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Existence of nonnegative solutions for a fractional m-point boundary value problem at resonance

Haidong Qu1* and Xuan Liu2

Author Affiliations

1 Department of Mathematics, Hanshan Normal University, Chaozhou, Guangdong, 521041, China

2 Department of Basic Education, Hanshan Normal University, Chaozhou, Guangdong, 521041, China

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Boundary Value Problems 2013, 2013:127  doi:10.1186/1687-2770-2013-127

The electronic version of this article is the complete one and can be found online at: http://www.boundaryvalueproblems.com/content/2013/1/127


Received:11 January 2013
Accepted:26 April 2013
Published:16 May 2013

© 2013 Qu and Liu; licensee Springer

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

Abstract

We consider the fractional differential equation

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/127/mathml/M1','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/127/mathml/M1">View MathML</a>

satisfying the boundary conditions

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/127/mathml/M2','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/127/mathml/M2">View MathML</a>

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/127/mathml/M3','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/127/mathml/M3">View MathML</a> is the Riemann-Liouville fractional order derivative. The parameters in the multi-point boundary conditions are such that the corresponding differential operator is a Fredholm map of index zero. As a result, the minimal and maximal nonnegative solutions for the problem are obtained by using a fixed point theorem of increasing operators.

MSC: 26A33, 34A08.

Keywords:
fractional order; coincidence degree; at resonance

1 Introduction

Let us consider the fractional differential equation

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/127/mathml/M4','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/127/mathml/M4">View MathML</a>

(1.1)

with the boundary conditions (BCs)

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/127/mathml/M5','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/127/mathml/M5">View MathML</a>

(1.2)

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/127/mathml/M6','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/127/mathml/M6">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/127/mathml/M7','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/127/mathml/M7">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/127/mathml/M8','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/127/mathml/M8">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/127/mathml/M9','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/127/mathml/M9">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/127/mathml/M10','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/127/mathml/M10">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/127/mathml/M11','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/127/mathml/M11">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/127/mathml/M12','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/127/mathml/M12">View MathML</a>. We assume that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/127/mathml/M13','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/127/mathml/M13">View MathML</a> is continuous. A boundary value problem at resonance for ordinary or fractional differential equations has been studied by several authors, including the most recent works [1-7] and the references therein. In the most papers mentioned above, the coincidence degree theory was applied to establish existence theorems. But in [8], Wang obtained the minimal and maximal nonnegative solutions for a second-order m-point boundary value problem at resonance by using a new fixed point theorem of increasing operators, and in this paper we use this method of Wang to establish the existence theorem of equations (1.1) and (1.2).

For the convenience of the reader, we briefly recall some notations.

Let X, Z be real Banach spaces, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/127/mathml/M14','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/127/mathml/M14">View MathML</a> be a Fredholm map of index zero and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/127/mathml/M15','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/127/mathml/M15">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/127/mathml/M16','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/127/mathml/M16">View MathML</a> be continuous projectors such that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/127/mathml/M17','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/127/mathml/M17">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/127/mathml/M18','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/127/mathml/M18">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/127/mathml/M19','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/127/mathml/M19">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/127/mathml/M20','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/127/mathml/M20">View MathML</a>. It follows that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/127/mathml/M21','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/127/mathml/M21">View MathML</a> is invertible. We denote the inverse of the map by <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/127/mathml/M22','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/127/mathml/M22">View MathML</a>. Since <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/127/mathml/M23','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/127/mathml/M23">View MathML</a>, there exists an isomorphism <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/127/mathml/M24','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/127/mathml/M24">View MathML</a>. Let Ω be an open bounded subset of X. The map <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/127/mathml/M25','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/127/mathml/M25">View MathML</a> will be called L-compact on <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/127/mathml/M26','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/127/mathml/M26">View MathML</a> if <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/127/mathml/M27','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/127/mathml/M27">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/127/mathml/M28','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/127/mathml/M28">View MathML</a> are compact. We take <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/127/mathml/M29','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/127/mathml/M29">View MathML</a>, then <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/127/mathml/M30','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/127/mathml/M30">View MathML</a> is a linear bijection with bounded inverse and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/127/mathml/M31','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/127/mathml/M31">View MathML</a>. We know from [9] that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/127/mathml/M32','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/127/mathml/M32">View MathML</a> is a cone in Z.

