SpringerOpen Newsletter

Receive periodic news and updates relating to SpringerOpen.

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

Open Access Research

Existence and multiplicity of positive solutions for one-dimensional prescribed mean curvature equations

Yanqiong Lu, Ruyun Ma* and Hongliang Gao

Author Affiliations

Department of Mathematics, Northwest Normal University, Lanzhou, 730070, P.R. China

For all author emails, please log on.

Boundary Value Problems 2014, 2014:120  doi:10.1186/1687-2770-2014-120


Dedicated to Professor Ivan T. Kiguradze for his merits in the theory of differential equations

The electronic version of this article is the complete one and can be found online at: http://www.boundaryvalueproblems.com/content/2014/1/120


Received:3 January 2014
Accepted:6 May 2014
Published:16 May 2014

© 2014 Lu et al.; licensee Springer.

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

Abstract

In this work, we establish the existence and multiplicity results of positive solutions for one-dimensional prescribed mean curvature equations. Our approach is based on fixed point index theory for completely continuous operators which leave invariant a suitable cone in a Banach space of continuous functions.

MSC: 34B10, 34B18.

Keywords:
mean curvature equations; positive solutions; existence; fixed point index

1 Introduction

The prescribed mean curvature problems like

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/120/mathml/M1','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/120/mathml/M1">View MathML</a>

have attracted much attention in recent years, see [1-4] and the references therein. Since the problem is quasilinear non-uniformly elliptic, it is more difficult to study the existence of classical solutions. The greatest obstacle is the lack of gradient estimate, such kind of estimate does not hold in general and boundary gradient blow-up may occur. This leads to some new phenomena very different from those in semilinear problems. Many well-known results of semilinear problems have to be reconsidered for this quasilinear problem. Motivated by the search for solutions of the above problem, many authors (see [5-13]) studied the existence of (positive) solutions for one-dimensional prescribed mean curvature equations with Dirichlet boundary conditions

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/120/mathml/M2','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/120/mathml/M2">View MathML</a>

(1.1)

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/120/mathml/M3','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/120/mathml/M3">View MathML</a> is a constant, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/120/mathml/M4','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/120/mathml/M4">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/120/mathml/M5','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/120/mathml/M5">View MathML</a> for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/120/mathml/M6','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/120/mathml/M6">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/120/mathml/M7','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/120/mathml/M7">View MathML</a>.

Note that if <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/120/mathml/M8','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/120/mathml/M8">View MathML</a>, problem (1.1) is degenerate to the second-order ordinary differential equation boundary value problems

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/120/mathml/M9','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/120/mathml/M9">View MathML</a>

(1.2)

The existence of (positive) solutions of (1.1) has been well known with various qualitative assumptions of the nonlinearity f, see [14,15] and the references therein.

If <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/120/mathml/M10','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/120/mathml/M10">View MathML</a>, Bonheure et al.[6], Habets and Omari [9], Kusahara and Usami [10], Pan and Xing [12,13] studied the existence of (positive) solutions of (1.2) by using the variational method, lower and upper solutions method and time mapping method, respectively.

However, to the best of our knowledge, the existence and multiplicity of positive solutions for (1.1) are relatively few by the fixed point index theory. In this paper, based on the fixed point index theory, we shall investigate the existence and multiplicity of positive solution of (1.1) when f is ϕ-superlinear and ϕ-sublinear at 0 and ∞, respectively, here <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/120/mathml/M11','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/120/mathml/M11">View MathML</a>.

Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/120/mathml/M11','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/120/mathml/M11">View MathML</a>. Then (1.1) can be rewritten as

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/120/mathml/M13','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/120/mathml/M13">View MathML</a>

(1.3)

Obviously, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/120/mathml/M14','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/120/mathml/M14">View MathML</a> is an odd, increasing homeomorphism with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/120/mathml/M15','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/120/mathml/M15">View MathML</a>.

For convenience, we introduce some notations

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/120/mathml/M16','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/120/mathml/M16">View MathML</a>

We will also need the function <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/120/mathml/M17','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/120/mathml/M17">View MathML</a>, and let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/120/mathml/M18','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/120/mathml/M18">View MathML</a>. By a similar method in [[14], Lemma 2.8], it is not difficult to verify that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/120/mathml/M19','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/120/mathml/M19">View MathML</a>.

In the rest of this paper, we shall study the existence of positive solutions of (1.3) by using the fixed point index theory to give a brief and clear proof for the existence of positive solutions of (1.1). More concretely, we shall prove the following.

Theorem 1.1Assume that<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/120/mathml/M20','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/120/mathml/M20">View MathML</a>and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/120/mathml/M5','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/120/mathml/M5">View MathML</a>for<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/120/mathml/M6','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/120/mathml/M6">View MathML</a>.

(i) If<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/120/mathml/M23','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/120/mathml/M23">View MathML</a>, then there exists<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/120/mathml/M24','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/120/mathml/M24">View MathML</a>such that (1.1) has a positive solution for<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/120/mathml/M25','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/120/mathml/M25">View MathML</a>.

(ii) If<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/120/mathml/M26','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/120/mathml/M26">View MathML</a>, then there exists<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/120/mathml/M27','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/120/mathml/M27">View MathML</a>such that (1.1) has a positive solution for<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/120/mathml/M28','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/120/mathml/M28">View MathML</a>.

