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The solvability of nonhomogeneous boundary value problems with ϕ-Laplacian operator

Sha-Sha Chen1 and Zhi-Hong Ma2*

Author Affiliations

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

2 Basic Science Department, Tianjin Agricultural University, Tianjin, 300384, P.R. China

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Boundary Value Problems 2014, 2014:82  doi:10.1186/1687-2770-2014-82

The electronic version of this article is the complete one and can be found online at: http://www.boundaryvalueproblems.com/content/2014/1/82


Received:13 October 2013
Accepted:1 April 2014
Published:10 April 2014

© 2014 Chen and Ma; 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

We treat the nonhomogeneous boundary value problems with ϕ-Laplacian operator <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/82/mathml/M1','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/82/mathml/M1">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/82/mathml/M2','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/82/mathml/M2">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/82/mathml/M3','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/82/mathml/M3">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/82/mathml/M4','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/82/mathml/M4">View MathML</a>, where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/82/mathml/M5','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/82/mathml/M5">View MathML</a> (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/82/mathml/M6','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/82/mathml/M6">View MathML</a>) is an increasing homeomorphism such that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/82/mathml/M7','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/82/mathml/M7">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/82/mathml/M8','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/82/mathml/M8">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/82/mathml/M9','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/82/mathml/M9">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/82/mathml/M10','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/82/mathml/M10">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/82/mathml/M11','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/82/mathml/M11">View MathML</a>, and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/82/mathml/M12','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/82/mathml/M12">View MathML</a> is continuous. We will show that even if some of the τ and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/82/mathml/M13','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/82/mathml/M13">View MathML</a> are negative, the boundary value problem with singular ϕ-Laplacian operator is always solvable, and the problem with a bounded ϕ-Laplacian operator has at least one positive solution.

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

1 Introduction

We are concerned with the nonhomogeneous boundary value problem with ϕ-Laplacian operator

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/82/mathml/M14','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/82/mathml/M14">View MathML</a>

(1.1)

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/82/mathml/M5','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/82/mathml/M5">View MathML</a> (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/82/mathml/M6','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/82/mathml/M6">View MathML</a>) is an increasing homeomorphism such that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/82/mathml/M7','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/82/mathml/M7">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/82/mathml/M8','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/82/mathml/M8">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/82/mathml/M19','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/82/mathml/M19">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/82/mathml/M10','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/82/mathml/M10">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/82/mathml/M11','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/82/mathml/M11">View MathML</a>, and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/82/mathml/M12','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/82/mathml/M12">View MathML</a> is continuous.

According to the related literature [1-5], a ϕ-Laplacian operator is said to be singular when the domain of ϕ is finite (i.e., <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/82/mathml/M23','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/82/mathml/M23">View MathML</a>), on the contrary the operator is called regular. On the other hand, we say that ϕ is bounded if its range is finite (i.e., <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/82/mathml/M24','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/82/mathml/M24">View MathML</a>) and unbounded in other case. There are three paradigmatic models in this context:

(1) <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/82/mathml/M25','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/82/mathml/M25">View MathML</a> (regular unbounded): The p-Laplacian operator

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/82/mathml/M26','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/82/mathml/M26">View MathML</a>

(2) <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/82/mathml/M23','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/82/mathml/M23">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/82/mathml/M28','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/82/mathml/M28">View MathML</a> (singular unbounded): The relativistic operator

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/82/mathml/M29','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/82/mathml/M29">View MathML</a>

(3) <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/82/mathml/M30','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/82/mathml/M30">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/82/mathml/M31','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/82/mathml/M31">View MathML</a> (regular bounded): The one dimensional mean curvature operator

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/82/mathml/M32','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/82/mathml/M32">View MathML</a>

The study of the ϕ-Laplacian equations is a classical topic that has attracted the attention of many experts because of its interest in applications. Since 2004, in a number of papers, Bereanu and Mawhin have considered such problems with Dirichlet, Neumann or periodic boundary conditions (see, for example, [1-3] and the references therein). In these papers, the various boundary value problems are reduced to the search for fixed point of some nonlinear operators defined on Banach spaces. In particular, they have also considered some boundary value problems with nonhomogeneous boundary conditions, and they obtained the existence of solutions by the use of the Schauder fixed point theorem (see [2,3]). Recently, Torres [4] proved the existence of a solution of a forced Liénard differential equation with ϕ-Laplacian by means of Schauder fixed point theorem.