Theorem 1.1[9]

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/127/mathml/M33','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/127/mathml/M33">View MathML</a>, where

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/127/mathml/M34','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/127/mathml/M34">View MathML</a>

and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/127/mathml/M35','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/127/mathml/M35">View MathML</a>is uniquely determined.

From the above theorem, the author [9] obtained that the assertions

(i) <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/127/mathml/M36','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/127/mathml/M36">View MathML</a> and

(ii) <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/127/mathml/M37','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/127/mathml/M37">View MathML</a> are equivalent.

We also need the following definition and theorem.

Definition 1.1[8]

Let K be a normal cone in a Banach space X, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/127/mathml/M38','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/127/mathml/M38">View MathML</a>, and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/127/mathml/M39','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/127/mathml/M39">View MathML</a> are said to be coupled lower and upper solutions of the equation <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/127/mathml/M40','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/127/mathml/M40">View MathML</a> if

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/127/mathml/M41','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/127/mathml/M41">View MathML</a>

Theorem 1.2[8]

Let<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/127/mathml/M14','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/127/mathml/M14">View MathML</a>be a Fredholm operator of index zero, Kbe a normal cone in a Banach spaceX, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/127/mathml/M39','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/127/mathml/M39">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/127/mathml/M38','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/127/mathml/M38">View MathML</a>, and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/127/mathml/M45','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/127/mathml/M45">View MathML</a>beL-compact and continuous. Suppose that the following conditions are satisfied:

(C1) <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/127/mathml/M46','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/127/mathml/M46">View MathML</a>and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/127/mathml/M47','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/127/mathml/M47">View MathML</a>are coupled lower and upper solutions of the equation<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/127/mathml/M40','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/127/mathml/M40">View MathML</a>;

(C2) <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/127/mathml/M49','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/127/mathml/M49">View MathML</a>is an increasing operator.

Then the equation<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/127/mathml/M40','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/127/mathml/M40">View MathML</a>has a minimal solution<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/127/mathml/M51','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/127/mathml/M51">View MathML</a>and a maximal solution<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/127/mathml/M52','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/127/mathml/M52">View MathML</a>in<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/127/mathml/M53','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/127/mathml/M53">View MathML</a>. Moreover,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/127/mathml/M54','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/127/mathml/M54">View MathML</a>

where

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/127/mathml/M55','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/127/mathml/M55">View MathML</a>

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/127/mathml/M56','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/127/mathml/M56">View MathML</a>and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/127/mathml/M57','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/127/mathml/M57">View MathML</a>.

2 Preliminaries

In this section, we present some necessary basic knowledge and definitions about fractional calculus theory.

Definition 2.1 (see Equation 2.1.1 in [10])

The R-L fractional integral <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/127/mathml/M58','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/127/mathml/M58">View MathML</a> of order <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/127/mathml/M59','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/127/mathml/M59">View MathML</a> (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/127/mathml/M60','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/127/mathml/M60">View MathML</a>) is defined by

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/127/mathml/M61','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/127/mathml/M61">View MathML</a>

Here <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/127/mathml/M62','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/127/mathml/M62">View MathML</a> is the gamma function.

Definition 2.2 (see Equation 2.1.5 in [10])

The R-L fractional derivative <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/127/mathml/M63','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/127/mathml/M63">View MathML</a> of order <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/127/mathml/M59','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/127/mathml/M59">View MathML</a> (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/127/mathml/M60','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/127/mathml/M60">View MathML</a>) is defined by

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/127/mathml/M66','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/127/mathml/M66">View MathML</a>

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/127/mathml/M67','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/127/mathml/M67">View MathML</a> means the integral part of q.

