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Global structure of positive solutions for three-point boundary value problems

Jia-Ping Gu1, Liang-Gen Hu1 and Huai-Nian Zhang2*

Author Affiliations

1 Department of Mathematics, Ningbo University, Ningbo, 315211, P.R. China

2 Department of Mathematics and Physics, Beijing Institute of Petrochemical Technology, Beijing, 102617, P.R. China

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Boundary Value Problems 2013, 2013:174  doi:10.1186/1687-2770-2013-174


The electronic version of this article is the complete one and can be found online at: http://www.boundaryvalueproblems.com/content/2013/1/174


Received:29 April 2013
Accepted:8 July 2013
Published:25 July 2013

© 2013 Gu et al.; licensee Springer

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

Abstract

In this paper, we are concerned with the three-point boundary value problem for second-order differential equations

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/174/mathml/M1','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/174/mathml/M1">View MathML</a>

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/174/mathml/M2','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/174/mathml/M2">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/174/mathml/M3','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/174/mathml/M3">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/174/mathml/M4','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/174/mathml/M4">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/174/mathml/M5','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/174/mathml/M5">View MathML</a>; <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/174/mathml/M6','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/174/mathml/M6">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/174/mathml/M7','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/174/mathml/M7">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/174/mathml/M8','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/174/mathml/M8">View MathML</a> satisfies <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/174/mathml/M9','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/174/mathml/M9">View MathML</a> for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/174/mathml/M10','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/174/mathml/M10">View MathML</a>. The existence of the continuum of a positive solution is established by utilizing the Leray-Schauder global continuation principle. Furthermore, the interval of α about the nonexistence of a positive solution is also given.

MSC: 34B10, 34B18, 34G20.

Keywords:
positive solution; global continuous theorem; continuum; differential equation

1 Introduction

In this paper, we consider the following three-point boundary value problem for second-order differential equations:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/174/mathml/M11','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/174/mathml/M11">View MathML</a>

(1.1)

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/174/mathml/M2','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/174/mathml/M2">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/174/mathml/M3','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/174/mathml/M3">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/174/mathml/M4','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/174/mathml/M4">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/174/mathml/M5','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/174/mathml/M5">View MathML</a>; <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/174/mathml/M6','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/174/mathml/M6">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/174/mathml/M7','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/174/mathml/M7">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/174/mathml/M8','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/174/mathml/M8">View MathML</a> satisfies <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/174/mathml/M9','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/174/mathml/M9">View MathML</a> for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/174/mathml/M10','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/174/mathml/M10">View MathML</a>.

The existence and multiplicity of positive solutions for multi-point boundary value problems have been studied by several authors and many nice results have been obtained; see, for example, [1-6] and the references therein for more information on this problem. The multi-point boundary conditions of ordinary differential equations arose in different areas of applied mathematics and physics. In addition, they are often used to model many physical phenomena which include gas diffusion through porous media, nonlinear diffusion generated by nonlinear sources, chemically reacting systems, infectious diseases as well as concentration in chemical or biological problems. In all these problems, only positive solutions are very meaningful.

In 2009, Sun et al.[1] studied the three-point boundary value problem

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/174/mathml/M21','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/174/mathml/M21">View MathML</a>

(1.2)

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/174/mathml/M22','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/174/mathml/M22">View MathML</a> is a parameter, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/174/mathml/M2','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/174/mathml/M2">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/174/mathml/M3','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/174/mathml/M3">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/174/mathml/M4','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/174/mathml/M4">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/174/mathml/M26','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/174/mathml/M26">View MathML</a>. Based on Krein-Rutmann theorems and the fixed point index theory, they not only established the criteria of the existence and multiplicity of a positive solution, but also obtained the parameter μ in relation with the nonlinear term f and the first eigenvalue of the linear operator.

On the other hand, we note that the nice results in [1] only gave the existence and multiplicity of positive solutions, and if the parameter α is regarded as a variable, then an interesting problem as to what happens to the global structure of positive solutions of (1.2) was not considered. However, this relationship is very useful for computing the numerical solution of (1.2) as it can be used to guide the numerical work. For example, the global bifurcation of solutions for second-order differential equations has been extensively studied in the literature, see [4,7,8].

Motivated by this, in this paper, we consider the three-point boundary value problem for second-order differential equations (1.1) and make use of the Leray-Schauder global continuation theorem in the frame of techniques nicely employed by Ma and Thompson [4] and convex analysis technique. We consider two cases <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/174/mathml/M27','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/174/mathml/M27">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/174/mathml/M28','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/174/mathml/M28">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/174/mathml/M29','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/174/mathml/M29">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/174/mathml/M30','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/174/mathml/M30">View MathML</a>, and establish the existence of continuum of positive solutions, where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/174/mathml/M31','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/174/mathml/M31">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/174/mathml/M32','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/174/mathml/M32">View MathML</a>. Moreover, the interval of parameter α about the nonexistence of positive solutions is also given. Our main results extend and improve the corresponding results [1,3,4]. In contrast to [[1], Theorem 3.1 and Theorem 3.2], we obtain the global structure and behavior of positive solutions, where the parameter α is regarded as a variable.

The rest of this paper is arranged as follows. In Section 2, we give Green’s function and some lemmas. In Section 3, we consider the case <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/174/mathml/M27','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/174/mathml/M27">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/174/mathml/M28','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/174/mathml/M28">View MathML</a>, and give the existence of the continuum of positive solutions and the interval of parameter α about the nonexistence of positive solutions. In Section 4, we study the case <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/174/mathml/M29','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/174/mathml/M29">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/174/mathml/M30','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/174/mathml/M30">View MathML</a>, and give the existence of global continuum of positive solutions.