(iii) If<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/120/mathml/M29','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/120/mathml/M29">View MathML</a>, then there exists<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/120/mathml/M24','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/120/mathml/M24">View MathML</a>such that (1.1) has at least two positive solutions for<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/120/mathml/M25','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/120/mathml/M25">View MathML</a>.

(iv) If<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/120/mathml/M32','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/120/mathml/M32">View MathML</a>, then there exists<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/120/mathml/M24','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/120/mathml/M24">View MathML</a>such that (1.1) has at least two positive solutions for<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/120/mathml/M25','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/120/mathml/M25">View MathML</a>.

Corollary 1.2Assume that<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/120/mathml/M4','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/120/mathml/M4">View MathML</a>and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/120/mathml/M5','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/120/mathml/M5">View MathML</a>for<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/120/mathml/M6','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/120/mathml/M6">View MathML</a>.

(a) If<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/120/mathml/M23','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/120/mathml/M23">View MathML</a>and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/120/mathml/M39','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/120/mathml/M39">View MathML</a>, then there exists<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/120/mathml/M24','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/120/mathml/M24">View MathML</a>such that (1.1) has a positive solution for<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/120/mathml/M25','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/120/mathml/M25">View MathML</a>.

(b) If<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/120/mathml/M26','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/120/mathml/M26">View MathML</a>and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/120/mathml/M43','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/120/mathml/M43">View MathML</a>, then there exists<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/120/mathml/M27','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/120/mathml/M27">View MathML</a>such that (1.1) has a positive solution for<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/120/mathml/M28','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/120/mathml/M28">View MathML</a>.

Remark 1.1 The results of Theorem 1.1 and Corollary 1.2 are different with the case <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/120/mathml/M8','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/120/mathml/M8">View MathML</a> which is the classical Dirichlet boundary value problem (1.2). This phenomenon is a striking feature of problem (1.1), which is just the reason why we study the existence of positive solutions of problem (1.1). It is pointed out that in equation of (1.1), having replaced <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/120/mathml/M47','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/120/mathml/M47">View MathML</a> with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/120/mathml/M48','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/120/mathml/M48">View MathML</a>, Theorem 1.1 and Corollary 1.2 also hold as well as all of the proofs with obvious changes.

2 Preliminaries

Throughout the paper <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/120/mathml/M49','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/120/mathml/M49">View MathML</a> will denote absolute value, and let

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/120/mathml/M50','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/120/mathml/M50">View MathML</a>

Then E is a Banach space endowed with the norm <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/120/mathml/M51','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/120/mathml/M51">View MathML</a>.

We first establish some preliminary results to prove our main result. An easy but useful property of ϕ and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/120/mathml/M52','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/120/mathml/M52">View MathML</a> is the following one.

Lemma 2.1Let<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/120/mathml/M11','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/120/mathml/M11">View MathML</a>. Then<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/120/mathml/M54','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/120/mathml/M54">View MathML</a>is an odd, increasing homeomorphism with<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/120/mathml/M15','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/120/mathml/M15">View MathML</a>. Moreover, ϕhas the following properties:

(i) ϕis convex up on<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/120/mathml/M56','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/120/mathml/M56">View MathML</a>and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/120/mathml/M52','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/120/mathml/M52">View MathML</a>is concave up on<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/120/mathml/M58','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/120/mathml/M58">View MathML</a>.

(ii) For each<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/120/mathml/M59','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/120/mathml/M59">View MathML</a>, there exists<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/120/mathml/M60','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/120/mathml/M60">View MathML</a>such that<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/120/mathml/M61','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/120/mathml/M61">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/120/mathml/M62','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/120/mathml/M62">View MathML</a>with<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/120/mathml/M63','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/120/mathml/M63">View MathML</a>and for each<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/120/mathml/M64','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/120/mathml/M64">View MathML</a>, there exists<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/120/mathml/M65','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/120/mathml/M65">View MathML</a>such that<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/120/mathml/M66','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/120/mathml/M66">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/120/mathml/M62','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/120/mathml/M62">View MathML</a>with<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/120/mathml/M68','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/120/mathml/M68">View MathML</a>.

(iii) For each<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/120/mathml/M64','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/120/mathml/M64">View MathML</a>, there exists<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/120/mathml/M70','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/120/mathml/M70">View MathML</a>such that<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/120/mathml/M71','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/120/mathml/M71">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/120/mathml/M72','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/120/mathml/M72">View MathML</a>. For each<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/120/mathml/M59','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/120/mathml/M59">View MathML</a>with<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/120/mathml/M74','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/120/mathml/M74">View MathML</a>, there exists<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/120/mathml/M75','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/120/mathml/M75">View MathML</a>such that<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/120/mathml/M76','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/120/mathml/M76">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/120/mathml/M72','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/120/mathml/M72">View MathML</a>.

