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This article is part of the series Recent Advances in Operator Equations, Boundary Value Problems, Fixed Point Theory and Applications, and General Inequalities.

Open Access Research

Existence of three solutions for a nonlocal elliptic system of ( p , q ) -Kirchhoff type

Guang-Sheng Chen1, Hui-Yu Tang1*, De-Quan Zhang2, Yun-Xiu Jiao3 and Hao-Xiang Wang4

Author Affiliations

1 Department of Construction and Information Engineering, Guangxi Modern Vocational Technology College, Hechi, Guangxi, 547000, China

2 Faculty of Science, Guilin University of Aerospace Industry, Guilin, Guangxi, 541004, China

3 Department of Common Courses, Xinxiang Polytechnic College, Xinxiang, Henan, 453006, China

4 School of Electronic Engineering, Xidian University, Xi’an, Shanxi, 710126, China

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Boundary Value Problems 2013, 2013:175  doi:10.1186/1687-2770-2013-175

The electronic version of this article is the complete one and can be found online at: http://www.boundaryvalueproblems.com/content/2013/1/175


Received:1 May 2013
Accepted:10 July 2013
Published:25 July 2013

© 2013 Chen et al.; licensee Springer

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

Abstract

In this paper, we study the solutions of a nonlocal elliptic system of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/175/mathml/M1','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/175/mathml/M1">View MathML</a>-Kirchhoff type on a bounded domain based on the three critical points theorem introduced by Ricceri. Firstly, we establish the existence of three weak solutions under appropriate hypotheses; then, we prove the existence of at least three weak solutions for the nonlocal elliptic system of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/175/mathml/M1','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/175/mathml/M1">View MathML</a>-Kirchhoff type.

Keywords:
<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/175/mathml/M1','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/175/mathml/M1">View MathML</a>-Kirchhoff type system; multiple solutions; three critical points theory

1 Introduction and main results

We consider the boundary problem involving <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/175/mathml/M1','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/175/mathml/M1">View MathML</a>-Kirchhoff

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/175/mathml/M6','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/175/mathml/M6">View MathML</a>

(1.1)

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/175/mathml/M7','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/175/mathml/M7">View MathML</a> (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/175/mathml/M8','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/175/mathml/M8">View MathML</a>) is a bounded smooth domain, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/175/mathml/M9','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/175/mathml/M9">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/175/mathml/M10','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/175/mathml/M10">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/175/mathml/M11','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/175/mathml/M11">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/175/mathml/M12','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/175/mathml/M12">View MathML</a> is the p-Laplacian operator <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/175/mathml/M13','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/175/mathml/M13">View MathML</a>. <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/175/mathml/M14','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/175/mathml/M14">View MathML</a> are functions such that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/175/mathml/M15','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/175/mathml/M15">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/175/mathml/M16','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/175/mathml/M16">View MathML</a> are measurable in Ω for all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/175/mathml/M17','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/175/mathml/M17">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/175/mathml/M18','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/175/mathml/M18">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/175/mathml/M19','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/175/mathml/M19">View MathML</a> are continuously differentiable in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/175/mathml/M20','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/175/mathml/M20">View MathML</a> for a.e. <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/175/mathml/M21','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/175/mathml/M21">View MathML</a>. <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/175/mathml/M22','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/175/mathml/M22">View MathML</a> is the partial derivative of F with respect to i, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/175/mathml/M23','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/175/mathml/M23">View MathML</a>, so is <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/175/mathml/M24','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/175/mathml/M24">View MathML</a>. <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/175/mathml/M25','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/175/mathml/M25">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/175/mathml/M26','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/175/mathml/M26">View MathML</a>, are continuous functions which satisfy the following bounded conditions.