2 Preliminaries and lemmas

Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/174/mathml/M37','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/174/mathml/M37">View MathML</a> denote the Banach space of a continuous function with the maximum norm

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/174/mathml/M38','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/174/mathml/M38">View MathML</a>

Define a set by

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/174/mathml/M39','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/174/mathml/M39">View MathML</a>

then P is a cone.

We assume that

(H0) <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/174/mathml/M2','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/174/mathml/M2">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/174/mathml/M3','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/174/mathml/M3">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/174/mathml/M4','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/174/mathml/M4">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/174/mathml/M43','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/174/mathml/M43">View MathML</a>.

Lemma 2.1 (see [[1], Lemma 2.1])

Suppose that condition (H0) holds and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/174/mathml/M44','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/174/mathml/M44">View MathML</a>. Then the following linear differential equation

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/174/mathml/M45','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/174/mathml/M45">View MathML</a>

has a unique solution

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/174/mathml/M46','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/174/mathml/M46">View MathML</a>

where<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/174/mathml/M47','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/174/mathml/M47">View MathML</a>is defined by

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/174/mathml/M48','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/174/mathml/M48">View MathML</a>

(2.1)

For the sake of convenience, we list the following hypotheses:

(H1) <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/174/mathml/M6','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/174/mathml/M6">View MathML</a>.

(H2) <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/174/mathml/M7','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/174/mathml/M7">View MathML</a> satisfies <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/174/mathml/M9','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/174/mathml/M9">View MathML</a> for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/174/mathml/M10','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/174/mathml/M10">View MathML</a>.

(H3) <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/174/mathml/M53','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/174/mathml/M53">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/174/mathml/M54','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/174/mathml/M54">View MathML</a> (superlinear).

(H4) <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/174/mathml/M55','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/174/mathml/M55">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/174/mathml/M56','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/174/mathml/M56">View MathML</a> (sublinear).

Lemma 2.2Assume that (H0) holds. Let<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/174/mathml/M57','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/174/mathml/M57">View MathML</a>with<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/174/mathml/M58','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/174/mathml/M58">View MathML</a>for<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/174/mathml/M59','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/174/mathml/M59">View MathML</a>and letube a solution of

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/174/mathml/M60','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/174/mathml/M60">View MathML</a>

(2.2)

Then<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/174/mathml/M61','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/174/mathml/M61">View MathML</a>for<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/174/mathml/M59','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/174/mathml/M59">View MathML</a>. Moreover, if<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/174/mathml/M63','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/174/mathml/M63">View MathML</a>for some<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/174/mathml/M64','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/174/mathml/M64">View MathML</a>, then<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/174/mathml/M65','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/174/mathml/M65">View MathML</a>for all<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/174/mathml/M66','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/174/mathml/M66">View MathML</a>.

Proof We only show that if <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/174/mathml/M63','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/174/mathml/M63">View MathML</a> for some <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/174/mathml/M68','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/174/mathml/M68">View MathML</a>, then <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/174/mathml/M65','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/174/mathml/M65">View MathML</a> for all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/174/mathml/M66','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/174/mathml/M66">View MathML</a>.

If <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/174/mathml/M71','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/174/mathml/M71">View MathML</a>, then we have from [[3], Lemma 2] that the results hold.

Next, we consider the case <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/174/mathml/M72','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/174/mathml/M72">View MathML</a>. If it is not true, then there exists some <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/174/mathml/M73','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/174/mathml/M73">View MathML</a> such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/174/mathml/M74','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/174/mathml/M74">View MathML</a>

(2.3)

We separate the proof into two cases: Case I: <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/174/mathml/M75','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/174/mathml/M75">View MathML</a> and Case II: <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/174/mathml/M76','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/174/mathml/M76">View MathML</a>.

Case I. If <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/174/mathml/M75','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/174/mathml/M75">View MathML</a>, then <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/174/mathml/M78','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/174/mathml/M78">View MathML</a>. Since u is concave down in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/174/mathml/M79','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/174/mathml/M79">View MathML</a>, we obtain that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/174/mathml/M80','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/174/mathml/M80">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/174/mathml/M81','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/174/mathml/M81">View MathML</a> for all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/174/mathml/M59','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/174/mathml/M59">View MathML</a>. Set

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/174/mathml/M83','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/174/mathml/M83">View MathML</a>

then we find from the boundary conditions in (2.2) that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/174/mathml/M84','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/174/mathml/M84">View MathML</a>

This together with the concavity of u leads to

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/174/mathml/M85','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/174/mathml/M85">View MathML</a>

This contradicts the hypothesis <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/174/mathml/M43','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/174/mathml/M43">View MathML</a>.

Case II. Consider the case <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/174/mathml/M87','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/174/mathml/M87">View MathML</a>.

(1) If <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/174/mathml/M88','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/174/mathml/M88">View MathML</a>, then <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/174/mathml/M89','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/174/mathml/M89">View MathML</a> and the concavity of u imply that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/174/mathml/M90','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/174/mathml/M90">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/174/mathml/M91','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/174/mathml/M91">View MathML</a>. Hence, we get that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/174/mathml/M92','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/174/mathml/M92">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/174/mathml/M93','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/174/mathml/M93">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/174/mathml/M94','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/174/mathml/M94">View MathML</a> (since <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/174/mathml/M95','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/174/mathml/M95">View MathML</a>, we have that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/174/mathml/M96','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/174/mathml/M96">View MathML</a> leads to <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/174/mathml/M97','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/174/mathml/M97">View MathML</a>. This contradicts (2.3)). Again, since u is concave, we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/174/mathml/M98','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/174/mathml/M98">View MathML</a>

Consequently, we obtain that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/174/mathml/M99','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/174/mathml/M99">View MathML</a> contradicts the condition <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/174/mathml/M100','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/174/mathml/M100">View MathML</a>.