We note here that many nonlinear differential problems require the search of positive, meaningful, solutions. The existence of positive solutions for ordinary differential equations and p-Laplacian equations have been studied by several authors and many interesting results has been obtained (only to mention some of them; see [5-7], and the references therein). If the coefficient occurring in the boundary conditions takes a negative value, then the existence of a positive solution for a BVP with ϕ-Laplacian operator is less considered because it is sometimes difficult to construct a corresponding cone for applying the fixed point theorem.

The purpose of this paper is to study the nonhomogeneous boundary value problem (1.1) with ϕ-Laplacian operator even in the case where some of the τ and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/82/mathml/M13','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/82/mathml/M13">View MathML</a> are negative. Firstly, we show that the problem with singular ϕ-Laplacian operator is always solvable by the use of Schauder fixed point theorem. Secondly, we prove that the problem with bounded ϕ-Laplacian operator has at least one positive solution by means of a change of variable and the Krasnosel’skii fixed point theorem. As we will see, the interesting points of the paper are the following:

(1) Some of the τ and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/82/mathml/M13','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/82/mathml/M13">View MathML</a> coefficients in (1.1) are allowed to take a negative value.

(2) In order to obtain a positive solution of the problem (1.1), we make a change of variable and generate two first-order differential equations and the corresponding nonlinear operator B. The new method can be used for the differential domains and ranges of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/82/mathml/M35','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/82/mathml/M35">View MathML</a> and give an a priori estimate of the solution.

This paper is organized as follows. In Section 2, we give some preliminaries and lemmas. In Section 3, we show some theorems on the existence of (positive) solutions of the differential equations (1.1). Moreover, an example is given to illustrate our results.

2 Preliminaries and lemmas

Firstly, we consider the following ϕ-Laplacian differential equation:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/82/mathml/M36','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/82/mathml/M36">View MathML</a>

(2.1)

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/82/mathml/M37','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/82/mathml/M37">View MathML</a>.

Next we make the change of variable

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/82/mathml/M38','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/82/mathml/M38">View MathML</a>

(2.2)

i.e.,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/82/mathml/M39','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/82/mathml/M39">View MathML</a>

(2.3)

Together with (2.1), we get the following two first-order differential equations:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/82/mathml/M40','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/82/mathml/M40">View MathML</a>

(2.4)

and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/82/mathml/M41','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/82/mathml/M41">View MathML</a>

(2.5)

Integration in (2.4) yields

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/82/mathml/M42','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/82/mathml/M42">View MathML</a>

(2.6)

Then we get

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/82/mathml/M43','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/82/mathml/M43">View MathML</a>

Thus, together with the boundary condition in (2.5), we obtain

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/82/mathml/M44','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/82/mathml/M44">View MathML</a>

where we denote <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/82/mathml/M45','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/82/mathml/M45">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/82/mathml/M46','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/82/mathml/M46">View MathML</a>. Thus (2.5) can be rewritten as follows:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/82/mathml/M47','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/82/mathml/M47">View MathML</a>

Integration leads to

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/82/mathml/M48','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/82/mathml/M48">View MathML</a>

Now, we define two nonlinear operators B and S, respectively, by

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/82/mathml/M49','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/82/mathml/M49">View MathML</a>

(2.7)

and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/82/mathml/M50','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/82/mathml/M50">View MathML</a>

(2.8)

Remark 2.1 From the deduction of (2.2)-(2.7), we find that if <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/82/mathml/M51','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/82/mathml/M51">View MathML</a> is well defined and x is a fixed point of the nonlinear operator B, then <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/82/mathml/M52','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/82/mathml/M52">View MathML</a> in (2.6) is a solution of the problem (1.1).