(M) There exist two positive constants <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/175/mathml/M27','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/175/mathml/M27">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/175/mathml/M28','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/175/mathml/M28">View MathML</a> such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/175/mathml/M29','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/175/mathml/M29">View MathML</a>

(1.2)

Here and in the sequel, X denotes the Cartesian product of two Sobolev spaces <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/175/mathml/M30','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/175/mathml/M30">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/175/mathml/M31','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/175/mathml/M31">View MathML</a>, i.e., <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/175/mathml/M32','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/175/mathml/M32">View MathML</a>. The reflexive real Banach space X is endowed with the norm

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/175/mathml/M33','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/175/mathml/M33">View MathML</a>

Since <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/175/mathml/M10','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/175/mathml/M10">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/175/mathml/M11','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/175/mathml/M11">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/175/mathml/M30','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/175/mathml/M30">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/175/mathml/M31','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/175/mathml/M31">View MathML</a> are compactly embedded in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/175/mathml/M38','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/175/mathml/M38">View MathML</a>. Let

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/175/mathml/M39','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/175/mathml/M39">View MathML</a>

(1.3)

then one has <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/175/mathml/M40','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/175/mathml/M40">View MathML</a>. Furthermore, it is known from [1] that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/175/mathml/M41','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/175/mathml/M41">View MathML</a>

where Γ is the gamma function and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/175/mathml/M42','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/175/mathml/M42">View MathML</a> is the Lebesgue measure of Ω. As usual, by a weak solution of system (1.1), we mean any <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/175/mathml/M43','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/175/mathml/M43">View MathML</a> such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/175/mathml/M44','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/175/mathml/M44">View MathML</a>

(1.4)

for all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/175/mathml/M45','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/175/mathml/M45">View MathML</a>.

System (1.1) is related to the stationary version of a model established by Kirchhoff [2]. More precisely, Kirchhoff proposed the following model:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/175/mathml/M46','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/175/mathml/M46">View MathML</a>

(1.5)

which extends D’Alembert’s wave equation with free vibrations of elastic strings, where ρ denotes the mass density, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/175/mathml/M47','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/175/mathml/M47">View MathML</a> denotes the initial tension, h denotes the area of the cross-section, E denotes the Young modulus of the material, and L denotes the length of the string. Kirchhoff’s model considers the changes in length of the string produced during the vibrations.

Later, (1.1) was developed into the following form:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/175/mathml/M48','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/175/mathml/M48">View MathML</a>

(1.6)

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/175/mathml/M49','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/175/mathml/M49">View MathML</a> is a given function. After that, many authors studied the following problem:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/175/mathml/M50','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/175/mathml/M50">View MathML</a>

(1.7)

which is the stationary counterpart of (1.6). By applying variational methods and other techniques, many results of (1.7) were obtained, the reader is referred to [3-13] and the references therein. In particular, Alves et al. [[3], Theorem 4] supposed that M satisfies bounded condition (M) and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/175/mathml/M51','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/175/mathml/M51">View MathML</a> satisfies the condition

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/175/mathml/M52','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/175/mathml/M52">View MathML</a>

(AR)

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/175/mathml/M53','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/175/mathml/M53">View MathML</a>; one positive solution for (1.7) was given.

In [14], using Ekeland’s variational principle, Corrêa and Nascimento proved the existence of a weak solution for the boundary problem associated with the nonlocal elliptic system of p-Kirchhoff type

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/175/mathml/M54','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/175/mathml/M54">View MathML</a>

(1.8)

where η is the unit exterior vector on Ω, and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/175/mathml/M55','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/175/mathml/M55">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/175/mathml/M56','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/175/mathml/M56">View MathML</a> (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/175/mathml/M26','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/175/mathml/M26">View MathML</a>), f, g satisfy suitable assumptions.

In [15], when <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/175/mathml/M58','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/175/mathml/M58">View MathML</a> in (1.1), Bitao Cheng et al. studied the existence of two solutions and three solutions of the following nonlocal elliptic system:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/175/mathml/M59','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/175/mathml/M59">View MathML</a>

(1.9)

In this paper, our objective is to prove the existence of three solutions of problem (1.1) by applying the three critical points theorem established by Ricceri [16]. Our result, under appropriate assumptions, ensures the existence of an open interval <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/175/mathml/M60','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/175/mathml/M60">View MathML</a> and a positive real number ρ such that, for each <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/175/mathml/M61','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/175/mathml/M61">View MathML</a>, problem (1.1) admits at least three weak solutions whose norms in X are less than ρ. The purpose of the present paper is to generalize the main result of [15].

