Open Access Research

Structure of positive solution sets of differential boundary value problems

Xu Xian and Zhu Xunxia*

Author Affiliations

Department of Mathematics, Jiangsu Normal University, Xuzhou, Jiangsu, 221116, P.R. China

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Boundary Value Problems 2013, 2013:100  doi:10.1186/1687-2770-2013-100


The electronic version of this article is the complete one and can be found online at: http://www.boundaryvalueproblems.com/content/2013/1/100


Received:26 September 2012
Accepted:8 March 2013
Published:22 April 2013

© 2013 Xian and Xunxia; licensee Springer

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

Abstract

In this paper, we first obtain some results on the structure of positive solution sets of differential boundary value problems. Then by using the results, we obtain an existence result for differential boundary value problems. The method used to show the main result is the global bifurcation theory.

Keywords:
structure of positive solution sets; differential boundary value problems; bifurcation theory

1 Introduction

This paper considers the differential boundary value problem

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/100/mathml/M1','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/100/mathml/M1">View MathML</a>

(1.1)

where f is ϕ-superlinear at ∞ and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/100/mathml/M2','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/100/mathml/M2">View MathML</a> maybe negative and p is a positive continuous function, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/100/mathml/M3','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/100/mathml/M3">View MathML</a> is a parameter.

Equations of form (1.1) occur in the study for the p-Laplacian equation, non-Newtonial fluid theory and the turbulent flow of a gas in a porous medium. The case where

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/100/mathml/M4','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/100/mathml/M4">View MathML</a>

i.e., perturbations of the p-Laplacian, has received much attention in the recent literature. Also, problem (1.1) with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/100/mathml/M5','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/100/mathml/M5">View MathML</a> has been studied by several authors in recent years (see [1] and the references therein). Here, we are interested in the case when <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/100/mathml/M6','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/100/mathml/M6">View MathML</a> may be negative (the so-called semipositone case) (see [2] and its references for a review). As pointed out by Lions in [3], semi-positone problems are mathematically very challenging. During the last ten years, finding positive solutions to semi-positone problems has been actively pursued and significant progress on semi-positone problems has taken place; see [4-8] and the references therein. For instance, Hai et al.[9] considered the existence positive solution of (1.1). Under some super-linear conditions on the non-linear term f, they proved that there exists <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/100/mathml/M7','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/100/mathml/M7">View MathML</a> such that (1.1) has one positive solution for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/100/mathml/M8','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/100/mathml/M8">View MathML</a>. The main method in [9] used to show the main result are the fixed-point theorems.

The main purpose of this paper is going to study the structure of the positive set of (1.1). Rabinowitz [10] gave the first important results on the structure of the solution sets of non-linear equations and obtained by the degree theoretic method. Amamn [11] studied the structure of the positive solution set of non-linear equations; the reader is referred to [12,13] for other results concerning the structure of solution sets of non-linear equations. In our paper, we will study the existence results for an unbounded connected component of a positive solution set for the differential boundary value problem of (1.1). This paper generalizes some results from the literature [9]. The paper is arranged as follows. In Section 2, we will give some preliminary lemmas. The main results will be given in Section 3.

2 Some lemmas

For convenience, we make the following assumptions:

(A1) ϕ is an odd, increasing homeomorphism on R with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/100/mathml/M9','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/100/mathml/M9">View MathML</a> concave on <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/100/mathml/M10','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/100/mathml/M10">View MathML</a>.

(A2) For each <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/100/mathml/M11','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/100/mathml/M11">View MathML</a>, there exists <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/100/mathml/M12','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/100/mathml/M12">View MathML</a> such that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/100/mathml/M13','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/100/mathml/M13">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/100/mathml/M14','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/100/mathml/M14">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/100/mathml/M15','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/100/mathml/M15">View MathML</a> (note that (A2) implies the existence of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/100/mathml/M16','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/100/mathml/M16">View MathML</a> such that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/100/mathml/M17','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/100/mathml/M17">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/100/mathml/M14','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/100/mathml/M14">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/100/mathml/M19','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/100/mathml/M19">View MathML</a>).

(A3) <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/100/mathml/M20','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/100/mathml/M20">View MathML</a> is continuous.

(A4) <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/100/mathml/M21','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/100/mathml/M21">View MathML</a> is continuous and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/100/mathml/M22','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/100/mathml/M22">View MathML</a>

uniformly for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/100/mathml/M23','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/100/mathml/M23">View MathML</a>.

Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/100/mathml/M24','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/100/mathml/M24">View MathML</a>, the usual real Banach space of continuous functions with the maximum norm <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/100/mathml/M25','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/100/mathml/M25">View MathML</a>. Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/100/mathml/M26','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/100/mathml/M26">View MathML</a> for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/100/mathml/M23','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/100/mathml/M23">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/100/mathml/M28','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/100/mathml/M28">View MathML</a>. Then P is a cone of E. Define

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/100/mathml/M29','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/100/mathml/M29">View MathML</a>

Then <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/100/mathml/M30','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/100/mathml/M30">View MathML</a> for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/100/mathml/M31','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/100/mathml/M31">View MathML</a> (here <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/100/mathml/M32','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/100/mathml/M32">View MathML</a> is a constant). For <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/100/mathml/M33','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/100/mathml/M33">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/100/mathml/M34','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/100/mathml/M34">View MathML</a>, define

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/100/mathml/M35','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/100/mathml/M35">View MathML</a>

here C is a constant such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/100/mathml/M36','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/100/mathml/M36">View MathML</a>

We know that C exists and is unique for every <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/100/mathml/M37','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/100/mathml/M37">View MathML</a> (see [14]). Then <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/100/mathml/M38','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/100/mathml/M38">View MathML</a> if and only if u is a solution of

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/100/mathml/M39','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/100/mathml/M39">View MathML</a>

From [9], we have the following Lemmas 2.1 and 2.2.

Lemma 2.1Letϕsatisfy (A1) and (A2). Then for each<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/100/mathml/M40','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/100/mathml/M40">View MathML</a>, there exist constants<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/100/mathml/M41','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/100/mathml/M41">View MathML</a>and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/100/mathml/M42','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/100/mathml/M42">View MathML</a>such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/100/mathml/M43','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/100/mathml/M43">View MathML</a>

Further, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/100/mathml/M44','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/100/mathml/M44">View MathML</a>as<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/100/mathml/M45','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/100/mathml/M45">View MathML</a>and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/100/mathml/M46','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/100/mathml/M46">View MathML</a>as<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/100/mathml/M47','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/100/mathml/M47">View MathML</a>.

Lemma 2.2Letϕbe as in Lemma 2.1. Then there exist<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/100/mathml/M48','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/100/mathml/M48">View MathML</a>and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/100/mathml/M49','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/100/mathml/M49">View MathML</a>such that<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/100/mathml/M50','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/100/mathml/M50">View MathML</a>for<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/100/mathml/M51','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/100/mathml/M51">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/100/mathml/M52','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/100/mathml/M52">View MathML</a>.

Lemma 2.3Let<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/100/mathml/M53','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/100/mathml/M53">View MathML</a>such that<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/100/mathml/M54','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/100/mathml/M54">View MathML</a>for<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/100/mathml/M23','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/100/mathml/M23">View MathML</a>. Let<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/100/mathml/M56','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/100/mathml/M56">View MathML</a>andωbe a solution of

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/100/mathml/M57','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/100/mathml/M57">View MathML</a>

(2.1)

here<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/100/mathml/M58','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/100/mathml/M58">View MathML</a>is continuous function with<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/100/mathml/M59','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/100/mathml/M59">View MathML</a> (here<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/100/mathml/M32','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/100/mathml/M32">View MathML</a>is a constant) for<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/100/mathml/M23','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/100/mathml/M23">View MathML</a>, if<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/100/mathml/M62','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/100/mathml/M62">View MathML</a>, then

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/100/mathml/M63','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/100/mathml/M63">View MathML</a>

where<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/100/mathml/M64','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/100/mathml/M64">View MathML</a>.

Proof By integrating, it follows that (3.1) has the unique solution given by

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/100/mathml/M65','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/100/mathml/M65">View MathML</a>

where C is such that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/100/mathml/M66','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/100/mathml/M66">View MathML</a>. Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/100/mathml/M67','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/100/mathml/M67">View MathML</a> for some <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/100/mathml/M68','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/100/mathml/M68">View MathML</a>. Then

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/100/mathml/M69','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/100/mathml/M69">View MathML</a>

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/100/mathml/M70','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/100/mathml/M70">View MathML</a>. By Lemma 2.2, we get

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/100/mathml/M71','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/100/mathml/M71">View MathML</a>

Now

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/100/mathml/M72','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/100/mathml/M72">View MathML</a>

