.

(b) There exists some <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/103/mathml/M196','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/103/mathml/M196">View MathML</a> such that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/103/mathml/M388','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/103/mathml/M388">View MathML</a>.

For case (a), we assume <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/103/mathml/M389','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/103/mathml/M389">View MathML</a>.

Since <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/103/mathml/M390','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/103/mathml/M390">View MathML</a>. Thus we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/103/mathml/M391','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/103/mathml/M391">View MathML</a>

(3.21)

If <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/103/mathml/M392','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/103/mathml/M392">View MathML</a>, then there exists a sufficiently small neighborhood of τ such that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/103/mathml/M205','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/103/mathml/M205">View MathML</a>. Thus <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/103/mathml/M394','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/103/mathml/M394">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/103/mathml/M395','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/103/mathml/M395">View MathML</a>. However, it follows from (3.7) and (3.21) that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/103/mathml/M396','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/103/mathml/M396">View MathML</a>

(3.22)

which is a contradiction.

If <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/103/mathml/M397','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/103/mathml/M397">View MathML</a>, then there exists a sufficiently small neighborhood of τ such that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/103/mathml/M213','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/103/mathml/M213">View MathML</a>. Thus <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/103/mathml/M399','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/103/mathml/M399">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/103/mathml/M400','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/103/mathml/M400">View MathML</a>. However, it follows from (3.8) and (3.21) that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/103/mathml/M401','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/103/mathml/M401">View MathML</a>

a contradiction.

If <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/103/mathml/M402','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/103/mathml/M402">View MathML</a>, then there exists a sufficiently small neighborhood of τ such that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/103/mathml/M205','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/103/mathml/M205">View MathML</a>. Thus <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/103/mathml/M394','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/103/mathml/M394">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/103/mathml/M400','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/103/mathml/M400">View MathML</a>. By the similarly argument as in (3.22), we reach a contradiction.

Hence <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/103/mathml/M406','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/103/mathml/M406">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/103/mathml/M407','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/103/mathml/M407">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/103/mathml/M408','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/103/mathml/M408">View MathML</a>. Similarly, we can prove <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/103/mathml/M409','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/103/mathml/M409">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/103/mathml/M410','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/103/mathml/M410">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/103/mathml/M411','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/103/mathml/M411">View MathML</a>.

Likewise, we can show that case (b) is also impossible, and thus (3.20) holds. Combining the above results, we see that if <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/103/mathml/M412','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/103/mathml/M412">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/103/mathml/M413','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/103/mathml/M413">View MathML</a>, then <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/103/mathml/M414','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/103/mathml/M414">View MathML</a>. Hence (3.17) is proved. From the property of coincidence, we know that (3.16) holds. Since in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/103/mathml/M293','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/103/mathml/M293">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/103/mathml/M416','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/103/mathml/M416">View MathML</a>, we have <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/103/mathml/M417','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/103/mathml/M417">View MathML</a>. The proof is complete. □

4 Main results

We are now in a position to prove our main result on the existence of at least <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/103/mathml/M418','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/103/mathml/M418">View MathML</a> solutions of boundary value problem (1.1)-(1.2).

Theorem 4.1Assume that

(H1) there existn (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/103/mathml/M35','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/103/mathml/M35">View MathML</a>and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/103/mathml/M36','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/103/mathml/M36">View MathML</a>) pairs of strict lower and upper solutions<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/103/mathml/M421','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/103/mathml/M421">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/103/mathml/M422','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/103/mathml/M422">View MathML</a>of the problem (1.1)-(1.2) such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/103/mathml/M423','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/103/mathml/M423">View MathML</a>

(H2) <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/103/mathml/M5','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/103/mathml/M5">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/103/mathml/M425','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/103/mathml/M425">View MathML</a>is continuous on<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/103/mathml/M426','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/103/mathml/M426">View MathML</a>and has property (H) relative to<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/103/mathml/M427','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/103/mathml/M427">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/103/mathml/M428','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/103/mathml/M428">View MathML</a>;

(H3) <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/103/mathml/M279','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/103/mathml/M279">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/103/mathml/M280','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/103/mathml/M280">View MathML</a>are continuous for each<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/103/mathml/M281','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/103/mathml/M281">View MathML</a>, and satisfy

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/103/mathml/M432','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/103/mathml/M432">View MathML</a>

Then BVP (1.1)-(1.2) has at least<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/103/mathml/M418','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/103/mathml/M418">View MathML</a>solutions<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/103/mathml/M434','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/103/mathml/M434">View MathML</a>such that<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/103/mathml/M435','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/103/mathml/M435">View MathML</a>, and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/103/mathml/M436','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/103/mathml/M436">View MathML</a>

Proof Choose <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/103/mathml/M110','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/103/mathml/M110">View MathML</a> large enough such that, for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/103/mathml/M438','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/103/mathml/M438">View MathML</a>,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/103/mathml/M439','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/103/mathml/M439">View MathML</a>

and let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/103/mathml/M135','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/103/mathml/M135">View MathML</a> be defined in (3.10), then it follows from Lemma 3.4 that we can take <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/103/mathml/M291','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/103/mathml/M291">View MathML</a> large enough such that, for all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/103/mathml/M442','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/103/mathml/M442">View MathML</a>,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/103/mathml/M443','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/103/mathml/M443">View MathML</a>