Proof By a simple computation, it follows that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/120/mathml/M78','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/120/mathml/M78">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/120/mathml/M79','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/120/mathml/M79">View MathML</a>. So ϕ is an odd, increasing homeomorphism with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/120/mathml/M15','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/120/mathml/M15">View MathML</a>. Moreover, from <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/120/mathml/M81','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/120/mathml/M81">View MathML</a>, we get that ϕ is convex up on <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/120/mathml/M56','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/120/mathml/M56">View MathML</a>. Notice that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/120/mathml/M83','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/120/mathml/M83">View MathML</a> is also an odd, increasing homeomorphism with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/120/mathml/M15','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/120/mathml/M15">View MathML</a>. It is easy to verify that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/120/mathml/M52','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/120/mathml/M52">View MathML</a> is concave up on <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/120/mathml/M58','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/120/mathml/M58">View MathML</a>.

(ii) For each <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/120/mathml/M64','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/120/mathml/M64">View MathML</a>, there exists <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/120/mathml/M88','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/120/mathml/M88">View MathML</a> such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/120/mathml/M89','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/120/mathml/M89">View MathML</a>

and for each <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/120/mathml/M59','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/120/mathml/M59">View MathML</a>, there exists <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/120/mathml/M60','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/120/mathml/M60">View MathML</a> such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/120/mathml/M92','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/120/mathml/M92">View MathML</a>

(iii) By a similar argument, it is not difficult to compute that for each <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/120/mathml/M64','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/120/mathml/M64">View MathML</a>, there exists <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/120/mathml/M70','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/120/mathml/M70">View MathML</a> such that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/120/mathml/M71','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/120/mathml/M71">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/120/mathml/M72','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/120/mathml/M72">View MathML</a>. For each <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/120/mathml/M59','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/120/mathml/M59">View MathML</a> with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/120/mathml/M98','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/120/mathml/M98">View MathML</a>, there exists <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/120/mathml/M75','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/120/mathml/M75">View MathML</a> such that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/120/mathml/M76','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/120/mathml/M76">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/120/mathml/M72','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/120/mathml/M72">View MathML</a>. □

Lemma 2.2Let<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/120/mathml/M102','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/120/mathml/M102">View MathML</a>with<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/120/mathml/M103','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/120/mathml/M103">View MathML</a>. Assume thatwis the solution of

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/120/mathml/M104','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/120/mathml/M104">View MathML</a>

(2.1)

Then<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/120/mathml/M105','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/120/mathml/M105">View MathML</a>on<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/120/mathml/M106','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/120/mathml/M106">View MathML</a>and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/120/mathml/M107','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/120/mathml/M107">View MathML</a>, where<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/120/mathml/M108','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/120/mathml/M108">View MathML</a>.

Proof By integrating, it follows that (2.1) has the unique solution given by

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/120/mathml/M109','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/120/mathml/M109">View MathML</a>

where C is such that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/120/mathml/M110','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/120/mathml/M110">View MathML</a>. Hence we must have <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/120/mathml/M111','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/120/mathml/M111">View MathML</a>. Further, since <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/120/mathml/M112','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/120/mathml/M112">View MathML</a>, by using <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/120/mathml/M113','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/120/mathml/M113">View MathML</a>, we obtain <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/120/mathml/M114','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/120/mathml/M114">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/120/mathml/M115','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/120/mathml/M115">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/120/mathml/M107','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/120/mathml/M107">View MathML</a> follows, here <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/120/mathml/M108','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/120/mathml/M108">View MathML</a>.

Since <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/120/mathml/M118','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/120/mathml/M118">View MathML</a>, there exists <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/120/mathml/M119','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/120/mathml/M119">View MathML</a> such that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/120/mathml/M120','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/120/mathml/M120">View MathML</a> and it follows from <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/120/mathml/M121','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/120/mathml/M121">View MathML</a> that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/120/mathml/M122','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/120/mathml/M122">View MathML</a> is decreasing on <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/120/mathml/M106','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/120/mathml/M106">View MathML</a>. Then <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/120/mathml/M124','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/120/mathml/M124">View MathML</a> for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/120/mathml/M125','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/120/mathml/M125">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/120/mathml/M126','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/120/mathml/M126">View MathML</a> for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/120/mathml/M127','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/120/mathml/M127">View MathML</a>. Hence, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/120/mathml/M105','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/120/mathml/M105">View MathML</a> on <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/120/mathml/M106','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/120/mathml/M106">View MathML</a>. □

Note that from Lemma 2.2, there exists <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/120/mathml/M130','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/120/mathml/M130">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/120/mathml/M131','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/120/mathml/M131">View MathML</a>, such that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/120/mathml/M132','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/120/mathml/M132">View MathML</a> with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/120/mathml/M133','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/120/mathml/M133">View MathML</a> depending on <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/120/mathml/M134','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/120/mathml/M134">View MathML</a>. Define the cone P in E by

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/120/mathml/M135','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/120/mathml/M135">View MathML</a>

and for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/120/mathml/M136','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/120/mathml/M136">View MathML</a>, let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/120/mathml/M137','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/120/mathml/M137">View MathML</a>.

Lemma 2.3 ([[11], Lemma 4.1 and Lemma 4.2])

For each<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/120/mathml/M138','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/120/mathml/M138">View MathML</a>, (2.1) has a unique solution given by

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/120/mathml/M139','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/120/mathml/M139">View MathML</a>

whereCis such that<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/120/mathml/M140','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/120/mathml/M140">View MathML</a>with<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/120/mathml/M111','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/120/mathml/M111">View MathML</a>. Moreover, the operator<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/120/mathml/M142','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/120/mathml/M142">View MathML</a>is continuous and sends equicontinuous sets in<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/120/mathml/M143','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/120/mathml/M143">View MathML</a>into a relatively compact set inE.