(2) If <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/174/mathml/M101','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/174/mathml/M101">View MathML</a>, then, adopting the same proof as in Case I, we get a contradiction.

(3) If <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/174/mathml/M102','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/174/mathml/M102">View MathML</a>, then it follows that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/174/mathml/M103','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/174/mathml/M103">View MathML</a>. In light of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/174/mathml/M102','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/174/mathml/M102">View MathML</a> and the concavity of u, we get from (2.3) that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/174/mathml/M105','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/174/mathml/M105">View MathML</a>

and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/174/mathml/M106','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/174/mathml/M106">View MathML</a>

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/174/mathml/M107','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/174/mathml/M107">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/174/mathml/M108','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/174/mathml/M108">View MathML</a>. Hence, we get from the boundary condition of (2.2) that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/174/mathml/M109','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/174/mathml/M109">View MathML</a>

leads to

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/174/mathml/M110','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/174/mathml/M110">View MathML</a>

This is a contradiction.

Consequently, we get from Case I and Case II that the conclusion holds. □

Remark 2.1 If <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/174/mathml/M111','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/174/mathml/M111">View MathML</a> is positive, then we know from the proof in Lemma 2.2 that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/174/mathml/M111','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/174/mathml/M111">View MathML</a> may only have zero point at <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/174/mathml/M113','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/174/mathml/M113">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/174/mathml/M114','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/174/mathml/M114">View MathML</a>.

Lemma 2.3Let (H0) hold and let<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/174/mathml/M115','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/174/mathml/M115">View MathML</a>be a function satisfying

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/174/mathml/M116','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/174/mathml/M116">View MathML</a>

and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/174/mathml/M117','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/174/mathml/M117">View MathML</a>

Then there exists<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/174/mathml/M118','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/174/mathml/M118">View MathML</a>such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/174/mathml/M119','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/174/mathml/M119">View MathML</a>

Proof If <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/174/mathml/M71','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/174/mathml/M71">View MathML</a>, then from [[4], Lemma 3.3] the conclusion holds.

Next, we consider the case <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/174/mathml/M72','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/174/mathml/M72">View MathML</a>. Clearly, it is easy to see from Lemma 2.2 that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/174/mathml/M65','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/174/mathml/M65">View MathML</a> for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/174/mathml/M66','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/174/mathml/M66">View MathML</a>.

(1) If <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/174/mathml/M124','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/174/mathml/M124">View MathML</a>, then <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/174/mathml/M125','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/174/mathml/M125">View MathML</a>. This together with the concavity of u yields <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/174/mathml/M126','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/174/mathml/M126">View MathML</a>. Since <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/174/mathml/M127','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/174/mathml/M127">View MathML</a>, there exists <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/174/mathml/M118','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/174/mathml/M118">View MathML</a> such that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/174/mathml/M129','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/174/mathml/M129">View MathML</a>.

(2) If <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/174/mathml/M130','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/174/mathml/M130">View MathML</a>, then <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/174/mathml/M131','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/174/mathml/M131">View MathML</a>. Consequently, there exists <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/174/mathml/M132','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/174/mathml/M132">View MathML</a> such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/174/mathml/M133','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/174/mathml/M133">View MathML</a>

and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/174/mathml/M134','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/174/mathml/M134">View MathML</a>

The assumption <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/174/mathml/M135','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/174/mathml/M135">View MathML</a> and the concavity of u in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/174/mathml/M79','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/174/mathml/M79">View MathML</a> imply that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/174/mathml/M137','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/174/mathml/M137">View MathML</a>

This completes the proof. □

From (2.1), we define an operator <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/174/mathml/M138','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/174/mathml/M138">View MathML</a> as follows:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/174/mathml/M139','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/174/mathml/M139">View MathML</a>

(2.4)

Assume that (H0)-(H2) hold, then it is easy to verify that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/174/mathml/M140','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/174/mathml/M140">View MathML</a> is well defined and completely continuous. We note that u is a positive solution of problem (1.1) if and only if <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/174/mathml/M141','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/174/mathml/M141">View MathML</a> on P.

By a positive solution of (1.1) we mean a solution of (1.1) which is positive on <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/174/mathml/M142','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/174/mathml/M142">View MathML</a>.

Denote by the closure of the set

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/174/mathml/M144','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/174/mathml/M144">View MathML</a>

in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/174/mathml/M145','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/174/mathml/M145">View MathML</a>.

Using the Leray-Schauder global continuation theorem [[8], Theorem 14.C], Ma and Thompson [[4], Lemma 2.2] obtained the following result.

Lemma 2.4LetPbe a cone in a Banach spaceX. Let<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/174/mathml/M146','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/174/mathml/M146">View MathML</a>be a bounded and open subset inXwith respect to the topology induced by<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/174/mathml/M147','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/174/mathml/M147">View MathML</a>onP. Assume that the operator<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/174/mathml/M148','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/174/mathml/M148">View MathML</a>is a continuous, compact map satisfying:

(1) the equation<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/174/mathml/M141','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/174/mathml/M141">View MathML</a>has no solution on<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/174/mathml/M150','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/174/mathml/M150">View MathML</a>;

(2) <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/174/mathml/M151','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/174/mathml/M151">View MathML</a>with<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/174/mathml/M152','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/174/mathml/M152">View MathML</a>.