Define the set <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/103/mathml/M444','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/103/mathml/M444">View MathML</a> (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/103/mathml/M445','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/103/mathml/M445">View MathML</a>) as

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/103/mathml/M446','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/103/mathml/M446">View MathML</a>

Also

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/103/mathml/M447','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/103/mathml/M447">View MathML</a>

For <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/103/mathml/M448','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/103/mathml/M448">View MathML</a>, we define the sets

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/103/mathml/M449','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/103/mathml/M449">View MathML</a>

Then by Lemma 3.4, we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/103/mathml/M450','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/103/mathml/M450">View MathML</a>

From the additive property of coincidence degree, we obtain

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/103/mathml/M451','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/103/mathml/M451">View MathML</a>

Since <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/103/mathml/M36','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/103/mathml/M36">View MathML</a> is arbitrary, we first deal with the case <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/103/mathml/M453','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/103/mathml/M453">View MathML</a>, the above discussion implies that the equation <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/103/mathml/M454','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/103/mathml/M454">View MathML</a>, that is, the problem (1.1)-(1.2) has at least one solution in the set <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/103/mathml/M455','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/103/mathml/M455">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/103/mathml/M456','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/103/mathml/M456">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/103/mathml/M457','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/103/mathml/M457">View MathML</a> respectively. That is, there exist at least three different solutions <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/103/mathml/M458','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/103/mathml/M458">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/103/mathml/M459','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/103/mathml/M459">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/103/mathml/M460','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/103/mathml/M460">View MathML</a> such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/103/mathml/M461','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/103/mathml/M461">View MathML</a>

and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/103/mathml/M462','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/103/mathml/M462">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/103/mathml/M463','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/103/mathml/M463">View MathML</a>.

For <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/103/mathml/M464','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/103/mathml/M464">View MathML</a>, replacing <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/103/mathml/M465','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/103/mathml/M465">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/103/mathml/M466','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/103/mathml/M466">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/103/mathml/M467','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/103/mathml/M467">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/103/mathml/M468','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/103/mathml/M468">View MathML</a> by <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/103/mathml/M466','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/103/mathml/M466">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/103/mathml/M470','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/103/mathml/M470">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/103/mathml/M468','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/103/mathml/M468">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/103/mathml/M472','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/103/mathml/M472">View MathML</a> respectively, then we can obtain another two different solutions <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/103/mathml/M473','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/103/mathml/M473">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/103/mathml/M474','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/103/mathml/M474">View MathML</a> of the problem (1.1)-(1.2) such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/103/mathml/M475','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/103/mathml/M475">View MathML</a>

Along this way, we can complete the proof by the induction method. □

We now present an example to illustrate that the assumptions of our theorem can be verified.

Example 1 Consider the following boundary value problem:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/103/mathml/M476','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/103/mathml/M476">View MathML</a>

(4.1)

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/103/mathml/M477','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/103/mathml/M477">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/103/mathml/M478','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/103/mathml/M478">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/103/mathml/M479','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/103/mathml/M479">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/103/mathml/M6','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/103/mathml/M6">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/103/mathml/M481','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/103/mathml/M481">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/103/mathml/M10','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/103/mathml/M10">View MathML</a>.

First, we have <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/103/mathml/M483','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/103/mathml/M483">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/103/mathml/M484','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/103/mathml/M484">View MathML</a>, thus <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/103/mathml/M485','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/103/mathml/M485">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/103/mathml/M486','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/103/mathml/M486">View MathML</a>.

Next, it is clear that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/103/mathml/M487','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/103/mathml/M487">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/103/mathml/M488','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/103/mathml/M488">View MathML</a> are n (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/103/mathml/M35','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/103/mathml/M35">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/103/mathml/M36','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/103/mathml/M36">View MathML</a>) pairs of strict lower and upper solutions of the problem (4.1), respectively. It can be seen that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/103/mathml/M491','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/103/mathml/M491">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/103/mathml/M492','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/103/mathml/M492">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/103/mathml/M493','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/103/mathml/M493">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/103/mathml/M494','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/103/mathml/M494">View MathML</a>. Thus conditions (H1) and (H3) of Theorem (4.1) hold.

Since <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/103/mathml/M495','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/103/mathml/M495">View MathML</a>, it follows that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/103/mathml/M496','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/103/mathml/M496">View MathML</a>

For all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/103/mathml/M497','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/103/mathml/M497">View MathML</a>, we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/103/mathml/M498','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/103/mathml/M498">View MathML</a>

Take <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/103/mathml/M499','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/103/mathml/M499">View MathML</a>, then

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/103/mathml/M500','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/103/mathml/M500">View MathML</a>

Hence f has property (H). It follows from Theorem 4.1 that the problem (4.1) has at least <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/103/mathml/M418','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/103/mathml/M418">View MathML</a> different solutions <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/103/mathml/M502','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/103/mathml/M502">View MathML</a> such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/103/mathml/M503','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/103/mathml/M503">View MathML</a>

Competing interests

The authors declare that they have no competing interests.

Authors’ contributions

The authors declare that the study was realized in collaboration with the same responsibility. All authors read and approved the final manuscript.

Acknowledgements

The authors are highly grateful to the referees for careful reading and comments on this paper. The research is supported by Hunan Provincial Natural Science Foundation of China (No. 13JJ3106, and 12JJ2004); it is also supported by the National Natural Science Foundation of China (No. 61074067, 11271372, and 11201138).

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