We next state the fixed point index theorem which will be used to prove our results.

Lemma 2.4 ([[16], Chapter 6])

LetEbe a Banach space andPbe a cone inE. Assume that Ω is a bounded open subset ofEwith<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/120/mathml/M144','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/120/mathml/M144">View MathML</a>, and let<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/120/mathml/M145','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/120/mathml/M145">View MathML</a>be a completely continuous operator such that<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/120/mathml/M146','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/120/mathml/M146">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/120/mathml/M147','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/120/mathml/M147">View MathML</a>.

(i) If<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/120/mathml/M148','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/120/mathml/M148">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/120/mathml/M147','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/120/mathml/M147">View MathML</a>, then<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/120/mathml/M150','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/120/mathml/M150">View MathML</a>.

(ii) If<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/120/mathml/M151','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/120/mathml/M151">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/120/mathml/M147','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/120/mathml/M147">View MathML</a>, then<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/120/mathml/M153','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/120/mathml/M153">View MathML</a>.

From Lemma 2.2, problem (1.3) is equivalent to the fixed point problem

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/120/mathml/M154','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/120/mathml/M154">View MathML</a>

in the space E, where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/120/mathml/M155','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/120/mathml/M155">View MathML</a> is such that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/120/mathml/M140','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/120/mathml/M140">View MathML</a> with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/120/mathml/M157','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/120/mathml/M157">View MathML</a>, since otherwise,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/120/mathml/M158','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/120/mathml/M158">View MathML</a>

which is a contradiction. This together with Lemma 2.3 implies that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/120/mathml/M159','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/120/mathml/M159">View MathML</a> is a completely continuous mapping. Moreover, for any fixed <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/120/mathml/M160','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/120/mathml/M160">View MathML</a>, we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/120/mathml/M161','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/120/mathml/M161">View MathML</a>

(2.2)

and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/120/mathml/M162','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/120/mathml/M162">View MathML</a>. In addition, from Lemma 2.2, it follows that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/120/mathml/M163','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/120/mathml/M163">View MathML</a> on <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/120/mathml/M106','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/120/mathml/M106">View MathML</a> and there exist <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/120/mathml/M130','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/120/mathml/M130">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/120/mathml/M131','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/120/mathml/M131">View MathML</a>, such that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/120/mathml/M167','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/120/mathml/M167">View MathML</a>. So <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/120/mathml/M168','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/120/mathml/M168">View MathML</a> is a completely continuous operator.

Lemma 2.5Let<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/120/mathml/M136','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/120/mathml/M136">View MathML</a>be given. If there exists<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/120/mathml/M170','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/120/mathml/M170">View MathML</a>small enough with<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/120/mathml/M171','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/120/mathml/M171">View MathML</a>such that<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/120/mathml/M172','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/120/mathml/M172">View MathML</a>, then

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/120/mathml/M173','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/120/mathml/M173">View MathML</a>

where<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/120/mathml/M174','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/120/mathml/M174">View MathML</a>is defined as in Lemma 2.1(iii).

Proof From the definition of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/120/mathml/M175','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/120/mathml/M175">View MathML</a>, for any <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/120/mathml/M176','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/120/mathml/M176">View MathML</a>, we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/120/mathml/M177','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/120/mathml/M177">View MathML</a>

 □

Lemma 2.6Let<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/120/mathml/M178','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/120/mathml/M178">View MathML</a>be given. If<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/120/mathml/M160','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/120/mathml/M160">View MathML</a>and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/120/mathml/M180','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/120/mathml/M180">View MathML</a>for<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/120/mathml/M7','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/120/mathml/M7">View MathML</a>, then

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/120/mathml/M182','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/120/mathml/M182">View MathML</a>

where<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/120/mathml/M183','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/120/mathml/M183">View MathML</a>and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/120/mathml/M184','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/120/mathml/M184">View MathML</a>.

Proof From problem (1.3), since <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/120/mathml/M185','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/120/mathml/M185">View MathML</a>, it follows that there exists <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/120/mathml/M119','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/120/mathml/M119">View MathML</a> such that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/120/mathml/M187','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/120/mathml/M187">View MathML</a>. Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/120/mathml/M188','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/120/mathml/M188">View MathML</a>. Then u satisfies the following boundary value problem:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/120/mathml/M189','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/120/mathml/M189">View MathML</a>

(2.3)

Let v be the solution of the problem

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/120/mathml/M190','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/120/mathml/M190">View MathML</a>

(2.4)

Then we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/120/mathml/M191','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/120/mathml/M191">View MathML</a>

(2.5)

and by a comparison argument, we get that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/120/mathml/M192','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/120/mathml/M192">View MathML</a> on <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/120/mathml/M193','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/120/mathml/M193">View MathML</a>. In fact, from <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/120/mathml/M194','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/120/mathml/M194">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/120/mathml/M195','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/120/mathml/M195">View MathML</a>, there exists <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/120/mathml/M196','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/120/mathml/M196">View MathML</a> such that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/120/mathml/M197','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/120/mathml/M197">View MathML</a>, i.e., <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/120/mathml/M198','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/120/mathml/M198">View MathML</a>. Thus, by a simple computation, we have that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/120/mathml/M199','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/120/mathml/M199">View MathML</a>

and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/120/mathml/M200','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/120/mathml/M200">View MathML</a>