And so

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/100/mathml/M73','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/100/mathml/M73">View MathML</a>

(2.2)

here

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/100/mathml/M74','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/100/mathml/M74">View MathML</a>

Note that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/100/mathml/M75','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/100/mathml/M75">View MathML</a> satisfies

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/100/mathml/M76','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/100/mathml/M76">View MathML</a>

In fact, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/100/mathml/M77','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/100/mathml/M77">View MathML</a> for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/100/mathml/M78','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/100/mathml/M78">View MathML</a>. We next prove that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/100/mathml/M79','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/100/mathml/M79">View MathML</a> for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/100/mathml/M78','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/100/mathml/M78">View MathML</a>, here ν satisfies

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/100/mathml/M81','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/100/mathml/M81">View MathML</a>

Suppose it is not true, then <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/100/mathml/M82','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/100/mathml/M82">View MathML</a> has a negative absolute minimum at <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/100/mathml/M83','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/100/mathml/M83">View MathML</a>. Since

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/100/mathml/M84','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/100/mathml/M84">View MathML</a>

there exist <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/100/mathml/M85','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/100/mathml/M85">View MathML</a> such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/100/mathml/M86','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/100/mathml/M86">View MathML</a>

and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/100/mathml/M87','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/100/mathml/M87">View MathML</a>

Then

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/100/mathml/M88','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/100/mathml/M88">View MathML</a>

Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/100/mathml/M89','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/100/mathml/M89">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/100/mathml/M90','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/100/mathml/M90">View MathML</a>, then

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/100/mathml/M91','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/100/mathml/M91">View MathML</a>

On the other hand, using the inequality

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/100/mathml/M92','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/100/mathml/M92">View MathML</a>

and the fact that there exists <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/100/mathml/M93','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/100/mathml/M93">View MathML</a> such that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/100/mathml/M94','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/100/mathml/M94">View MathML</a>, we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/100/mathml/M95','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/100/mathml/M95">View MathML</a>

which is a contradiction. So, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/100/mathml/M79','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/100/mathml/M79">View MathML</a> for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/100/mathml/M78','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/100/mathml/M78">View MathML</a>. Obviously, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/100/mathml/M98','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/100/mathml/M98">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/100/mathml/M78','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/100/mathml/M78">View MathML</a>, since <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/100/mathml/M100','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/100/mathml/M100">View MathML</a> for each <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/100/mathml/M78','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/100/mathml/M78">View MathML</a>. From (2.2), we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/100/mathml/M102','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/100/mathml/M102">View MathML</a>

Similarly,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/100/mathml/M103','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/100/mathml/M103">View MathML</a>

If <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/100/mathml/M104','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/100/mathml/M104">View MathML</a>, then

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/100/mathml/M105','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/100/mathml/M105">View MathML</a>

The proof is complete. □

Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/100/mathml/M106','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/100/mathml/M106">View MathML</a> for each <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/100/mathml/M34','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/100/mathml/M34">View MathML</a>, where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/100/mathml/M108','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/100/mathml/M108">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/100/mathml/M109','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/100/mathml/M109">View MathML</a> are defined as that in Lemma 2.2 and Lemma 2.3, respectively. Then <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/100/mathml/M110','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/100/mathml/M110">View MathML</a> is also a cone of E. From Lemma 2.3, we know that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/100/mathml/M111','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/100/mathml/M111">View MathML</a> is completely continuous.

Let

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/100/mathml/M112','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/100/mathml/M112">View MathML</a>

From [[15], Lemma 29.1], we have Lemma 2.4.

Lemma 2.4LetXbe a compact metric space. Assume thatAandBare two disjoint closed subsets ofX. Then either there exist a connected component ofXmeeting bothAandBor<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/100/mathml/M113','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/100/mathml/M113">View MathML</a>where<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/100/mathml/M114','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/100/mathml/M114">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/100/mathml/M115','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/100/mathml/M115">View MathML</a>are disjoint compact subsets ofXcontainingAandB, respectively.