Now, for every <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/175/mathml/M62','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/175/mathml/M62">View MathML</a> and choosing <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/175/mathml/M63','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/175/mathml/M63">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/175/mathml/M64','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/175/mathml/M64">View MathML</a> with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/175/mathml/M65','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/175/mathml/M65">View MathML</a>, such that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/175/mathml/M66','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/175/mathml/M66">View MathML</a>, where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/175/mathml/M67','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/175/mathml/M67">View MathML</a>, put

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/175/mathml/M68','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/175/mathml/M68">View MathML</a>

(1.10)

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/175/mathml/M69','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/175/mathml/M69">View MathML</a>

(1.11)

Moreover, let a, c be positive constants and define

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/175/mathml/M70','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/175/mathml/M70">View MathML</a>

Our main result is stated as follows.

Theorem 1.1Assume that<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/175/mathml/M71','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/175/mathml/M71">View MathML</a>such that<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/175/mathml/M72','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/175/mathml/M72">View MathML</a>, and suppose that there exist four positive constantsa, b, γandβwith<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/175/mathml/M73','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/175/mathml/M73">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/175/mathml/M74','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/175/mathml/M74">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/175/mathml/M75','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/175/mathml/M75">View MathML</a>, and a function<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/175/mathml/M76','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/175/mathml/M76">View MathML</a>such that

(j1) <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/175/mathml/M77','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/175/mathml/M77">View MathML</a>for a.e. <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/175/mathml/M78','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/175/mathml/M78">View MathML</a>and all<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/175/mathml/M79','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/175/mathml/M79">View MathML</a>;

(j2) <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/175/mathml/M80','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/175/mathml/M80">View MathML</a>;

(j3) <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/175/mathml/M81','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/175/mathml/M81">View MathML</a>for a.e. <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/175/mathml/M82','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/175/mathml/M82">View MathML</a>and all<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/175/mathml/M83','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/175/mathml/M83">View MathML</a>;

(j4) <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/175/mathml/M84','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/175/mathml/M84">View MathML</a>for a.e. <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/175/mathml/M82','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/175/mathml/M82">View MathML</a>.

Then there exist an open interval<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/175/mathml/M86','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/175/mathml/M86">View MathML</a>and a positive real numberρwith the following property: for each<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/175/mathml/M87','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/175/mathml/M87">View MathML</a>and for two Carathéodory functions<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/175/mathml/M88','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/175/mathml/M88">View MathML</a>satisfying

(j5) <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/175/mathml/M89','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/175/mathml/M89">View MathML</a>for all<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/175/mathml/M90','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/175/mathml/M90">View MathML</a>,

there exists<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/175/mathml/M91','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/175/mathml/M91">View MathML</a>such that, for each<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/175/mathml/M92','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/175/mathml/M92">View MathML</a>, problem (1.1) has at least three weak solutions<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/175/mathml/M93','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/175/mathml/M93">View MathML</a> (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/175/mathml/M94','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/175/mathml/M94">View MathML</a>) whose norms<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/175/mathml/M95','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/175/mathml/M95">View MathML</a>are less thanρ.

2 Proof of the main result

First we recall the modified form of Ricceri’s three critical points theorem (Theorem 1 in [16]) and Proposition 3.1 of [17], which is our primary tool in proving our main result.

Theorem 2.1 ([16], Theorem 1)

Suppose that X is a reflexive real Banach space and that<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/175/mathml/M96','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/175/mathml/M96">View MathML</a>is a continuously Gâteaux differentiable and sequentially weakly lower semicontinuous functional whose Gâteaux derivative admits a continuous inverse on<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/175/mathml/M97','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/175/mathml/M97">View MathML</a>, and that Φ is bounded on each bounded subset ofX; <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/175/mathml/M98','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/175/mathml/M98">View MathML</a>is a continuously Gâteaux differentiable functional whose Gâteaux derivative is compact; <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/175/mathml/M99','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/175/mathml/M99">View MathML</a>is an interval. Suppose that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/175/mathml/M100','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/175/mathml/M100">View MathML</a>

for all<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/175/mathml/M101','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/175/mathml/M101">View MathML</a>, and that there exists<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/175/mathml/M102','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/175/mathml/M102">View MathML</a>such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/175/mathml/M103','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/175/mathml/M103">View MathML</a>