This together with ϕ is an increasing homeomorphism implies that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/120/mathml/M201','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/120/mathml/M201">View MathML</a>

(2.6)

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/120/mathml/M202','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/120/mathml/M202">View MathML</a>

(2.7)

Integrating from 0 to x for (2.6) and integrating from x to <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/120/mathml/M203','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/120/mathml/M203">View MathML</a> for (2.7), respectively, we have that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/120/mathml/M204','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/120/mathml/M204">View MathML</a> for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/120/mathml/M205','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/120/mathml/M205">View MathML</a>.

Note that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/120/mathml/M206','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/120/mathml/M206">View MathML</a>

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/120/mathml/M207','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/120/mathml/M207">View MathML</a> is such that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/120/mathml/M208','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/120/mathml/M208">View MathML</a>, and hence <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/120/mathml/M209','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/120/mathml/M209">View MathML</a>. If <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/120/mathml/M210','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/120/mathml/M210">View MathML</a>, then it follows that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/120/mathml/M211','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/120/mathml/M211">View MathML</a>

which is a contradiction. Thus, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/120/mathml/M212','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/120/mathml/M212">View MathML</a>. Moreover, we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/120/mathml/M213','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/120/mathml/M213">View MathML</a>

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/120/mathml/M214','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/120/mathml/M214">View MathML</a>. Consequently, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/120/mathml/M215','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/120/mathml/M215">View MathML</a> for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/120/mathml/M216','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/120/mathml/M216">View MathML</a>. Obviously,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/120/mathml/M217','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/120/mathml/M217">View MathML</a>

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/120/mathml/M218','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/120/mathml/M218">View MathML</a> satisfies <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/120/mathml/M187','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/120/mathml/M187">View MathML</a>. It follows from <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/120/mathml/M220','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/120/mathml/M220">View MathML</a> that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/120/mathml/M221','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/120/mathml/M221">View MathML</a>. Therefore,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/120/mathml/M222','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/120/mathml/M222">View MathML</a>

(2.8)

If <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/120/mathml/M223','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/120/mathml/M223">View MathML</a>, then

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/120/mathml/M224','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/120/mathml/M224">View MathML</a>

If <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/120/mathml/M225','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/120/mathml/M225">View MathML</a>, let w be the solution of

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/120/mathml/M226','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/120/mathml/M226">View MathML</a>

(2.9)

Then

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/120/mathml/M227','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/120/mathml/M227">View MathML</a>

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/120/mathml/M228','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/120/mathml/M228">View MathML</a> satisfies <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/120/mathml/M110','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/120/mathml/M110">View MathML</a>, i.e., <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/120/mathml/M230','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/120/mathml/M230">View MathML</a>. If <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/120/mathml/M231','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/120/mathml/M231">View MathML</a>, then

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/120/mathml/M232','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/120/mathml/M232">View MathML</a>

which is contradiction. Hence <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/120/mathml/M233','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/120/mathml/M233">View MathML</a>.

By a similar argument as before, it follows that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/120/mathml/M234','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/120/mathml/M234">View MathML</a> on <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/120/mathml/M235','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/120/mathml/M235">View MathML</a>. Moreover,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/120/mathml/M236','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/120/mathml/M236">View MathML</a>

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/120/mathml/M237','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/120/mathml/M237">View MathML</a>. So <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/120/mathml/M238','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/120/mathml/M238">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/120/mathml/M239','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/120/mathml/M239">View MathML</a>. Therefore, we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/120/mathml/M240','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/120/mathml/M240">View MathML</a>

(2.10)

and subsequently,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/120/mathml/M241','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/120/mathml/M241">View MathML</a>

Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/120/mathml/M183','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/120/mathml/M183">View MathML</a>. Then

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/120/mathml/M243','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/120/mathml/M243">View MathML</a>

 □

Lemma 2.7Let<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/120/mathml/M136','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/120/mathml/M136">View MathML</a>be given. If<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/120/mathml/M245','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/120/mathml/M245">View MathML</a>, then

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/120/mathml/M246','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/120/mathml/M246">View MathML</a>

where<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/120/mathml/M247','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/120/mathml/M247">View MathML</a>.

Proof Obviously, for any <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/120/mathml/M176','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/120/mathml/M176">View MathML</a>, it follows that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/120/mathml/M249','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/120/mathml/M249">View MathML</a> for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/120/mathml/M7','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/120/mathml/M7">View MathML</a>. So we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/120/mathml/M251','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/120/mathml/M251">View MathML</a>

 □

Lemma 2.8Let<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/120/mathml/M136','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/120/mathml/M136">View MathML</a>be given. If<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/120/mathml/M245','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/120/mathml/M245">View MathML</a>, then

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/120/mathml/M254','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/120/mathml/M254">View MathML</a>

where<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/120/mathml/M255','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/120/mathml/M255">View MathML</a>and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/120/mathml/M183','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/120/mathml/M183">View MathML</a>.