Lemma 2.1[11]

If<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/127/mathml/M68','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/127/mathml/M68">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/127/mathml/M69','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/127/mathml/M69">View MathML</a>, then, for<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/127/mathml/M70','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/127/mathml/M70">View MathML</a>, the relations

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/127/mathml/M71','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/127/mathml/M71">View MathML</a>

and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/127/mathml/M72','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/127/mathml/M72">View MathML</a>

hold a.e. on<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/127/mathml/M73','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/127/mathml/M73">View MathML</a>.

Lemma 2.2 (see [11])

Let<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/127/mathml/M69','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/127/mathml/M69">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/127/mathml/M75','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/127/mathml/M75">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/127/mathml/M76','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/127/mathml/M76">View MathML</a>, then we have the equality

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/127/mathml/M77','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/127/mathml/M77">View MathML</a>

where<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/127/mathml/M78','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/127/mathml/M78">View MathML</a> (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/127/mathml/M79','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/127/mathml/M79">View MathML</a>) are some constants.

Lemma 2.3 (see Corollary 2.1 in [10])

Let<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/127/mathml/M69','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/127/mathml/M69">View MathML</a>and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/127/mathml/M75','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/127/mathml/M75">View MathML</a>, the equation<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/127/mathml/M82','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/127/mathml/M82">View MathML</a>is valid if and only if<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/127/mathml/M83','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/127/mathml/M83">View MathML</a>, where<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/127/mathml/M78','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/127/mathml/M78">View MathML</a> (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/127/mathml/M85','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/127/mathml/M85">View MathML</a>) are arbitrary constants.

Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/127/mathml/M86','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/127/mathml/M86">View MathML</a> with the norm <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/127/mathml/M87','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/127/mathml/M87">View MathML</a>, then X and Z are Banach spaces.

Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/127/mathml/M88','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/127/mathml/M88">View MathML</a>. It follows from Theorem 1.1.1 in [12] that K is a normal cone.

Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/127/mathml/M89','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/127/mathml/M89">View MathML</a>.

We define the operators <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/127/mathml/M90','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/127/mathml/M90">View MathML</a> by

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/127/mathml/M91','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/127/mathml/M91">View MathML</a>

(2.1)

and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/127/mathml/M92','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/127/mathml/M92">View MathML</a> by

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/127/mathml/M93','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/127/mathml/M93">View MathML</a>

then BVPs (1.1) and (1.2) can be written as <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/127/mathml/M94','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/127/mathml/M94">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/127/mathml/M95','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/127/mathml/M95">View MathML</a>.

Lemma 2.4If the operatorLis defined in (2.1), then

(i) <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/127/mathml/M96','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/127/mathml/M96">View MathML</a>,

(ii) <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/127/mathml/M97','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/127/mathml/M97">View MathML</a>.

Proof (i) It can be seen from Lemma 2.3 and BCs (1.2) that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/127/mathml/M98','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/127/mathml/M98">View MathML</a>.

(ii) If <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/127/mathml/M99','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/127/mathml/M99">View MathML</a>, then there exists a function <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/127/mathml/M100','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/127/mathml/M100">View MathML</a> such that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/127/mathml/M101','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/127/mathml/M101">View MathML</a>, by Lemma 2.2, we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/127/mathml/M102','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/127/mathml/M102">View MathML</a>

It follows from BCs (1.2) and the equation <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/127/mathml/M103','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/127/mathml/M103">View MathML</a> that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/127/mathml/M104','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/127/mathml/M104">View MathML</a>

and noting the definition of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/127/mathml/M105','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/127/mathml/M105">View MathML</a>, we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/127/mathml/M106','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/127/mathml/M106">View MathML</a>

Thus,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/127/mathml/M107','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/127/mathml/M107">View MathML</a>

which is

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/127/mathml/M108','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/127/mathml/M108">View MathML</a>

Then <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/127/mathml/M109','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/127/mathml/M109">View MathML</a>, hence <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/127/mathml/M110','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/127/mathml/M110">View MathML</a>.