LetUbe an open and bounded subset of the metric space<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/100/mathml/M116','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/100/mathml/M116">View MathML</a>. We set<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/100/mathml/M117','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/100/mathml/M117">View MathML</a>, whose boundary is denoted by<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/100/mathml/M118','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/100/mathml/M118">View MathML</a>. Consider a map<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/100/mathml/M119','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/100/mathml/M119">View MathML</a>, such that<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/100/mathml/M120','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/100/mathml/M120">View MathML</a>is compact and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/100/mathml/M121','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/100/mathml/M121">View MathML</a>. Such a maphwill also be called an admissible homotopy onU. Ifhis an admissible homotopy, for every<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/100/mathml/M122','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/100/mathml/M122">View MathML</a>and every<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/100/mathml/M123','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/100/mathml/M123">View MathML</a>, one has that<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/100/mathml/M124','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/100/mathml/M124">View MathML</a>and it makes sense to evaluate<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/100/mathml/M125','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/100/mathml/M125">View MathML</a>.

Lemma 2.5Ifhis an admissible homotopy on<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/100/mathml/M126','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/100/mathml/M126">View MathML</a>, the<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/100/mathml/M127','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/100/mathml/M127">View MathML</a>is constant for all<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/100/mathml/M122','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/100/mathml/M122">View MathML</a>.

Lemma 2.6Let<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/100/mathml/M129','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/100/mathml/M129">View MathML</a>such that<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/100/mathml/M130','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/100/mathml/M130">View MathML</a>. Then for arbitrary<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/100/mathml/M131','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/100/mathml/M131">View MathML</a>, there exists<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/100/mathml/M132','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/100/mathml/M132">View MathML</a>such that for each<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/100/mathml/M133','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/100/mathml/M133">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/100/mathml/M134','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/100/mathml/M134">View MathML</a>and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/100/mathml/M135','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/100/mathml/M135">View MathML</a>,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/100/mathml/M136','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/100/mathml/M136">View MathML</a>

where<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/100/mathml/M137','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/100/mathml/M137">View MathML</a>.

Proof From (A4), for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/100/mathml/M138','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/100/mathml/M138">View MathML</a>, such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/100/mathml/M139','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/100/mathml/M139">View MathML</a>

there exists <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/100/mathml/M140','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/100/mathml/M140">View MathML</a> such that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/100/mathml/M141','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/100/mathml/M141">View MathML</a> for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/100/mathml/M142','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/100/mathml/M142">View MathML</a>. Let

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/100/mathml/M143','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/100/mathml/M143">View MathML</a>

Assume by contradiction that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/100/mathml/M144','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/100/mathml/M144">View MathML</a>

(2.3)

for some <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/100/mathml/M134','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/100/mathml/M134">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/100/mathml/M146','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/100/mathml/M146">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/100/mathml/M147','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/100/mathml/M147">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/100/mathml/M148','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/100/mathml/M148">View MathML</a>. Let

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/100/mathml/M149','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/100/mathml/M149">View MathML</a>

From Lemma 2.3, we know that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/100/mathml/M150','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/100/mathml/M150">View MathML</a>. Namely,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/100/mathml/M151','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/100/mathml/M151">View MathML</a>

(2.4)

From (2.3) and (2.4), we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/100/mathml/M152','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/100/mathml/M152">View MathML</a>

so <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/100/mathml/M153','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/100/mathml/M153">View MathML</a>. For <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/100/mathml/M147','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/100/mathml/M147">View MathML</a>, we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/100/mathml/M155','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/100/mathml/M155">View MathML</a>

for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/100/mathml/M156','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/100/mathml/M156">View MathML</a>. Therefore, let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/100/mathml/M157','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/100/mathml/M157">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/100/mathml/M158','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/100/mathml/M158">View MathML</a>, assume that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/100/mathml/M159','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/100/mathml/M159">View MathML</a>, then

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/100/mathml/M160','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/100/mathml/M160">View MathML</a>

where

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/100/mathml/M161','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/100/mathml/M161">View MathML</a>

and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/100/mathml/M162','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/100/mathml/M162">View MathML</a>

Thus,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/100/mathml/M163','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/100/mathml/M163">View MathML</a>

so <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/100/mathml/M164','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/100/mathml/M164">View MathML</a>, which is a contradiction. Then (2.3) holds. The proof is complete. □

3 Main results

For convenience, let us introduce the following symbols. For any r,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/100/mathml/M165','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/100/mathml/M165">View MathML</a>

Now we give our main results of this paper.