Proof By using a similar argument of the proof of Lemma 2.6, we have that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/120/mathml/M257','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/120/mathml/M257">View MathML</a> for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/120/mathml/M258','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/120/mathml/M258">View MathML</a>. Meanwhile, (2.8) is true. If <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/120/mathml/M223','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/120/mathml/M223">View MathML</a>, then we can get that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/120/mathml/M260','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/120/mathml/M260">View MathML</a>

If <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/120/mathml/M225','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/120/mathml/M225">View MathML</a>, then (2.10) holds and it follows that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/120/mathml/M262','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/120/mathml/M262">View MathML</a>

Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/120/mathml/M263','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/120/mathml/M263">View MathML</a>, then

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/120/mathml/M264','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/120/mathml/M264">View MathML</a>

 □

3 Proof of the main results

Proof of Theorem 1.1 (i) Choose a suitable number <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/120/mathml/M265','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/120/mathml/M265">View MathML</a>. By Lemma 2.8, we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/120/mathml/M266','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/120/mathml/M266">View MathML</a>

where

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/120/mathml/M267','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/120/mathml/M267">View MathML</a>

If <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/120/mathml/M23','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/120/mathml/M23">View MathML</a>, then <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/120/mathml/M269','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/120/mathml/M269">View MathML</a>, and so we can choose <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/120/mathml/M270','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/120/mathml/M270">View MathML</a> such that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/120/mathml/M271','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/120/mathml/M271">View MathML</a>, where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/120/mathml/M170','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/120/mathml/M170">View MathML</a> small enough satisfies

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/120/mathml/M273','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/120/mathml/M273">View MathML</a>

(3.1)

Then Lemma 2.5 implies that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/120/mathml/M274','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/120/mathml/M274">View MathML</a>

From Lemma 2.4, it follows that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/120/mathml/M275','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/120/mathml/M275">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/120/mathml/M276','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/120/mathml/M276">View MathML</a>. By using the additivity-excision property of the fixed point index [16], we have that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/120/mathml/M277','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/120/mathml/M277">View MathML</a>

Therefore, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/120/mathml/M175','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/120/mathml/M175">View MathML</a> has a fixed point in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/120/mathml/M279','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/120/mathml/M279">View MathML</a>. Consequently, (1.1) has a positive solution for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/120/mathml/M25','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/120/mathml/M25">View MathML</a>.

(ii) Choose a suitable number <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/120/mathml/M265','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/120/mathml/M265">View MathML</a>. By Lemma 2.7, there exists

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/120/mathml/M282','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/120/mathml/M282">View MathML</a>

such that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/120/mathml/M283','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/120/mathml/M283">View MathML</a> for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/120/mathml/M284','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/120/mathml/M284">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/120/mathml/M28','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/120/mathml/M28">View MathML</a>. That is, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/120/mathml/M286','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/120/mathml/M286">View MathML</a>.

If <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/120/mathml/M26','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/120/mathml/M26">View MathML</a>, then there exists <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/120/mathml/M270','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/120/mathml/M270">View MathML</a> such that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/120/mathml/M289','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/120/mathml/M289">View MathML</a> for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/120/mathml/M290','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/120/mathml/M290">View MathML</a>, where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/120/mathml/M178','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/120/mathml/M178">View MathML</a> is chosen large enough so that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/120/mathml/M292','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/120/mathml/M292">View MathML</a>

(3.2)

Clearly, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/120/mathml/M293','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/120/mathml/M293">View MathML</a> for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/120/mathml/M294','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/120/mathml/M294">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/120/mathml/M7','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/120/mathml/M7">View MathML</a>. From Lemma 2.6, we get that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/120/mathml/M296','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/120/mathml/M296">View MathML</a>

This together with Lemma 2.4 implies <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/120/mathml/M297','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/120/mathml/M297">View MathML</a>. By using the additivity-excision property of the fixed point index [16], we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/120/mathml/M298','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/120/mathml/M298">View MathML</a>

Therefore, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/120/mathml/M175','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/120/mathml/M175">View MathML</a> has a fixed point in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/120/mathml/M279','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/120/mathml/M279">View MathML</a>. Consequently, (1.1) has a positive solution for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/120/mathml/M28','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/120/mathml/M28">View MathML</a>.

(iii) Since ϕ is a bounded operator, multiplying (1.3) by <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/120/mathml/M302','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/120/mathml/M302">View MathML</a> and integrating from 0 to <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/120/mathml/M203','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/120/mathml/M203">View MathML</a>, we get that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/120/mathml/M304','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/120/mathml/M304">View MathML</a>

Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/120/mathml/M305','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/120/mathml/M305">View MathML</a> be the solution of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/120/mathml/M306','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/120/mathml/M306">View MathML</a>. Then there exists <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/120/mathml/M307','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/120/mathml/M307">View MathML</a> such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/120/mathml/M308','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/120/mathml/M308">View MathML</a>

(3.3)

with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/120/mathml/M309','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/120/mathml/M309">View MathML</a>.

Choose two numbers <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/120/mathml/M310','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/120/mathml/M310">View MathML</a> satisfying

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/120/mathml/M311','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/120/mathml/M311">View MathML</a>

By Lemma 2.8, there exist

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/120/mathml/M312','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/120/mathml/M312">View MathML</a>

and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/120/mathml/M313','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/120/mathml/M313">View MathML</a>

such that for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/120/mathml/M25','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/120/mathml/M25">View MathML</a>, we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/120/mathml/M315','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/120/mathml/M315">View MathML</a>

This together with Lemma 2.4 implies <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/120/mathml/M316','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/120/mathml/M316">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/120/mathml/M317','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/120/mathml/M317">View MathML</a>.

