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This article is part of the series Jean Mawhin’s Achievements in Nonlinear Analysis.

Open Access Research

Positive solutions of nonhomogeneous boundary value problems for some nonlinear equation with ϕ-Laplacian

Liang-Gen Hu* and Jing Xu

Author Affiliations

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

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Boundary Value Problems 2012, 2012:130  doi:10.1186/1687-2770-2012-130


The electronic version of this article is the complete one and can be found online at: http://www.boundaryvalueproblems.com/content/2012/1/130


Received:10 June 2012
Accepted:23 October 2012
Published:12 November 2012

© 2012 Hu and Xu; licensee Springer

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

Abstract

We will consider the nonhomogeneous ϕ-Laplacian differential equation

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/130/mathml/M1','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/130/mathml/M1">View MathML</a>

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/130/mathml/M2','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/130/mathml/M2">View MathML</a> (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/130/mathml/M3','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/130/mathml/M3">View MathML</a>) is an increasing homeomorphism such that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/130/mathml/M4','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/130/mathml/M4">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/130/mathml/M5','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/130/mathml/M5">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/130/mathml/M6','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/130/mathml/M6">View MathML</a> are continuous, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/130/mathml/M7','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/130/mathml/M7">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/130/mathml/M8','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/130/mathml/M8">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/130/mathml/M9','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/130/mathml/M9">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/130/mathml/M10','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/130/mathml/M10">View MathML</a>. Based on the Krasnosel’skii fixed point theorem, the existence of a positive solution is obtained, even if some of the <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/130/mathml/M11','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/130/mathml/M11">View MathML</a> coefficients are negative. Two examples are also given to illustrate our main results.

Keywords:
nonhomogeneous; ϕ-Laplacian; positive solution; fixed point; negative coefficient

1 Introduction

We are concerned with the ϕ-Laplacian differential equation with the nonhomogeneous Dirichlet-Neumann boundary conditions

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/130/mathml/M12','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/130/mathml/M12">View MathML</a>

(1.1)

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/130/mathml/M2','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/130/mathml/M2">View MathML</a> (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/130/mathml/M14','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/130/mathml/M14">View MathML</a>) is an increasing homeomorphism such that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/130/mathml/M4','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/130/mathml/M4">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/130/mathml/M5','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/130/mathml/M5">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/130/mathml/M6','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/130/mathml/M6">View MathML</a> are continuous (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/130/mathml/M18','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/130/mathml/M18">View MathML</a>), <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/130/mathml/M7','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/130/mathml/M7">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/130/mathml/M9','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/130/mathml/M9">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/130/mathml/M8','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/130/mathml/M8">View MathML</a>, for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/130/mathml/M22','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/130/mathml/M22">View MathML</a>.

Boundary value problems, including the ϕ-Laplacian operator, have received a lot of attention with respect to the existence and multiplicity of solutions. Since 2004, with a number of papers, Bereanu and Mawhin have considered such problems with Dirichlet, Neumann or periodic boundary conditions (see [1-4] and the references therein). In these papers, the various boundary value problems are reduced to the search for fixed points of some nonlinear operators defined on Banach spaces. In particular, they have studied some boundary value problems with nonhomogeneous boundary conditions and obtained the existence of solutions by the use of Schauder’s fixed point theorem (see [3,4]). Recently, Torres [5] has proved the existence of a solution of a forced Liénard differential equation with ϕ-Laplacian by means of Schauder’s fixed point theorem.

However, many nonlinear differential equations need to seek the existence of positive solutions because the positive solutions are very meaningful. The existence of positive solutions for homogeneous and nonhomogeneous boundary value problems have been studied by several authors and many interesting results have been obtained (only to mention some of them, see [6-9], their references and the papers citing them). The problems with negative coefficients for the boundary conditions (see [10-12]) often occur in some heat flow problems, the deflection of a beam, and Floquet theory of the beam equation and have been considered by some experts (see [7,8,10-12]). If the coefficient takes a negative value, then it is sometimes difficult to find an appropriate cone to guarantee the existence of a positive solution of the corresponding differential equation. Comparing with the previous result [2-4], the cone may be smaller.