Then the set

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/174/mathml/M153','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/174/mathml/M153">View MathML</a>

has a continuumof solutions in<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/174/mathml/M154','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/174/mathml/M154">View MathML</a>, which connects the set<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/174/mathml/M155','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/174/mathml/M155">View MathML</a>with the set<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/174/mathml/M156','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/174/mathml/M156">View MathML</a>.

3 The superlinear case

For a given <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/174/mathml/M157','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/174/mathml/M157">View MathML</a>, we let

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/174/mathml/M158','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/174/mathml/M158">View MathML</a>

From (H0)-(H2) and Lemma 2.2, we get that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/174/mathml/M159','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/174/mathml/M159">View MathML</a>

(3.1)

For any <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/174/mathml/M157','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/174/mathml/M157">View MathML</a>, we denote

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/174/mathml/M161','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/174/mathml/M161">View MathML</a>

From (H0)-(H2) and Lemma 2.2, we have that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/174/mathml/M162','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/174/mathml/M162">View MathML</a>

(3.2)

Lemma 3.1 [[1], Theorem 3.2]

Let conditions (H0)-(H3) hold. Then there exist two constants<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/174/mathml/M163','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/174/mathml/M163">View MathML</a>and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/174/mathml/M164','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/174/mathml/M164">View MathML</a>with<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/174/mathml/M165','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/174/mathml/M165">View MathML</a>such that problem (1.1) has at least one positive solution<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/174/mathml/M166','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/174/mathml/M166">View MathML</a>with<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/174/mathml/M167','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/174/mathml/M167">View MathML</a>. Furthermore,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/174/mathml/M168','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/174/mathml/M168">View MathML</a>

where<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/174/mathml/M169','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/174/mathml/M169">View MathML</a>.

Proof Since <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/174/mathml/M29','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/174/mathml/M29">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/174/mathml/M30','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/174/mathml/M30">View MathML</a>, we take <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/174/mathml/M172','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/174/mathml/M172">View MathML</a> in [[1], Theorem 3.2] and all the conditions in [[1], Theorem 3.2] are satisfied. Therefore, the conclusion holds. □

Lemma 3.2Assume that (H0)-(H3) hold. Let<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/174/mathml/M163','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/174/mathml/M163">View MathML</a>and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/174/mathml/M164','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/174/mathml/M164">View MathML</a>be the constants as in Lemma 3.1. Then there exists a positive number<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/174/mathml/M175','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/174/mathml/M175">View MathML</a>with<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/174/mathml/M176','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/174/mathml/M176">View MathML</a>such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/174/mathml/M177','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/174/mathml/M177">View MathML</a>

where<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/174/mathml/M178','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/174/mathml/M178">View MathML</a>is defined by (3.2).

Proof First we claim that if <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/174/mathml/M27','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/174/mathml/M27">View MathML</a>, then there exists a positive number <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/174/mathml/M180','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/174/mathml/M180">View MathML</a> such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/174/mathml/M181','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/174/mathml/M181">View MathML</a>

(3.3)

Suppose this fails, that is, there exists a sequence <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/174/mathml/M182','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/174/mathml/M182">View MathML</a> with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/174/mathml/M183','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/174/mathml/M183">View MathML</a> such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/174/mathml/M184','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/174/mathml/M184">View MathML</a>

We may suppose that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/174/mathml/M185','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/174/mathml/M185">View MathML</a> for some <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/174/mathml/M186','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/174/mathml/M186">View MathML</a>. For the sake of convenience, we denote <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/174/mathml/M187','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/174/mathml/M187">View MathML</a>. From (H1), (H2) and Lemma 2.2, we see that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/174/mathml/M188','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/174/mathml/M188">View MathML</a> for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/174/mathml/M66','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/174/mathml/M66">View MathML</a>. Set <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/174/mathml/M190','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/174/mathml/M190">View MathML</a>, then we get

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/174/mathml/M191','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/174/mathml/M191">View MathML</a>

(3.4)

where

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/174/mathml/M192','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/174/mathml/M192">View MathML</a>

From Remark 2.1, we know that the set <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/174/mathml/M193','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/174/mathml/M193">View MathML</a> has at most two points <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/174/mathml/M113','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/174/mathml/M113">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/174/mathml/M114','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/174/mathml/M114">View MathML</a>. Therefore, from the hypothesis of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/174/mathml/M27','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/174/mathml/M27">View MathML</a>, it follows that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/174/mathml/M197','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/174/mathml/M197">View MathML</a> is continuous in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/174/mathml/M79','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/174/mathml/M79">View MathML</a> and there exists a constant <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/174/mathml/M199','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/174/mathml/M199">View MathML</a>, independent of n, such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/174/mathml/M200','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/174/mathml/M200">View MathML</a>

This together with (3.4) yields

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/174/mathml/M201','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/174/mathml/M201">View MathML</a>

(3.5)

In light of Lemma 2.3, there exists <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/174/mathml/M202','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/174/mathml/M202">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/174/mathml/M203','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/174/mathml/M203">View MathML</a> , such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/174/mathml/M204','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/174/mathml/M204">View MathML</a>

(3.6)

Applying the Newton-Leibniz formula, we find

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/174/mathml/M205','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/174/mathml/M205">View MathML</a>

Consequently, combining (3.5) and (3.6), we conclude that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/174/mathml/M206','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/174/mathml/M206">View MathML</a>

for some constant <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/174/mathml/M207','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/174/mathml/M207">View MathML</a> independent of n. Utilizing the Ascoli-Arzela theorem, we have that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/174/mathml/M208','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/174/mathml/M208">View MathML</a> is a relatively compact set on <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/174/mathml/M209','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/174/mathml/M209">View MathML</a>. Assume, taking a subsequence if necessary, that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/174/mathml/M210','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/174/mathml/M210">View MathML</a> in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/174/mathml/M209','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/174/mathml/M209">View MathML</a>. Then <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/174/mathml/M212','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/174/mathml/M212">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/174/mathml/M213','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/174/mathml/M213">View MathML</a> in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/174/mathml/M79','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/174/mathml/M79">View MathML</a>.