Since <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/120/mathml/M23','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/120/mathml/M23">View MathML</a>, from the proof of the case (i), it follows that we can choose <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/120/mathml/M319','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/120/mathml/M319">View MathML</a> such that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/120/mathml/M320','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/120/mathml/M320">View MathML</a>. Subsequently,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/120/mathml/M321','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/120/mathml/M321">View MathML</a>

On the other hand, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/120/mathml/M43','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/120/mathml/M43">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/120/mathml/M323','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/120/mathml/M323">View MathML</a> and (3.3) together with Lemma 2.7 implies

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/120/mathml/M324','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/120/mathml/M324">View MathML</a>

That is, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/120/mathml/M325','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/120/mathml/M325">View MathML</a>. Subsequently,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/120/mathml/M326','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/120/mathml/M326">View MathML</a>

Therefore, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/120/mathml/M175','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/120/mathml/M175">View MathML</a> has two fixed points <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/120/mathml/M328','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/120/mathml/M328">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/120/mathml/M329','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/120/mathml/M329">View MathML</a> such that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/120/mathml/M330','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/120/mathml/M330">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/120/mathml/M331','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/120/mathml/M331">View MathML</a>. These are the desired distinct positive solutions of (1.1) for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/120/mathml/M25','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/120/mathml/M25">View MathML</a> satisfying

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/120/mathml/M333','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/120/mathml/M333">View MathML</a>

(3.4)

(iv) Choose two numbers <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/120/mathml/M310','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/120/mathml/M310">View MathML</a> satisfying

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/120/mathml/M335','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/120/mathml/M335">View MathML</a>

By Lemma 2.7, there exists <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/120/mathml/M336','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/120/mathml/M336">View MathML</a> such that for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/120/mathml/M28','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/120/mathml/M28">View MathML</a>, we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/120/mathml/M338','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/120/mathml/M338">View MathML</a>

That is, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/120/mathml/M339','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/120/mathml/M339">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/120/mathml/M340','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/120/mathml/M340">View MathML</a>.

Since <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/120/mathml/M26','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/120/mathml/M26">View MathML</a>, from the proof of the case (ii), choose <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/120/mathml/M319','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/120/mathml/M319">View MathML</a> such that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/120/mathml/M343','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/120/mathml/M343">View MathML</a>. Consequently,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/120/mathml/M344','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/120/mathml/M344">View MathML</a>

On the other hand, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/120/mathml/M39','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/120/mathml/M39">View MathML</a> and (3.3) together with Lemma 2.8 implies that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/120/mathml/M346','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/120/mathml/M346">View MathML</a>

Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/120/mathml/M347','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/120/mathml/M347">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/120/mathml/M348','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/120/mathml/M348">View MathML</a>. Then, for any <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/120/mathml/M25','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/120/mathml/M25">View MathML</a>, we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/120/mathml/M350','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/120/mathml/M350">View MathML</a>

That is, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/120/mathml/M351','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/120/mathml/M351">View MathML</a>. Subsequently,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/120/mathml/M352','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/120/mathml/M352">View MathML</a>

Therefore, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/120/mathml/M175','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/120/mathml/M175">View MathML</a> has two fixed points <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/120/mathml/M328','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/120/mathml/M328">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/120/mathml/M329','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/120/mathml/M329">View MathML</a> such that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/120/mathml/M356','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/120/mathml/M356">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/120/mathml/M331','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/120/mathml/M331">View MathML</a>. These are the desired distinct positive solutions of (1.1) for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/120/mathml/M25','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/120/mathml/M25">View MathML</a> satisfying (3.4). □

Proof of Corollary 1.2 It is easy to show by the result of Theorem 1.1(i) and (ii). □

Example 3.1 Let us consider the following problem:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/120/mathml/M359','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/120/mathml/M359">View MathML</a>

(3.5)

Obviously, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/120/mathml/M360','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/120/mathml/M360">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/120/mathml/M361','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/120/mathml/M361">View MathML</a>. From Theorem 1.1(iv), there exists <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/120/mathml/M24','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/120/mathml/M24">View MathML</a> such that (3.5) has at least two positive solutions for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/120/mathml/M25','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/120/mathml/M25">View MathML</a>.

Example 3.2 Let us consider the following problem:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/120/mathml/M364','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/120/mathml/M364">View MathML</a>

(3.6)

Obviously, we divided the discussion into two cases as follows.

Case 1. <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/120/mathml/M365','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/120/mathml/M365">View MathML</a>

In this case, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/120/mathml/M366','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/120/mathml/M366">View MathML</a> is convex, and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/120/mathml/M367','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/120/mathml/M367">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/120/mathml/M368','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/120/mathml/M368">View MathML</a>. From Corollary 1.2(a), there exists <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/120/mathml/M24','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/120/mathml/M24">View MathML</a> such that (3.6) has at least one positive solution for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/120/mathml/M25','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/120/mathml/M25">View MathML</a>.