The purpose of this paper is to establish the criteria of the existence of a positive solution to the problem (1.1) by utilizing the Krasnosel’skii fixed point theorem, even if some of the <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/130/mathml/M11','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/130/mathml/M11">View MathML</a> coefficients are negative. The method of proof is inspired by the ideas exposed in [3-5,7,8]. As we will see, our results are new, and the interesting points of those results are the following two aspects: (i) Some of the <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/130/mathml/M11','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/130/mathml/M11">View MathML</a> coefficients appearing in (1.1) are allowed to take a negative value. (ii) The existence of a positive solution for the class of ϕ-Laplacian differential equations with a nonhomogeneous boundary condition is proved. Notice that the existence of a positive solution for the class of ϕ-Laplacian equations has been less studied in the related literature.

This paper is organized as follows. In Section 2, we give some lemmas, which play an important role in the proof of the main theorem. In Section 3, we obtain the existence of a positive solution to the problem (1.1). Moreover, two examples are also given to illustrate the main results.

2 Preliminaries and lemmas

Let X denote the Banach space <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/130/mathml/M25','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/130/mathml/M25">View MathML</a> of continuous functions endowed with the maximum norm <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/130/mathml/M26','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/130/mathml/M26">View MathML</a>. Define a nonlinear operator <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/130/mathml/M27','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/130/mathml/M27">View MathML</a> by

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/130/mathml/M28','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/130/mathml/M28">View MathML</a>

(2.1)

Deriving on both sides of (2.1) leads to

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/130/mathml/M29','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/130/mathml/M29">View MathML</a>

(2.2)

i.e.,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/130/mathml/M30','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/130/mathml/M30">View MathML</a>

(2.3)

Again, deriving in (2.3) implies

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/130/mathml/M31','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/130/mathml/M31">View MathML</a>

Moreover, from (2.1) and (2.3), we get that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/130/mathml/M32','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/130/mathml/M32">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/130/mathml/M33','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/130/mathml/M33">View MathML</a>. Therefore, the existence of a solution for Eq. (1.1) is equivalent to seeking a fixed point of the nonlinear operator S.

For the sake of convenience, we give the following conditions.

(A) Denote <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/130/mathml/M34','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/130/mathml/M34">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/130/mathml/M35','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/130/mathml/M35">View MathML</a>, and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/130/mathml/M11','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/130/mathml/M11">View MathML</a> satisfies the conditions

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/130/mathml/M37','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/130/mathml/M37">View MathML</a>

and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/130/mathml/M38','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/130/mathml/M38">View MathML</a>

(F) The function <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/130/mathml/M6','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/130/mathml/M6">View MathML</a> is continuous and satisfies <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/130/mathml/M40','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/130/mathml/M40">View MathML</a>, for any <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/130/mathml/M41','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/130/mathml/M41">View MathML</a>, where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/130/mathml/M42','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/130/mathml/M42">View MathML</a> are two constants.

(B) <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/130/mathml/M43','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/130/mathml/M43">View MathML</a> (for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/130/mathml/M44','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/130/mathml/M44">View MathML</a>).

(H) There exists a <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/130/mathml/M45','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/130/mathml/M45">View MathML</a> such that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/130/mathml/M46','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/130/mathml/M46">View MathML</a>, and let the inequality

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/130/mathml/M47','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/130/mathml/M47">View MathML</a>

be true.

For the unboundedϕ-Laplacian (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/130/mathml/M48','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/130/mathml/M48">View MathML</a>), we obtain the following results.