On the other hand, from <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/174/mathml/M27','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/174/mathml/M27">View MathML</a> and the fact <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/174/mathml/M216','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/174/mathml/M216">View MathML</a> as <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/174/mathml/M217','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/174/mathml/M217">View MathML</a>, it follows that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/174/mathml/M218','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/174/mathml/M218">View MathML</a>

(3.7)

uniformly holds for all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/174/mathml/M59','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/174/mathml/M59">View MathML</a>. From (2.4) and the fact <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/174/mathml/M220','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/174/mathml/M220">View MathML</a>, we get

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/174/mathml/M221','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/174/mathml/M221">View MathML</a>

This together with (3.7) and the Lebesgue dominated convergence theorem, we get

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/174/mathml/M222','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/174/mathml/M222">View MathML</a>

contradicts <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/174/mathml/M212','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/174/mathml/M212">View MathML</a>. Therefore, the claim (3.3) holds.

Next, we prove that if <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/174/mathml/M28','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/174/mathml/M28">View MathML</a>, then there is a positive number <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/174/mathml/M225','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/174/mathml/M225">View MathML</a> such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/174/mathml/M226','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/174/mathml/M226">View MathML</a>

(3.8)

Suppose on the contrary that there exists a sequence <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/174/mathml/M182','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/174/mathml/M182">View MathML</a> with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/174/mathml/M228','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/174/mathml/M228">View MathML</a> such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/174/mathml/M229','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/174/mathml/M229">View MathML</a>

We define <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/174/mathml/M187','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/174/mathml/M187">View MathML</a>. From (H0)-(H2), Lemma 2.2 and the concavity of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/174/mathml/M231','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/174/mathml/M231">View MathML</a>, it follows that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/174/mathml/M232','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/174/mathml/M232">View MathML</a>

and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/174/mathml/M233','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/174/mathml/M233">View MathML</a>

(3.9)

Take <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/174/mathml/M234','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/174/mathml/M234">View MathML</a>, where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/174/mathml/M235','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/174/mathml/M235">View MathML</a>. (H3) implies that there exists a constant <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/174/mathml/M236','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/174/mathml/M236">View MathML</a> such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/174/mathml/M237','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/174/mathml/M237">View MathML</a>

(3.10)

Since <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/174/mathml/M238','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/174/mathml/M238">View MathML</a>, we find a sufficiently large <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/174/mathml/M239','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/174/mathml/M239">View MathML</a> such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/174/mathml/M240','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/174/mathml/M240">View MathML</a>

This together with (3.9) and (3.10) yields

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/174/mathml/M241','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/174/mathml/M241">View MathML</a>

(3.11)

Put <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/174/mathml/M242','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/174/mathml/M242">View MathML</a> for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/174/mathml/M243','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/174/mathml/M243">View MathML</a>. Hence, we have from (3.11) that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/174/mathml/M244','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/174/mathml/M244">View MathML</a>

On the other hand, multiplying (1.1) by ψ and integrating by parts, we find

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/174/mathml/M245','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/174/mathml/M245">View MathML</a>

leads to

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/174/mathml/M246','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/174/mathml/M246">View MathML</a>

Then we obtain

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/174/mathml/M247','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/174/mathml/M247">View MathML</a>

This is a contradiction. Consequently, conclusion (3.8) holds.

Combining (3.3) and (3.8), we let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/174/mathml/M248','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/174/mathml/M248">View MathML</a>. Thus, the result holds. □

Theorem 3.1Assume that conditions (H0)-(H3) hold. Thencontains a continuum which joins<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/174/mathml/M250','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/174/mathml/M250">View MathML</a>with<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/174/mathml/M251','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/174/mathml/M251">View MathML</a>.

Proof We divide the proof into four steps.

Step 1. We construct a continuum.

For arbitrarily given <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/174/mathml/M252','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/174/mathml/M252">View MathML</a>, let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/174/mathml/M175','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/174/mathml/M175">View MathML</a> be as in Lemma 3.2. Define a set by

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/174/mathml/M254','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/174/mathml/M254">View MathML</a>

(3.12)

It follows from Lemma 3.1 and the excision property of the fixed point index that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/174/mathml/M255','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/174/mathml/M255">View MathML</a>

From Lemma 3.2, we know that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/174/mathml/M256','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/174/mathml/M256">View MathML</a> has no solutions in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/174/mathml/M257','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/174/mathml/M257">View MathML</a>. Therefore, from Lemma 2.4, there exists a continuum <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/174/mathml/M258','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/174/mathml/M258">View MathML</a> which joins <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/174/mathml/M259','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/174/mathml/M259">View MathML</a> with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/174/mathml/M260','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/174/mathml/M260">View MathML</a>. Here <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/174/mathml/M178','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/174/mathml/M178">View MathML</a> is defined by (3.2), and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/174/mathml/M259','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/174/mathml/M259">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/174/mathml/M260','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/174/mathml/M260">View MathML</a> are defined by (3.1).