Theorem 3.1Suppose (A1) to (A4) hold. Then<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/100/mathml/M166','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/100/mathml/M166">View MathML</a>possesses an unbounded connected component<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/100/mathml/M167','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/100/mathml/M167">View MathML</a>such that<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/100/mathml/M168','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/100/mathml/M168">View MathML</a>for some<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/100/mathml/M169','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/100/mathml/M169">View MathML</a>and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/100/mathml/M170','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/100/mathml/M170">View MathML</a>

where<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/100/mathml/M171','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/100/mathml/M171">View MathML</a>denotes the projection of<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/100/mathml/M167','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/100/mathml/M167">View MathML</a>onto theλ-axis.

Proof We divide our proof into four steps.

Step 1. Let

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/100/mathml/M173','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/100/mathml/M173">View MathML</a>

(3.1)

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/100/mathml/M174','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/100/mathml/M174">View MathML</a>. Obviously, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/100/mathml/M175','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/100/mathml/M175">View MathML</a> is a completely continuous operator.

Note that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/100/mathml/M176','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/100/mathml/M176">View MathML</a> for all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/100/mathml/M177','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/100/mathml/M177">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/100/mathml/M33','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/100/mathml/M33">View MathML</a> with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/100/mathml/M179','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/100/mathml/M179">View MathML</a>. From Lemma 2.5, there exists <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/100/mathml/M180','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/100/mathml/M180">View MathML</a> large enough such that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/100/mathml/M181','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/100/mathml/M181">View MathML</a> and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/100/mathml/M182','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/100/mathml/M182">View MathML</a>

(3.2)

Obviously,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/100/mathml/M183','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/100/mathml/M183">View MathML</a>

(3.3)

Therefore,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/100/mathml/M184','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/100/mathml/M184">View MathML</a>

(3.4)

So, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/100/mathml/M185','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/100/mathml/M185">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/100/mathml/M186','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/100/mathml/M186">View MathML</a>.

Step 2. Let

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/100/mathml/M187','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/100/mathml/M187">View MathML</a>

From Lemma 2.6, we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/100/mathml/M188','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/100/mathml/M188">View MathML</a>

This implies that T has no bifurcation point on <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/100/mathml/M189','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/100/mathml/M189">View MathML</a>. From step 1,we have <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/100/mathml/M190','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/100/mathml/M190">View MathML</a>, then for each <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/100/mathml/M191','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/100/mathml/M191">View MathML</a>, denote by <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/100/mathml/M192','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/100/mathml/M192">View MathML</a> the connected component of the metric space <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/100/mathml/M193','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/100/mathml/M193">View MathML</a> emitting from <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/100/mathml/M194','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/100/mathml/M194">View MathML</a>. Now we will show that, there must exist <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/100/mathml/M195','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/100/mathml/M195">View MathML</a> such that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/100/mathml/M196','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/100/mathml/M196">View MathML</a> is unbounded. Assume on the contrary that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/100/mathml/M192','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/100/mathml/M192">View MathML</a> is bounded for each <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/100/mathml/M198','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/100/mathml/M198">View MathML</a>. Take a bounded open neighborhood <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/100/mathml/M199','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/100/mathml/M199">View MathML</a> in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/100/mathml/M193','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/100/mathml/M193">View MathML</a> for each <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/100/mathml/M192','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/100/mathml/M192">View MathML</a> such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/100/mathml/M202','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/100/mathml/M202">View MathML</a>

(3.5)

and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/100/mathml/M203','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/100/mathml/M203">View MathML</a>, where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/100/mathml/M204','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/100/mathml/M204">View MathML</a> denotes the closure of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/100/mathml/M199','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/100/mathml/M199">View MathML</a> in the metric space <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/100/mathml/M206','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/100/mathml/M206">View MathML</a>. Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/100/mathml/M207','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/100/mathml/M207">View MathML</a> denote the boundary of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/100/mathml/M199','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/100/mathml/M199">View MathML</a> in the metric space <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/100/mathml/M206','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/100/mathml/M206">View MathML</a>. Obviously, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/100/mathml/M210','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/100/mathml/M210">View MathML</a> is a compact subset. Assume that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/100/mathml/M211','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/100/mathml/M211">View MathML</a>. From the maximal connectedness of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/100/mathml/M192','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/100/mathml/M192">View MathML</a>, there is no connected subset of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/100/mathml/M210','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/100/mathml/M210">View MathML</a> meeting both <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/100/mathml/M214','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/100/mathml/M214">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/100/mathml/M192','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/100/mathml/M192">View MathML</a>. From Lemma 2.4, there exist compact disjoint subsets <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/100/mathml/M216','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/100/mathml/M216">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/100/mathml/M217','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/100/mathml/M217">View MathML</a> of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/100/mathml/M218','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/100/mathml/M218">View MathML</a> such that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/100/mathml/M219','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/100/mathml/M219">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/100/mathml/M220','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/100/mathml/M220">View MathML</a>, and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/100/mathml/M221','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/100/mathml/M221">View MathML</a>. Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/100/mathml/M222','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/100/mathml/M222">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/100/mathml/M223','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/100/mathml/M223">View MathML</a> be the <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/100/mathml/M224','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/100/mathml/M224">View MathML</a>-neighborhood of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/100/mathml/M216','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/100/mathml/M216">View MathML</a> in the metric space <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/100/mathml/M206','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/100/mathml/M206">View MathML</a>. Set