Case 2. <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/120/mathml/M371','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/120/mathml/M371">View MathML</a>

In this case, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/120/mathml/M366','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/120/mathml/M366">View MathML</a> is concave, and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/120/mathml/M373','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/120/mathml/M373">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/120/mathml/M368','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/120/mathml/M368">View MathML</a>. From Theorem 1.1(iv), there exists <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/120/mathml/M24','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/120/mathml/M24">View MathML</a> such that (3.6) has at least two positive solutions for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/120/mathml/M25','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/120/mathml/M25">View MathML</a>.

Remark 3.1 It is worth to point out that our results only partly generalize the results of Habet and Omari [9] and Pan [12], since <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/120/mathml/M47','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/120/mathml/M47">View MathML</a> is more general than <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/120/mathml/M378','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/120/mathml/M378">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/120/mathml/M379','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/120/mathml/M379">View MathML</a> and due to the limitation of the fixed point index method.

Remark 3.2 Since <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/120/mathml/M47','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/120/mathml/M47">View MathML</a> in (1.1) is autonomous, it is not difficult to show that the solutions of (1.1) are symmetric around <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/120/mathml/M381','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/120/mathml/M381">View MathML</a>, see [2,12]. It follows that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/120/mathml/M382','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/120/mathml/M382">View MathML</a> in the proof of Theorem 1.1, and we can set <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/120/mathml/M383','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/120/mathml/M383">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/120/mathml/M131','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/120/mathml/M131">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/120/mathml/M385','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/120/mathml/M385">View MathML</a>.

Competing interests

The authors declare that they have no competing interests regarding the publication of this paper.

Authors’ contributions

YL and RM completed the main study, carried out the results of this article and drafted the paper. HG checked the proofs and verified the calculation. All the authors read and approved the final manuscript.

Acknowledgements

The authors are very grateful to the anonymous referee for his valuable suggestions. This work was supported by the NSFC (No. 11361054, No. 11201378), SRFDP (No. 20126203110004), Gansu provincial National Science Foundation of China (No. 1208RJZA258).

References

  1. Atkinson, FV, Peletier, LA, Serrin, J: Ground states for the prescribed mean curvature equation: the supercritical case. Nonlinear Diffusion Equations and Their Equilibrium States, I, pp. 51–74. Springer, New York (1988)

  2. Cano-Casanova, S, López-Gómez, J, Takimoto, K: A quasilinear parabolic perturbation of the linear heat equation. J. Differ. Equ.. 252, 323–343 (2012). Publisher Full Text OpenURL

  3. Coffman, C, Ziemer, W: A prescribed mean curvature problem on domains without radial symmetry. SIAM J. Math. Anal.. 22(4), 982–990 (1991). Publisher Full Text OpenURL

  4. Pan, H, Xing, R: Sub- and supersolution methods for prescribed mean curvature equations with Dirichlet boundary conditions. J. Differ. Equ.. 254, 1464–1499 (2013). Publisher Full Text OpenURL

  5. Bereanu, C, Mawhin, J: Boundary-value problems with non-surjective ϕ-Laplacian and one-sided bounded nonlinearity. Adv. Differ. Equ.. 11(1), 35–60 (2006)

  6. Bonheure, D, Habets, P, Obersnel, F, Omari, P: Classical and non-classical solutions of a prescribed curvature equation. J. Differ. Equ.. 243, 208–237 (2007). Publisher Full Text OpenURL

  7. Hai, DD, Schmitt, K, Shivaji, R: Positive solutions of quasilinear boundary value problems. J. Math. Anal. Appl.. 217, 672–686 (1998). Publisher Full Text OpenURL

  8. Hai, DD, Schmitt, K, Shivaji, R: On the number of solutions of boundary value problems involving the p-Laplacian. Electron. J. Differ. Equ.. 1, 1–9 (1996)

  9. Habets, P, Omari, P: Multiple positive solutions of a one-dimensional prescribed mean curvature problem. Commun. Contemp. Math.. 9(5), 701–730 (2007). Publisher Full Text OpenURL

  10. Kusahara, T, Usami, H: A barrier method for quasilinear ordinary differential equations of the curvature type. Czechoslov. Math. J.. 50(125), 185–196 (2000)

  11. Manásevich, R, Mawhin, J: Boundary value problems for nonlinear perturbations of vector p-Laplacian-like operators. J. Korean Math. Soc.. 37(5), 665–685 (2000)

  12. Pan, H: One-dimensional prescribed mean curvature equation with exponential nonlinearity. Nonlinear Anal.. 70, 999–1010 (2009). Publisher Full Text OpenURL

  13. Pan, H, Xing, R: Time maps and exact multiplicity results for one-dimensional prescribed mean curvature equations. Nonlinear Anal.. 74, 1234–1260 (2011). Publisher Full Text OpenURL

  14. Wang, H: On the number of positive solutions of nonlinear systems. J. Math. Anal. Appl.. 281, 287–306 (2003). Publisher Full Text OpenURL

  15. Wang, H: On the existence of positive solutions for semilinear elliptic equations in the annulus. J. Differ. Equ.. 109, 1–7 (1994). Publisher Full Text OpenURL

  16. Deimling, K: Nonlinear Functional Analysis, Springer, Berlin (1985)