Let be the closure of the set

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/174/mathml/M265','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/174/mathml/M265">View MathML</a>

(3.13)

and let

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/174/mathml/M266','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/174/mathml/M266">View MathML</a>

Since <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/174/mathml/M267','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/174/mathml/M267">View MathML</a>, we know that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/174/mathml/M268','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/174/mathml/M268">View MathML</a>.

Step 2. We show that there exists <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/174/mathml/M269','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/174/mathml/M269">View MathML</a> satisfying

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/174/mathml/M270','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/174/mathml/M270">View MathML</a>

(3.14)

If it is not true, then there exists <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/174/mathml/M271','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/174/mathml/M271">View MathML</a> such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/174/mathml/M272','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/174/mathml/M272">View MathML</a>

(3.15)

Taking <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/174/mathml/M273','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/174/mathml/M273">View MathML</a>, it follows that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/174/mathml/M274','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/174/mathml/M274">View MathML</a>. Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/174/mathml/M275','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/174/mathml/M275">View MathML</a> be a given number by Lemma 3.1 and the set

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/174/mathml/M276','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/174/mathml/M276">View MathML</a>

Then we know, from Lemma 3.1 and the excision property of the fixed point index, that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/174/mathml/M277','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/174/mathml/M277">View MathML</a>

From Lemma 3.2, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/174/mathml/M256','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/174/mathml/M256">View MathML</a> has no solution in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/174/mathml/M279','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/174/mathml/M279">View MathML</a>. Again, using Lemma 2.4, we find a continuum <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/174/mathml/M280','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/174/mathml/M280">View MathML</a> which joins <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/174/mathml/M259','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/174/mathml/M259">View MathML</a> with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/174/mathml/M282','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/174/mathml/M282">View MathML</a>. This contradicts (3.15). Therefore, the conclusion in (3.14) holds.

Step 3. Let ζ be a continuum satisfying (3.14). We claim that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/174/mathml/M283','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/174/mathml/M283">View MathML</a>

(3.16)

Suppose on the contrary that there exists a sequence <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/174/mathml/M284','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/174/mathml/M284">View MathML</a> such that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/174/mathml/M285','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/174/mathml/M285">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/174/mathml/M286','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/174/mathml/M286">View MathML</a>. From (H0)-(H2), Lemma 2.2 and the concavity of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/174/mathml/M231','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/174/mathml/M231">View MathML</a>, it follows that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/174/mathml/M288','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/174/mathml/M288">View MathML</a>

and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/174/mathml/M289','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/174/mathml/M289">View MathML</a>

Adopting the same proof as in the second step in Lemma 3.2, we can find a contradiction. Hence, the result in (3.16) holds.

Step 4. Let ζ be a continuum satisfying (3.14). Next we show that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/174/mathml/M290','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/174/mathml/M290">View MathML</a>

(3.17)

If it is not true, then we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/174/mathml/M291','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/174/mathml/M291">View MathML</a>

for some <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/174/mathml/M292','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/174/mathml/M292">View MathML</a>. Then there exists a sequence <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/174/mathml/M284','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/174/mathml/M284">View MathML</a> such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/174/mathml/M294','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/174/mathml/M294">View MathML</a>

and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/174/mathml/M295','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/174/mathml/M295">View MathML</a>

(3.18)

From conditions (H0)-(H2) and Lemma 2.2, we get that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/174/mathml/M296','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/174/mathml/M296">View MathML</a> on <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/174/mathml/M142','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/174/mathml/M142">View MathML</a> and the graph of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/174/mathml/M298','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/174/mathml/M298">View MathML</a> is strictly concave down on <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/174/mathml/M142','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/174/mathml/M142">View MathML</a>.

(1) If <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/174/mathml/M71','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/174/mathml/M71">View MathML</a>, then we obtain from the boundary condition of (3.18) that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/174/mathml/M301','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/174/mathml/M301">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/174/mathml/M302','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/174/mathml/M302">View MathML</a>. From the strict concavity of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/174/mathml/M298','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/174/mathml/M298">View MathML</a>, it follows that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/174/mathml/M304','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/174/mathml/M304">View MathML</a>

implies

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/174/mathml/M305','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/174/mathml/M305">View MathML</a>

This is a contradiction.

(2) If <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/174/mathml/M72','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/174/mathml/M72">View MathML</a>, then <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/174/mathml/M307','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/174/mathml/M307">View MathML</a> (since <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/174/mathml/M301','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/174/mathml/M301">View MathML</a>, we know that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/174/mathml/M309','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/174/mathml/M309">View MathML</a> and the strict concavity of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/174/mathml/M298','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/174/mathml/M298">View MathML</a> imply <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/174/mathml/M81','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/174/mathml/M81">View MathML</a> on <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/174/mathml/M142','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/174/mathml/M142">View MathML</a>, which is a contradiction). Put

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/174/mathml/M313','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/174/mathml/M313">View MathML</a>

Then

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/174/mathml/M314','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/174/mathml/M314">View MathML</a>

From the strict concavity of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/174/mathml/M298','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/174/mathml/M298">View MathML</a>, we get that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/174/mathml/M316','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/174/mathml/M316">View MathML</a>

a contradiction.

Consequently, the conclusion in (3.17) holds. □

Remark 3.1 In contrast to [[1], Theorem 3.2], we obtain the global structure and behavior of positive solutions, where the parameter α is regarded as a variable.