On the other hand, if <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/127/mathml/M109','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/127/mathml/M109">View MathML</a>, let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/127/mathml/M112','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/127/mathml/M112">View MathML</a>, then <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/127/mathml/M100','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/127/mathml/M100">View MathML</a>, and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/127/mathml/M114','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/127/mathml/M114">View MathML</a>, which implies that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/127/mathml/M99','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/127/mathml/M99">View MathML</a>, thus <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/127/mathml/M116','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/127/mathml/M116">View MathML</a>. In general <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/127/mathml/M117','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/127/mathml/M117">View MathML</a>. Clearly, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/127/mathml/M118','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/127/mathml/M118">View MathML</a> is closed in Z and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/127/mathml/M119','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/127/mathml/M119">View MathML</a>, thus L is a Fredholm operator of index zero. This completes the proof. □

In what follows, some property operators are defined. We define continuous projectors <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/127/mathml/M15','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/127/mathml/M15">View MathML</a> by

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/127/mathml/M121','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/127/mathml/M121">View MathML</a>

and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/127/mathml/M16','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/127/mathml/M16">View MathML</a> by

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/127/mathml/M123','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/127/mathml/M123">View MathML</a>

where

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/127/mathml/M124','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/127/mathml/M124">View MathML</a>

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/127/mathml/M125','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/127/mathml/M125">View MathML</a> is the beta function defined by

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/127/mathml/M126','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/127/mathml/M126">View MathML</a>

By calculating, we easily obtain <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/127/mathml/M127','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/127/mathml/M127">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/127/mathml/M128','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/127/mathml/M128">View MathML</a>, and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/127/mathml/M19','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/127/mathml/M19">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/127/mathml/M20','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/127/mathml/M20">View MathML</a>. We also define <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/127/mathml/M131','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/127/mathml/M131">View MathML</a> by

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/127/mathml/M132','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/127/mathml/M132">View MathML</a>

and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/127/mathml/M133','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/127/mathml/M133">View MathML</a> by

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/127/mathml/M134','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/127/mathml/M134">View MathML</a>

thus

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/127/mathml/M135','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/127/mathml/M135">View MathML</a>

and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/127/mathml/M136','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/127/mathml/M136">View MathML</a>

Lemma 2.5LetΩbe any open bounded subset of<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/127/mathml/M137','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/127/mathml/M137">View MathML</a>, then<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/127/mathml/M27','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/127/mathml/M27">View MathML</a>and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/127/mathml/M139','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/127/mathml/M139">View MathML</a>are compact, which implies thatNisL-compact on<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/127/mathml/M26','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/127/mathml/M26">View MathML</a>for any open bounded set<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/127/mathml/M141','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/127/mathml/M141">View MathML</a>.

Proof For a positive integer n, let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/127/mathml/M142','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/127/mathml/M142">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/127/mathml/M143','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/127/mathml/M143">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/127/mathml/M144','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/127/mathml/M144">View MathML</a>. It is easy to see that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/127/mathml/M27','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/127/mathml/M27">View MathML</a> is compact. Now, we prove that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/127/mathml/M139','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/127/mathml/M139">View MathML</a> is compact. For <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/127/mathml/M147','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/127/mathml/M147">View MathML</a>, we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/127/mathml/M148','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/127/mathml/M148">View MathML</a>

which implies that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/127/mathml/M139','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/127/mathml/M139">View MathML</a> is bounded.

Moreover, for each <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/127/mathml/M150','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/127/mathml/M150">View MathML</a>, let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/127/mathml/M151','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/127/mathml/M151">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/127/mathml/M152','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/127/mathml/M152">View MathML</a>, then

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/127/mathml/M153','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/127/mathml/M153">View MathML</a>

Thus

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/127/mathml/M154','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/127/mathml/M154">View MathML</a>

such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/127/mathml/M155','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/127/mathml/M155">View MathML</a>

for

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/127/mathml/M156','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/127/mathml/M156">View MathML</a>

and each

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/127/mathml/M157','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/127/mathml/M157">View MathML</a>

It is concluded that N is L-compact on <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/127/mathml/M26','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/127/mathml/M26">View MathML</a>. This completes the proof. □

3 Main result

In this section, we establish the existence of the nonnegative solution to equations (1.1) and (1.2).