To such a continuous function f, we associate its Nemytskii operator <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/82/mathml/M53','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/82/mathml/M53">View MathML</a> defined by

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/82/mathml/M54','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/82/mathml/M54">View MathML</a>

(2.9)

It is easy to verify that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/82/mathml/M55','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/82/mathml/M55">View MathML</a> is continuous and takes bounded sets into bounded sets.

Lemma 2.1 (See [8])

LetXbe a Banach space and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/82/mathml/M56','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/82/mathml/M56">View MathML</a>a cone. Suppose that<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/82/mathml/M57','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/82/mathml/M57">View MathML</a>and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/82/mathml/M58','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/82/mathml/M58">View MathML</a>are bounded open sets contained inXsuch that<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/82/mathml/M59','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/82/mathml/M59">View MathML</a>and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/82/mathml/M60','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/82/mathml/M60">View MathML</a>. Suppose further that<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/82/mathml/M61','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/82/mathml/M61">View MathML</a>is a completely continuous operator. If either

(i) <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/82/mathml/M62','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/82/mathml/M62">View MathML</a>for<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/82/mathml/M63','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/82/mathml/M63">View MathML</a>and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/82/mathml/M64','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/82/mathml/M64">View MathML</a>for<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/82/mathml/M65','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/82/mathml/M65">View MathML</a>, or

(ii) <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/82/mathml/M66','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/82/mathml/M66">View MathML</a>for<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/82/mathml/M63','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/82/mathml/M63">View MathML</a>and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/82/mathml/M68','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/82/mathml/M68">View MathML</a>for<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/82/mathml/M65','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/82/mathml/M65">View MathML</a>,

thenShas at least one fixed point in<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/82/mathml/M70','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/82/mathml/M70">View MathML</a>.

3 The main result

Let X be the Banach space <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/82/mathml/M71','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/82/mathml/M71">View MathML</a> with the maximum norm <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/82/mathml/M72','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/82/mathml/M72">View MathML</a>. Define a cone by

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/82/mathml/M73','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/82/mathml/M73">View MathML</a>

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

(i) The nonlinearity <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/82/mathml/M74','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/82/mathml/M74">View MathML</a> satisfies

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/82/mathml/M75','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/82/mathml/M75">View MathML</a>

for any <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/82/mathml/M76','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/82/mathml/M76">View MathML</a>;

(ii) <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/82/mathml/M77','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/82/mathml/M77">View MathML</a>;

(iii) There exists a positive constant r such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/82/mathml/M78','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/82/mathml/M78">View MathML</a>

(3.1)

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/82/mathml/M79','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/82/mathml/M79">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/82/mathml/M80','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/82/mathml/M80">View MathML</a>;

(iv) <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/82/mathml/M81','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/82/mathml/M81">View MathML</a>.

Remark 3.1

(1) If <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/82/mathml/M82','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/82/mathml/M82">View MathML</a>, for all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/82/mathml/M83','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/82/mathml/M83">View MathML</a>, then the condition (iv) clearly holds.

(2) If <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/82/mathml/M84','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/82/mathml/M84">View MathML</a>, instead of conditions (i) and (ii), we assume that for any <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/82/mathml/M76','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/82/mathml/M76">View MathML</a>, the following inequalities hold:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/82/mathml/M86','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/82/mathml/M86">View MathML</a>

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/82/mathml/M87','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/82/mathml/M87">View MathML</a> are two constants. Also, the condition (3.1) is replaced by

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/82/mathml/M88','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/82/mathml/M88">View MathML</a>

Case I. Singular ϕ-Laplacian operator: <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/82/mathml/M89','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/82/mathml/M89">View MathML</a> (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/82/mathml/M90','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/82/mathml/M90">View MathML</a>).

Theorem 3.1Iffis continuous, then the problem (1.1) has at least one solution.