Theorem 3.2Suppose that<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/174/mathml/M2','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/174/mathml/M2">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/174/mathml/M318','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/174/mathml/M318">View MathML</a>and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/174/mathml/M319','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/174/mathml/M319">View MathML</a>. Let condition (H1) and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/174/mathml/M320','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/174/mathml/M320">View MathML</a>hold, and let<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/174/mathml/M321','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/174/mathml/M321">View MathML</a>be a solution of

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/174/mathml/M322','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/174/mathml/M322">View MathML</a>

(3.19)

Then problem (3.19) has no positive solutions.

Proof If <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/174/mathml/M323','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/174/mathml/M323">View MathML</a>, then we know that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/174/mathml/M324','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/174/mathml/M324">View MathML</a> for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/174/mathml/M59','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/174/mathml/M59">View MathML</a> is a trivial solution of (3.19).

Suppose on the contrary that there exists a positive solution <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/174/mathml/M111','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/174/mathml/M111">View MathML</a> to equation (3.19), i.e.,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/174/mathml/M327','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/174/mathml/M327">View MathML</a>

(3.20)

Therefore, we know from equation (3.19) that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/174/mathml/M328','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/174/mathml/M328">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/174/mathml/M111','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/174/mathml/M111">View MathML</a> is concave down in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/174/mathml/M79','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/174/mathml/M79">View MathML</a>. Now, we consider two cases. Case I: <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/174/mathml/M71','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/174/mathml/M71">View MathML</a>; Case II: <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/174/mathml/M72','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/174/mathml/M72">View MathML</a>.

(1) Case I. <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/174/mathml/M71','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/174/mathml/M71">View MathML</a>.

Clearly, we know from the boundary conditions of (3.19) that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/174/mathml/M334','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/174/mathml/M334">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/174/mathml/M99','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/174/mathml/M99">View MathML</a>. Since u is concave down and u is a positive solution of equation (3.19), we obtain that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/174/mathml/M336','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/174/mathml/M336">View MathML</a>

implies

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/174/mathml/M337','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/174/mathml/M337">View MathML</a>

i.e.,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/174/mathml/M338','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/174/mathml/M338">View MathML</a>

contradicts the condition <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/174/mathml/M99','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/174/mathml/M99">View MathML</a>.

(2) Case II. <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/174/mathml/M72','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/174/mathml/M72">View MathML</a>.

If <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/174/mathml/M341','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/174/mathml/M341">View MathML</a>, then <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/174/mathml/M342','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/174/mathml/M342">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/174/mathml/M90','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/174/mathml/M90">View MathML</a> imply that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/174/mathml/M92','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/174/mathml/M92">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/174/mathml/M91','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/174/mathml/M91">View MathML</a>. This contradicts (3.20).

If

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/174/mathml/M346','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/174/mathml/M346">View MathML</a>

(3.21)

then the concavity of u implies that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/174/mathml/M347','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/174/mathml/M347">View MathML</a>

Thus

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/174/mathml/M348','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/174/mathml/M348">View MathML</a>

(3.22)

For <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/174/mathml/M349','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/174/mathml/M349">View MathML</a>, we obtain that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/174/mathml/M350','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/174/mathml/M350">View MathML</a>

contradicts (3.21).

For <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/174/mathml/M351','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/174/mathml/M351">View MathML</a>, we have <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/174/mathml/M352','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/174/mathml/M352">View MathML</a>. Since <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/174/mathml/M321','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/174/mathml/M321">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/174/mathml/M328','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/174/mathml/M328">View MathML</a>, we get from Taylor’s expansion that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/174/mathml/M355','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/174/mathml/M355">View MathML</a>

(3.23)

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/174/mathml/M356','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/174/mathml/M356">View MathML</a>. Substituting (3.23) into (3.22), we find that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/174/mathml/M357','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/174/mathml/M357">View MathML</a>

From the boundary condition of (3.19), it follows that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/174/mathml/M358','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/174/mathml/M358">View MathML</a>

leads to

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/174/mathml/M359','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/174/mathml/M359">View MathML</a>

This is a contradiction.

Therefore, we conclude that if <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/174/mathml/M319','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/174/mathml/M319">View MathML</a>, then problem (3.19) has no positive solutions. □

4 The sublinear case

Lemma 4.1 [[1], Theorem 3.1]

Let conditions (H0)-(H2) and (H4) hold. Then there exist two constants<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/174/mathml/M163','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/174/mathml/M163">View MathML</a>and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/174/mathml/M164','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/174/mathml/M164">View MathML</a>with<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/174/mathml/M165','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/174/mathml/M165">View MathML</a>such that problem (1.1) has at least one positive solution<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/174/mathml/M166','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/174/mathml/M166">View MathML</a>with<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/174/mathml/M167','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/174/mathml/M167">View MathML</a>. Furthermore,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/174/mathml/M366','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/174/mathml/M366">View MathML</a>

where<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/174/mathml/M169','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/174/mathml/M169">View MathML</a>.

Lemma 4.2Assume that (H0)-(H2) and (H4) hold. Let<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/174/mathml/M163','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/174/mathml/M163">View MathML</a>and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/174/mathml/M164','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/174/mathml/M164">View MathML</a>be the constants as in Lemma 4.1. Then there exists a positive number<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/174/mathml/M370','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/174/mathml/M370">View MathML</a>with<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/174/mathml/M371','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/174/mathml/M371">View MathML</a>such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/174/mathml/M372','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/174/mathml/M372">View MathML</a>

where<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/174/mathml/M178','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/174/mathml/M178">View MathML</a>is defined by (3.2).