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/100/mathml/M227','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/100/mathml/M227">View MathML</a>

(3.6)

Then we have <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/100/mathml/M228','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/100/mathml/M228">View MathML</a> and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/100/mathml/M229','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/100/mathml/M229">View MathML</a>

(3.7)

Obviously, the collection of the subsets

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/100/mathml/M230','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/100/mathml/M230">View MathML</a>

is an open cover of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/100/mathml/M193','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/100/mathml/M193">View MathML</a>. Since <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/100/mathml/M232','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/100/mathml/M232">View MathML</a> is compact, then there exist finite points, namely

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/100/mathml/M233','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/100/mathml/M233">View MathML</a>

such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/100/mathml/M234','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/100/mathml/M234">View MathML</a>

Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/100/mathml/M235','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/100/mathml/M235">View MathML</a>. Then U is a bounded open subset of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/100/mathml/M206','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/100/mathml/M206">View MathML</a>. From (3.7), we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/100/mathml/M237','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/100/mathml/M237">View MathML</a>

(3.8)

Thus,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/100/mathml/M238','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/100/mathml/M238">View MathML</a>

From (3.5) and (3.8), we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/100/mathml/M239','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/100/mathml/M239">View MathML</a>

Then from Lemma 2.5, we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/100/mathml/M240','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/100/mathml/M240">View MathML</a>

(3.9)

Since <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/100/mathml/M241','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/100/mathml/M241">View MathML</a>, then

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/100/mathml/M242','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/100/mathml/M242">View MathML</a>

(3.10)

Since <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/100/mathml/M243','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/100/mathml/M243">View MathML</a>, then from (3.4) we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/100/mathml/M244','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/100/mathml/M244">View MathML</a>

(3.11)

which contradicts to (3.10) and (3.11). Therefore, there must exist <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/100/mathml/M245','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/100/mathml/M245">View MathML</a> such that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/100/mathml/M196','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/100/mathml/M196">View MathML</a> is bounded.

Step 3. Obviously, the projection of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/100/mathml/M192','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/100/mathml/M192">View MathML</a> is a interval, denote it by <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/100/mathml/M248','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/100/mathml/M248">View MathML</a>, then <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/100/mathml/M249','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/100/mathml/M249">View MathML</a>. Then <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/100/mathml/M192','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/100/mathml/M192">View MathML</a> is a bounded connected component of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/100/mathml/M251','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/100/mathml/M251">View MathML</a>. Take <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/100/mathml/M252','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/100/mathml/M252">View MathML</a>, let

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/100/mathml/M253','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/100/mathml/M253">View MathML</a>

Obviously, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/100/mathml/M254','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/100/mathml/M254">View MathML</a>. For each <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/100/mathml/M255','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/100/mathml/M255">View MathML</a>, denote by <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/100/mathml/M256','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/100/mathml/M256">View MathML</a> the connected component of the metric space <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/100/mathml/M257','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/100/mathml/M257">View MathML</a>, which passes the point p. Now we shall prove that there must exist a <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/100/mathml/M258','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/100/mathml/M258">View MathML</a> such that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/100/mathml/M259','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/100/mathml/M259">View MathML</a> is an unbounded connected component of the metric space <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/100/mathml/M257','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/100/mathml/M257">View MathML</a>. On the contrary, assume that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/100/mathml/M256','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/100/mathml/M256">View MathML</a> is bounded for each <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/100/mathml/M255','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/100/mathml/M255">View MathML</a>. Then, for each <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/100/mathml/M255','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/100/mathml/M255">View MathML</a>, in the same way as in the construction of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/100/mathml/M264','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/100/mathml/M264">View MathML</a> in (3.6) we can show that there exists a neighborhood <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/100/mathml/M265','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/100/mathml/M265">View MathML</a> of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/100/mathml/M256','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/100/mathml/M256">View MathML</a> in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/100/mathml/M267','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/100/mathml/M267">View MathML</a> such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/100/mathml/M268','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/100/mathml/M268">View MathML</a>