(2.1)

Then there exist an open interval<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/175/mathml/M104','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/175/mathml/M104">View MathML</a>and a positive real numberρwith the following property: for every<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/175/mathml/M105','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/175/mathml/M105">View MathML</a>and every<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/175/mathml/M106','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/175/mathml/M106">View MathML</a>functional<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/175/mathml/M107','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/175/mathml/M107">View MathML</a>with compact derivative, there exists<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/175/mathml/M91','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/175/mathml/M91">View MathML</a>such that, for each<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/175/mathml/M92','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/175/mathml/M92">View MathML</a>, the equation

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/175/mathml/M110','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/175/mathml/M110">View MathML</a>

has at least three solutions inXwhose norms are less thanρ.

Proposition 2.1 ([17], Proposition 3.1)

Assume thatXis a nonempty set and Φ, Ψ are two real functions onX. Suppose that there are<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/175/mathml/M111','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/175/mathml/M111">View MathML</a>and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/175/mathml/M112','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/175/mathml/M112">View MathML</a>such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/175/mathml/M113','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/175/mathml/M113">View MathML</a>

Then, for eachhsatisfying

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/175/mathml/M114','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/175/mathml/M114">View MathML</a>

one has

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/175/mathml/M115','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/175/mathml/M115">View MathML</a>

Before proving Theorem 1.1, we define a functional and give a lemma.

The functional <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/175/mathml/M116','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/175/mathml/M116">View MathML</a> is defined by

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/175/mathml/M117','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/175/mathml/M117">View MathML</a>

(2.2)

for all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/175/mathml/M43','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/175/mathml/M43">View MathML</a>, where

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/175/mathml/M119','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/175/mathml/M119">View MathML</a>

(2.3)

By conditions (M) and (j3), it is clear that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/175/mathml/M120','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/175/mathml/M120">View MathML</a> and a critical point of H corresponds to a weak solution of system (1.1).

Lemma 2.2Assume that there exist two positive constantsa, bwith<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/175/mathml/M121','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/175/mathml/M121">View MathML</a>such that

(j1) <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/175/mathml/M77','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/175/mathml/M77">View MathML</a>, for a.e. <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/175/mathml/M78','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/175/mathml/M78">View MathML</a>and all<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/175/mathml/M79','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/175/mathml/M79">View MathML</a>;

(j2) <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/175/mathml/M80','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/175/mathml/M80">View MathML</a>.

Then there exist<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/175/mathml/M126','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/175/mathml/M126">View MathML</a>and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/175/mathml/M127','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/175/mathml/M127">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/175/mathml/M128','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/175/mathml/M128">View MathML</a>such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/175/mathml/M129','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/175/mathml/M129">View MathML</a>

and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/175/mathml/M130','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/175/mathml/M130">View MathML</a>

Proof We put

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/175/mathml/M131','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/175/mathml/M131">View MathML</a>

and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/175/mathml/M132','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/175/mathml/M132">View MathML</a>. Then we can verify easily <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/175/mathml/M133','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/175/mathml/M133">View MathML</a> and, in particular, we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/175/mathml/M134','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/175/mathml/M134">View MathML</a>

(2.4)

and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/175/mathml/M135','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/175/mathml/M135">View MathML</a>

(2.5)

Hence, we obtain from (1.10), (1.11), (2.4) and (2.5) that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/175/mathml/M136','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/175/mathml/M136">View MathML</a>

(2.6)

Under condition (M), by a simple computation, we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/175/mathml/M137','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/175/mathml/M137">View MathML</a>

(2.7)

Setting <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/175/mathml/M138','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/175/mathml/M138">View MathML</a> and applying the assumption of Lemma 2.2

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/175/mathml/M139','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/175/mathml/M139">View MathML</a>

from (2.6) and (2.7), we obtain

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/175/mathml/M140','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/175/mathml/M140">View MathML</a>

Since, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/175/mathml/M141','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/175/mathml/M141">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/175/mathml/M142','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/175/mathml/M142">View MathML</a> for each <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/175/mathml/M82','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/175/mathml/M82">View MathML</a>, from condition (j1) of Lemma 2.2, we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/175/mathml/M144','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/175/mathml/M144">View MathML</a>

Hence, based on condition (j2), we get

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/175/mathml/M145','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/175/mathml/M145">View MathML</a>

 □

Now, we can prove our main result.