Proof Define a set by

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/82/mathml/M91','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/82/mathml/M91">View MathML</a>

From the definition of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/82/mathml/M92','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/82/mathml/M92">View MathML</a>, (2.2), (2.7), and (2.8), we get, for any <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/82/mathml/M93','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/82/mathml/M93">View MathML</a>,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/82/mathml/M94','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/82/mathml/M94">View MathML</a>

and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/82/mathml/M95','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/82/mathml/M95">View MathML</a>

Thus <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/82/mathml/M96','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/82/mathml/M96">View MathML</a>. Utilizing (2.9) and Arzela-Ascoli theorem, it is easy to verify that the nonlinear operator S is a completely continuous operator. Therefore, the nonlinear operator S has at least one fixed point by Schauder fixed point theorem.

From the definition of S in (2.8), we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/82/mathml/M97','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/82/mathml/M97">View MathML</a>

i.e.,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/82/mathml/M98','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/82/mathml/M98">View MathML</a>

Then we obtain

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/82/mathml/M99','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/82/mathml/M99">View MathML</a>

and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/82/mathml/M100','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/82/mathml/M100">View MathML</a>

Consequently, we conclude that the fixed point of S is a solution of the problem (1.1). □

Remark 3.2 Theorem 3.1 shows that if ϕ is singular (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/82/mathml/M101','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/82/mathml/M101">View MathML</a>) and f is continuous on <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/82/mathml/M102','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/82/mathml/M102">View MathML</a>, then the problem (1.1) is always solvable.

Case II. Bounded ϕ-Laplacian operator: <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/82/mathml/M103','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/82/mathml/M103">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/82/mathml/M104','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/82/mathml/M104">View MathML</a>.

Theorem 3.2If the conditions (i)-(iv) hold, then the nonlinear operatorBdefined by (2.7) has at least one fixed point. Further, the problem (1.1) has at least one positive solution.

Proof Define a set by

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/82/mathml/M105','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/82/mathml/M105">View MathML</a>

From the conditions (ii) and (iv), we obtain, for any <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/82/mathml/M106','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/82/mathml/M106">View MathML</a>,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/82/mathml/M107','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/82/mathml/M107">View MathML</a>

Clearly, the nonlinear operator <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/82/mathml/M108','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/82/mathml/M108">View MathML</a> is well defined.

The condition (iii) implies that we find a constant <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/82/mathml/M109','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/82/mathml/M109">View MathML</a> such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/82/mathml/M110','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/82/mathml/M110">View MathML</a>

Define a set by

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/82/mathml/M111','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/82/mathml/M111">View MathML</a>

In virtue of the increasing property of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/82/mathml/M35','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/82/mathml/M35">View MathML</a>, we get, for any <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/82/mathml/M113','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/82/mathml/M113">View MathML</a>,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/82/mathml/M114','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/82/mathml/M114">View MathML</a>

Thus, for any <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/82/mathml/M113','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/82/mathml/M113">View MathML</a>, we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/82/mathml/M116','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/82/mathml/M116">View MathML</a>

We choose a small positive constant <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/82/mathml/M117','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/82/mathml/M117">View MathML</a> such that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/82/mathml/M118','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/82/mathml/M118">View MathML</a> and define <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/82/mathml/M119','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/82/mathml/M119">View MathML</a>. Then for any <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/82/mathml/M120','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/82/mathml/M120">View MathML</a>, we find

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/82/mathml/M121','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/82/mathml/M121">View MathML</a>

Thus, for any <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/82/mathml/M122','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/82/mathml/M122">View MathML</a>, we get

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/82/mathml/M123','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/82/mathml/M123">View MathML</a>

In addition, a standard argument involving the Arzela-Ascoli theorem implies that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/82/mathml/M124','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/82/mathml/M124">View MathML</a> is a completely continuous operator. Therefore, the nonlinear operator B has at least one fixed point by the use of Lemma 2.1. Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/82/mathml/M125','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/82/mathml/M125">View MathML</a> be a fixed point of B, then, from (2.6), we obtain

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/82/mathml/M126','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/82/mathml/M126">View MathML</a>

Consequently, we get from Remark 2.1 that the problem (1.1) has a positive solution <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/82/mathml/M52','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/82/mathml/M52">View MathML</a>. □

Remark 3.3 In order to prove the existence of a positive solution of the problem (1.1), we make a change of variable and introduce a first-order differential equation, and investigate the existence of a fixed point of the corresponding nonlinear operator B. The technique can be used for the different domains and ranges of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/82/mathml/M35','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/82/mathml/M35">View MathML</a> and give an a priori estimate of the solution.