Lemma 1Assume that the conditions (F) and (H) hold, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/130/mathml/M49','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/130/mathml/M49">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/130/mathml/M50','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/130/mathml/M50">View MathML</a>, and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/130/mathml/M51','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/130/mathml/M51">View MathML</a>. Then there exists a constant<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/130/mathml/M52','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/130/mathml/M52">View MathML</a>such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/130/mathml/M53','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/130/mathml/M53">View MathML</a>

Proof From the representation (2.2) and the conditions (F)-(H), we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/130/mathml/M54','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/130/mathml/M54">View MathML</a>

Again since <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/130/mathml/M7','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/130/mathml/M7">View MathML</a>, we get that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/130/mathml/M56','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/130/mathml/M56">View MathML</a>. Therefore, applying the condition (F) leads to

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/130/mathml/M57','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/130/mathml/M57">View MathML</a>

and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/130/mathml/M58','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/130/mathml/M58">View MathML</a>

where

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/130/mathml/M59','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/130/mathml/M59">View MathML</a>

This completes the proof. □

Next, let us define a cone by

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/130/mathml/M60','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/130/mathml/M60">View MathML</a>

The definition of the cone is inspired by the results in [7,8]. To show our main results, the following lemma is essential.

Lemma 2Let the conditions (A), (F), and (H) hold and the nonlinear operatorSbe defined by (2.1). Then<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/130/mathml/M61','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/130/mathml/M61">View MathML</a>.

Proof From the definition of the operator S, we find for any <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/130/mathml/M62','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/130/mathml/M62">View MathML</a> that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/130/mathml/M63','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/130/mathml/M63">View MathML</a>

The conditions (F) and (H) yield

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/130/mathml/M64','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/130/mathml/M64">View MathML</a>

Further, Lemma 1 shows

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/130/mathml/M65','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/130/mathml/M65">View MathML</a>

Consequently, we get that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/130/mathml/M61','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/130/mathml/M61">View MathML</a>. □

Remark 1 If the coefficients <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/130/mathml/M67','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/130/mathml/M67">View MathML</a> are nonnegative, then the conclusion in Lemma 2 also holds without the hypothesis (H).

Lemma 3If<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/130/mathml/M44','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/130/mathml/M44">View MathML</a>and, in addition, the assumptions of Lemma 2 and the condition (B) are satisfied, then the conclusions of Lemma 1 and Lemma 2 hold.

Lemma 4 (See [13])

LetXbe a Banach space and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/130/mathml/M69','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/130/mathml/M69">View MathML</a>be a cone. Suppose that<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/130/mathml/M70','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/130/mathml/M70">View MathML</a>and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/130/mathml/M71','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/130/mathml/M71">View MathML</a>are bounded open sets contained inXsuch that<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/130/mathml/M72','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/130/mathml/M72">View MathML</a>and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/130/mathml/M73','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/130/mathml/M73">View MathML</a>. Suppose further that<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/130/mathml/M74','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/130/mathml/M74">View MathML</a>is a completely continuous operator. If either

(i) <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/130/mathml/M75','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/130/mathml/M75">View MathML</a>for<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/130/mathml/M76','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/130/mathml/M76">View MathML</a>and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/130/mathml/M77','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/130/mathml/M77">View MathML</a>for<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/130/mathml/M78','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/130/mathml/M78">View MathML</a>or

(ii) <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/130/mathml/M77','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/130/mathml/M77">View MathML</a>for<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/130/mathml/M76','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/130/mathml/M76">View MathML</a>and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/130/mathml/M75','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/130/mathml/M75">View MathML</a>for<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/130/mathml/M78','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/130/mathml/M78">View MathML</a>,

thenShas at least one fixed point in<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/130/mathml/M83','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/130/mathml/M83">View MathML</a>.

3 The main result

Theorem 1Assume that the conditions (A), (F), and (H) hold and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/130/mathml/M48','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/130/mathml/M48">View MathML</a>. Then Eq. (1.1) has at least one positive solution.

Proof Lemma 2 shows that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/130/mathml/M85','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/130/mathml/M85">View MathML</a>. In addition, a standard argument involving the Arzela-Ascoli theorem implies that S is a completely continuous operator.