Proof First, we claim that if <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/174/mathml/M29','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/174/mathml/M29">View MathML</a>, then there exists a positive number <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/174/mathml/M375','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/174/mathml/M375">View MathML</a> such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/174/mathml/M376','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/174/mathml/M376">View MathML</a>

(4.1)

Suppose this fails, that is, there exists a sequence <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/174/mathml/M377','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/174/mathml/M377">View MathML</a> with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/174/mathml/M216','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/174/mathml/M216">View MathML</a> such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/174/mathml/M379','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/174/mathml/M379">View MathML</a>

From conditions (H0)-(H2) and Lemma 2.2, it follows that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/174/mathml/M380','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/174/mathml/M380">View MathML</a>

The concavity of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/174/mathml/M231','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/174/mathml/M231">View MathML</a> implies

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/174/mathml/M382','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/174/mathml/M382">View MathML</a>

Choose <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/174/mathml/M383','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/174/mathml/M383">View MathML</a>, where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/174/mathml/M384','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/174/mathml/M384">View MathML</a>. In light of (H4), we get that there exists a constant <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/174/mathml/M385','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/174/mathml/M385">View MathML</a> such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/174/mathml/M386','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/174/mathml/M386">View MathML</a>

Since <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/174/mathml/M387','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/174/mathml/M387">View MathML</a>, we get that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/174/mathml/M388','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/174/mathml/M388">View MathML</a>

implies

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/174/mathml/M389','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/174/mathml/M389">View MathML</a>

Adopting the same proof as in Lemma 3.2, we get a contradiction. Hence, conclusion (4.1) holds.

Now, we show that if <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/174/mathml/M30','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/174/mathml/M30">View MathML</a>, then there exists a positive number <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/174/mathml/M391','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/174/mathml/M391">View MathML</a> such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/174/mathml/M392','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/174/mathml/M392">View MathML</a>

(4.2)

Define the nondecreasing function <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/174/mathml/M393','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/174/mathml/M393">View MathML</a> by

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/174/mathml/M394','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/174/mathml/M394">View MathML</a>

Since <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/174/mathml/M30','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/174/mathml/M30">View MathML</a>, it follows that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/174/mathml/M396','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/174/mathml/M396">View MathML</a>

(4.3)

Suppose that conclusion (4.2) fails, that is, there exists a sequence <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/174/mathml/M377','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/174/mathml/M377">View MathML</a> with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/174/mathml/M286','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/174/mathml/M286">View MathML</a>. We may assume that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/174/mathml/M399','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/174/mathml/M399">View MathML</a>. From (H0)-(H2) and Lemma 2.2, we have that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/174/mathml/M400','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/174/mathml/M400">View MathML</a>

Define <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/174/mathml/M401','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/174/mathml/M401">View MathML</a>, which implies that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/174/mathml/M402','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/174/mathml/M402">View MathML</a>

Since <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/174/mathml/M403','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/174/mathml/M403">View MathML</a>, we get from (4.3) that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/174/mathml/M404','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/174/mathml/M404">View MathML</a>

uniformly holds for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/174/mathml/M59','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/174/mathml/M59">View MathML</a>. Again, applying the proof method as that in Lemma 3.2, we get a contradiction. Consequently, conclusion (4.2) holds.

If we let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/174/mathml/M406','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/174/mathml/M406">View MathML</a>, then combining (4.1) and (4.2), we have that the result holds. □

Theorem 4.1Let (H0)-(H2) and (H4) hold. Thencontains a continuum which joins<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/174/mathml/M250','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/174/mathml/M250">View MathML</a>with<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/174/mathml/M409','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/174/mathml/M409">View MathML</a>.

Proof Applying the method as in Theorem 3.1, we find from Lemma 2.2, Lemma 4.1 and Lemma 4.2 that there exists a continuum <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/174/mathml/M410','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/174/mathml/M410">View MathML</a> satisfying

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/174/mathml/M411','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/174/mathml/M411">View MathML</a>

and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/174/mathml/M412','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/174/mathml/M412">View MathML</a>

Next, we only show that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/174/mathml/M413','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/174/mathml/M413">View MathML</a>

Suppose on the contrary that there exists a sequence <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/174/mathml/M414','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/174/mathml/M414">View MathML</a> with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/174/mathml/M183','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/174/mathml/M183">View MathML</a> such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/174/mathml/M416','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/174/mathml/M416">View MathML</a>

Define <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/174/mathml/M187','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/174/mathml/M187">View MathML</a>. From conditions (H0)-(H2) and Lemma 2.2, it follows that for any <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/174/mathml/M418','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/174/mathml/M418">View MathML</a>,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/174/mathml/M419','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/174/mathml/M419">View MathML</a>

and the concavity of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/174/mathml/M231','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/174/mathml/M231">View MathML</a> leads to

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/174/mathml/M421','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/174/mathml/M421">View MathML</a>

Take <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/174/mathml/M383','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/174/mathml/M383">View MathML</a>, where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/174/mathml/M384','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/174/mathml/M384">View MathML</a>. In light of (H4), we get that there exists a constant <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/174/mathml/M385','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/174/mathml/M385">View MathML</a> such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/174/mathml/M425','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/174/mathml/M425">View MathML</a>

Using the same proof as in Lemma 3.2, we get a contradiction.

Hence, the conclusion holds. □

Remark 4.1 In contrast to [[1], Theorem 3.1], we obtain the global structure and behavior of positive solutions, where the parameter α is regarded as a variable.

Competing interests

The authors declare that they have no competing interests.

Authors’ contributions

All authors contributed significantly in writing this paper. All authors read and approved the final manuscript.

Acknowledgements

The work was supported partly by NSFC (No. 11201248), K.C. Wong Magna Fund of Ningbo University and Ningbo Natural Science Foundation (No. 2012A610031).

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