Theorem 3.3If<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/82/mathml/M129','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/82/mathml/M129">View MathML</a>andfsatisfies the condition

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/82/mathml/M130','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/82/mathml/M130">View MathML</a>

for any<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/82/mathml/M76','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/82/mathml/M76">View MathML</a>, then the problem (1.1) has at least one solution.

Proof Adopting a similar technique to (2.2)-(2.6), we define a nonlinear operator by

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/82/mathml/M133','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/82/mathml/M133">View MathML</a>

Then we get, for any <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/82/mathml/M134','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/82/mathml/M134">View MathML</a>,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/82/mathml/M135','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/82/mathml/M135">View MathML</a>

Then the operator <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/82/mathml/M136','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/82/mathml/M136">View MathML</a> has at least one fixed point by the use of Schauder fixed point theorem. Applying expression (2.6), we conclude that the problem (1.1) has at least one solution. □

Remark 3.4 Observe that the solution provided by Theorem 3.3 could be trivial or negative.

Theorem 3.4If the conditions (ii) and (iv) hold andfsatisfies the condition

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/82/mathml/M137','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/82/mathml/M137">View MathML</a>

(3.2)

for any<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/82/mathml/M138','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/82/mathml/M138">View MathML</a>, then problem (1.1) has no solution.

Proof Taking arbitrarily <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/82/mathml/M139','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/82/mathml/M139">View MathML</a>, we get from (3.2)

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/82/mathml/M140','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/82/mathml/M140">View MathML</a>

Thus, it implies that there exists a neighborhood <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/82/mathml/M141','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/82/mathml/M141">View MathML</a> such that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/82/mathml/M142','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/82/mathml/M142">View MathML</a> for any <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/82/mathml/M143','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/82/mathml/M143">View MathML</a>. This implies that the nonlinear operator <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/82/mathml/M144','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/82/mathml/M144">View MathML</a> is not well defined, since the domain of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/82/mathml/M35','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/82/mathml/M35">View MathML</a> is the interval <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/82/mathml/M146','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/82/mathml/M146">View MathML</a> and thus a solution of (1.1) cannot exist. □

Example 3.1 We consider the following nonhomogeneous boundary value problem with ϕ-Laplacian operator:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/82/mathml/M147','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/82/mathml/M147">View MathML</a>

(3.3)

where

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/82/mathml/M148','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/82/mathml/M148">View MathML</a>

It is easy to see that the nonlinearity f satisfies the condition (i), that is,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/82/mathml/M149','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/82/mathml/M149">View MathML</a>

for any <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/82/mathml/M150','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/82/mathml/M150">View MathML</a>. Computation yields

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/82/mathml/M151','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/82/mathml/M151">View MathML</a>

and there exists a positive constant <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/82/mathml/M152','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/82/mathml/M152">View MathML</a> such that the following inequalities hold:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/82/mathml/M153','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/82/mathml/M153">View MathML</a>

and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/82/mathml/M154','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/82/mathml/M154">View MathML</a>

Then conditions (ii)-(iv) also hold. Therefore, we find from Theorem 3.2 that the differential equation (3.3) has at least one positive solution.

Competing interests

The authors declare that they have no competing interests.

Authors’ contributions

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

Acknowledgements

The authors would wish to express their appreciations to the anonymous referee for his/her valuable suggestions, which have greatly improved this paper. The work was partially supported by NSFC of China (No. 11201248), K.C. Wong Fund of Ningbo University.

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