Now, we choose a positive constant <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/130/mathml/M86','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/130/mathml/M86">View MathML</a> such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/130/mathml/M87','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/130/mathml/M87">View MathML</a>

and define <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/130/mathml/M88','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/130/mathml/M88">View MathML</a>. For any <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/130/mathml/M76','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/130/mathml/M76">View MathML</a>, we get from the condition (F) that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/130/mathml/M90','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/130/mathml/M90">View MathML</a>

Thus, for any <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/130/mathml/M76','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/130/mathml/M76">View MathML</a>, we find that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/130/mathml/M92','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/130/mathml/M92">View MathML</a>

(3.1)

From the hypothesis (A), we can let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/130/mathml/M93','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/130/mathml/M93">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/130/mathml/M94','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/130/mathml/M94">View MathML</a>. Next, we choose a positive constant <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/130/mathml/M95','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/130/mathml/M95">View MathML</a> such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/130/mathml/M96','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/130/mathml/M96">View MathML</a>

and define <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/130/mathml/M97','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/130/mathml/M97">View MathML</a>. Clearly, for any <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/130/mathml/M78','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/130/mathml/M78">View MathML</a>, we obtain

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/130/mathml/M99','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/130/mathml/M99">View MathML</a>

Then, for any <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/130/mathml/M78','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/130/mathml/M78">View MathML</a>, it implies that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/130/mathml/M101','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/130/mathml/M101">View MathML</a>

(3.2)

Based on Lemma 4, we get from (3.1) and (3.2) that the operator S has at least one fixed point. Thus, it follows that Eq. (1.1) has at least one positive solution. □

Remark 2 If the coefficients <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/130/mathml/M102','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/130/mathml/M102">View MathML</a> are nonnegative, then the condition (A) is replaced with

(A′) <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/130/mathml/M103','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/130/mathml/M103">View MathML</a>.

Applying the results in Remark 1 and Theorem 1, we get the following result.

Corollary 1Assume that the conditions (A′) and (F) hold and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/130/mathml/M48','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/130/mathml/M48">View MathML</a>. Then Eq. (1.1) has at least one positive solution.

If<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/130/mathml/M2','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/130/mathml/M2">View MathML</a>(<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/130/mathml/M44','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/130/mathml/M44">View MathML</a>), then we have the following result.

Theorem 2Assume that the conditions (A), (F), (B), and (H) hold. Then Eq. (1.1) has at least one positive solution.

Proof Using Lemma 3 and the proof of Theorem 1, we get that the conclusion holds. □

Example 1 Consider the differential equation

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/130/mathml/M107','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/130/mathml/M107">View MathML</a>

(3.3)

subjected to the boundary conditions

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/130/mathml/M108','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/130/mathml/M108">View MathML</a>

(3.4)

Clearly, we find

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/130/mathml/M109','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/130/mathml/M109">View MathML</a>

Computing yields

Therefore, we conclude from Theorem 1 that Eq. (3.3)-(3.4) has at least one positive solution.

Example 2 Consider the differential equation

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/130/mathml/M111','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/130/mathml/M111">View MathML</a>

(3.5)

subjected to the boundary conditions

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/130/mathml/M112','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/130/mathml/M112">View MathML</a>

(3.6)

Obviously, we obtain

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/130/mathml/M113','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/130/mathml/M113">View MathML</a>

It is easy to verify that the conditions (B), (F), and (H) hold. Consequently, we get from Theorem 2 that the equation (3.5)-(3.6) has at least one positive solution.

Competing interests

The authors declare that they have no competing interests.

Authors’ contributions

The authors typed, read and approved the final manuscript.

Acknowledgements

The authors would like to thank the anonymous referees very much for helpful comments and suggestions which led to the improvement of the presentation and quality of the work. The work was supported partly by NSFC of Tianyuan Youth Foundation (No.11126125), K.C. Wong Magna Fund of Ningbo University and Ningbo Natural Science Foundation (2012A610031).

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