Proof of Theorem 1.1 For each <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/175/mathml/M146','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/175/mathml/M146">View MathML</a>, let

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/175/mathml/M147','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/175/mathml/M147">View MathML</a>

From the assumption of Theorem 1.1, we know that Φ is a continuously Gâteaux differentiable and sequentially weakly lower semicontinuous functional. Additionally, the Gâteaux derivative of Φ has a continuous inverse on <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/175/mathml/M97','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/175/mathml/M97">View MathML</a>. Since <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/175/mathml/M10','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/175/mathml/M10">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/175/mathml/M11','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/175/mathml/M11">View MathML</a>, Ψ and J are continuously Gâteaux differential functionals whose Gâteaux derivatives are compact. Obviously, Φ is bounded on each bounded subset of X. In particular, for each <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/175/mathml/M151','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/175/mathml/M151">View MathML</a>,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/175/mathml/M152','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/175/mathml/M152">View MathML</a>

Hence, the weak solutions of problem (1.1) are exactly the solutions of the following equation:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/175/mathml/M153','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/175/mathml/M153">View MathML</a>

From (j3), for each <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/175/mathml/M154','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/175/mathml/M154">View MathML</a>, one has

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/175/mathml/M155','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/175/mathml/M155">View MathML</a>

(2.8)

and so the first condition of Theorem 2.1 is satisfied. By Lemma 2.2, there exists <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/175/mathml/M156','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/175/mathml/M156">View MathML</a> such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/175/mathml/M157','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/175/mathml/M157">View MathML</a>

(2.9)

and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/175/mathml/M158','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/175/mathml/M158">View MathML</a>

(2.10)

From (1.3), we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/175/mathml/M159','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/175/mathml/M159">View MathML</a>

for each <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/175/mathml/M160','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/175/mathml/M160">View MathML</a>. We obtain

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/175/mathml/M161','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/175/mathml/M161">View MathML</a>

(2.11)

for each <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/175/mathml/M160','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/175/mathml/M160">View MathML</a>. Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/175/mathml/M138','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/175/mathml/M138">View MathML</a> for each <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/175/mathml/M164','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/175/mathml/M164">View MathML</a> such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/175/mathml/M165','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/175/mathml/M165">View MathML</a>

From (2.11), we get

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/175/mathml/M166','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/175/mathml/M166">View MathML</a>

(2.12)

Then, from (2.10) and (2.12), we find

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/175/mathml/M167','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/175/mathml/M167">View MathML</a>

Hence, we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/175/mathml/M168','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/175/mathml/M168">View MathML</a>

(2.13)

Fix h such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/175/mathml/M169','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/175/mathml/M169">View MathML</a>

by (2.9), (2.13) and Proposition 2.1, with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/175/mathml/M170','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/175/mathml/M170">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/175/mathml/M171','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/175/mathml/M171">View MathML</a>, we obtain

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/175/mathml/M172','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/175/mathml/M172">View MathML</a>

(2.14)

and so assumption (2.1) of Theorem 2.1 is satisfied.

Now, with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/175/mathml/M173','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/175/mathml/M173">View MathML</a>, from (2.8) and (2.14), all the assumptions of Theorem 2.1 hold. Hence, our conclusion follows from Theorem 2.1. □

Competing interests

The authors declare that they have no competing interests.

Authors’ contributions

This paper is the result of joint work of all authors who contributed equally to the final version of this paper. All authors read and approved the final manuscript.

Acknowledgements

The authors would like to thank the editors and the referees for their valuable suggestions to improve the quality of this paper. This work was supported by the Scientific Research Project of Guangxi Education Department (no. 201204LX672).

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