Theorem 3.1Suppose

(H1) There exist<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/127/mathml/M159','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/127/mathml/M159">View MathML</a>such that<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/127/mathml/M38','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/127/mathml/M38">View MathML</a>and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/127/mathml/M161','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/127/mathml/M161">View MathML</a>

(H2) For any<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/127/mathml/M162','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/127/mathml/M162">View MathML</a>, satisfying

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/127/mathml/M163','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/127/mathml/M163">View MathML</a>

where<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/127/mathml/M164','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/127/mathml/M164">View MathML</a>and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/127/mathml/M165','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/127/mathml/M165">View MathML</a>, then problems (1.1) and (1.2) have a minimal solution<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/127/mathml/M51','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/127/mathml/M51">View MathML</a>and a maximal solution<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/127/mathml/M52','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/127/mathml/M52">View MathML</a>in<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/127/mathml/M168','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/127/mathml/M168">View MathML</a>, respectively.

Proof By condition (H1), we know that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/127/mathml/M169','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/127/mathml/M169">View MathML</a>

so condition (C1) in Theorem 1.1 holds.

In addition, for each <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/127/mathml/M170','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/127/mathml/M170">View MathML</a>,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/127/mathml/M171','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/127/mathml/M171">View MathML</a>

Thus <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/127/mathml/M172','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/127/mathml/M172">View MathML</a>, that is, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/127/mathml/M49','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/127/mathml/M49">View MathML</a> by virtue of the equivalence. From condition (H2), we have that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/127/mathml/M49','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/127/mathml/M49">View MathML</a> is a monotone increasing operator. Then, in accordance with Lemma 2.5 and Theorem 1.2, we obtain a minimal solution <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/127/mathml/M51','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/127/mathml/M51">View MathML</a> and a maximal solution <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/127/mathml/M52','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/127/mathml/M52">View MathML</a> in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/127/mathml/M168','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/127/mathml/M168">View MathML</a> for problems (1.1) and (1.2). Thus we can define iterative sequences <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/127/mathml/M178','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/127/mathml/M178">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/127/mathml/M179','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/127/mathml/M179">View MathML</a> by

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/127/mathml/M180','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/127/mathml/M180">View MathML</a>

and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/127/mathml/M181','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/127/mathml/M181">View MathML</a>

Then from Theorem 1.2 we get <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/127/mathml/M182','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/127/mathml/M182">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/127/mathml/M183','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/127/mathml/M183">View MathML</a> converge uniformly to <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/127/mathml/M184','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/127/mathml/M184">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/127/mathml/M185','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/127/mathml/M185">View MathML</a>, respectively. Moreover,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/127/mathml/M186','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/127/mathml/M186">View MathML</a>

 □

4 Example

We consider the following problem:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/127/mathml/M187','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/127/mathml/M187">View MathML</a>

(4.1)

subject to BCs

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/127/mathml/M188','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/127/mathml/M188">View MathML</a>

(4.2)

We can choose

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/127/mathml/M189','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/127/mathml/M189">View MathML</a>

then

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/127/mathml/M190','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/127/mathml/M190">View MathML</a>

Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/127/mathml/M191','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/127/mathml/M191">View MathML</a>, then for any <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/127/mathml/M192','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/127/mathml/M192">View MathML</a>, we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/127/mathml/M193','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/127/mathml/M193">View MathML</a>

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/127/mathml/M165','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/127/mathml/M165">View MathML</a>. Finally, by Theorem 3.1, equation (4.1) with BCs (4.2) has a minimal solution <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/127/mathml/M51','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/127/mathml/M51">View MathML</a> and a maximal solution <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/127/mathml/M52','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/127/mathml/M52">View MathML</a> in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/127/mathml/M53','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/127/mathml/M53">View MathML</a>.

Competing interests

The authors declare that they have no competing interests.

Authors’ contributions

The authors declare that the study was realized in collaboration with the same responsibility. All authors read and approved the final manuscript.

Acknowledgements

The authors would like to thank the referees for their many constructive comments and suggestions to improve the paper.

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