(3.12)

Obviously, the set of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/100/mathml/M269','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/100/mathml/M269">View MathML</a> is an open cover of the set <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/100/mathml/M270','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/100/mathml/M270">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/100/mathml/M270','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/100/mathml/M270">View MathML</a> is a compact set. Thus, there exist finite subsets of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/100/mathml/M272','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/100/mathml/M272">View MathML</a>, say

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/100/mathml/M273','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/100/mathml/M273">View MathML</a>

which is also an open cover of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/100/mathml/M270','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/100/mathml/M270">View MathML</a>, that is,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/100/mathml/M275','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/100/mathml/M275">View MathML</a>

(3.13)

Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/100/mathml/M276','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/100/mathml/M276">View MathML</a>, then <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/100/mathml/M277','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/100/mathml/M277">View MathML</a> is a bounded open subset of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/100/mathml/M267','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/100/mathml/M267">View MathML</a>. Since

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/100/mathml/M279','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/100/mathml/M279">View MathML</a>

then by (3.12) we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/100/mathml/M280','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/100/mathml/M280">View MathML</a>

(3.14)

Let

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/100/mathml/M281','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/100/mathml/M281">View MathML</a>

and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/100/mathml/M282','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/100/mathml/M282">View MathML</a>. It is easy to see that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/100/mathml/M283','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/100/mathml/M283">View MathML</a>

From (3.12) and (3.13), we see that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/100/mathml/M284','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/100/mathml/M284">View MathML</a>. Obviously, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/100/mathml/M285','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/100/mathml/M285">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/100/mathml/M286','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/100/mathml/M286">View MathML</a>. Note the unboundedness of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/100/mathml/M192','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/100/mathml/M192">View MathML</a>, then <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/100/mathml/M288','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/100/mathml/M288">View MathML</a>. Now we have <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/100/mathml/M289','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/100/mathml/M289">View MathML</a>, which is a contradiction of the connectedness of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/100/mathml/M192','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/100/mathml/M192">View MathML</a>. Therefore, there must exist <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/100/mathml/M258','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/100/mathml/M258">View MathML</a> such that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/100/mathml/M259','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/100/mathml/M259">View MathML</a> is an unbounded connected component of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/100/mathml/M257','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/100/mathml/M257">View MathML</a>.

Step 4. Since <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/100/mathml/M294','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/100/mathml/M294">View MathML</a> for each <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/100/mathml/M295','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/100/mathml/M295">View MathML</a>, we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/100/mathml/M296','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/100/mathml/M296">View MathML</a>

So, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/100/mathml/M297','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/100/mathml/M297">View MathML</a> for each <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/100/mathml/M298','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/100/mathml/M298">View MathML</a>. This implies <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/100/mathml/M299','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/100/mathml/M299">View MathML</a> is an unbounded subset. Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/100/mathml/M167','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/100/mathml/M167">View MathML</a> be the connected component of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/100/mathml/M301','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/100/mathml/M301">View MathML</a> containing <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/100/mathml/M259','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/100/mathml/M259">View MathML</a>. Obviously, there exists <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/100/mathml/M169','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/100/mathml/M169">View MathML</a> such that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/100/mathml/M304','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/100/mathml/M304">View MathML</a>. As <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/100/mathml/M167','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/100/mathml/M167">View MathML</a> is unbounded, we easily see that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/100/mathml/M306','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/100/mathml/M306">View MathML</a>

The proof is complete. □

Corollary 3.1Let (A1) to (A4) hold. Then there exists<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/100/mathml/M307','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/100/mathml/M307">View MathML</a>such that problem (1.1) has a positive solution<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/100/mathml/M308','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/100/mathml/M308">View MathML</a>for<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/100/mathml/M309','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/100/mathml/M309">View MathML</a>with<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/100/mathml/M310','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/100/mathml/M310">View MathML</a>as<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/100/mathml/M311','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/100/mathml/M311">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, checked and approved the final manuscript.

Acknowledgements

This paper is supported by Innovation Project of Jiangsu Province postgraduate training project (CXLX12_0979).

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