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Nonlinear fractional differential equations with nonlocal fractional integro-differential boundary conditions

Bashir Ahmad and Ahmed Alsaedi*

Author Affiliations

Department of Mathematics, Faculty of Science, King Abdulaziz University, P.O. Box 80203, Jeddah, 21589, Saudi Arabia

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Boundary Value Problems 2012, 2012:124  doi:10.1186/1687-2770-2012-124

The electronic version of this article is the complete one and can be found online at: http://www.boundaryvalueproblems.com/content/2012/1/124


Received:23 July 2012
Accepted:4 October 2012
Published:24 October 2012

© 2012 Ahmad and Alsaedi; 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 study the existence of solutions for a class of nonlinear Caputo-type fractional boundary value problems with nonlocal fractional integro-differential boundary conditions. We apply some fixed point principles and Leray-Schauder degree theory to obtain the main results. Some examples are discussed for the illustration of the main work.

MSC: 34A08, 34A12, 34B15.

Keywords:
fractional differential equations; fractional boundary conditions; separated boundary conditions; fixed point theorems

1 Introduction

Nonlocal boundary value problems of fractional differential equations have been extensively studied in the recent years. In fact, the subject of fractional calculus has been quite attractive and exciting due to its applications in the modeling of many physical and engineering problems. For theoretical and practical development of the subject, we refer to the books [1-5]. Some recent results on fractional boundary value problems can be found in [6-14] and references therein. In [11], the authors studied a boundary value problem of fractional differential equations with fractional separated boundary conditions.

In this article, motivated by [11], we consider a fractional boundary value problem with fractional integro-differential boundary conditions given by

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/124/mathml/M1','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/124/mathml/M1">View MathML</a>

(1.1)

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/124/mathml/M2','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/124/mathml/M2">View MathML</a> denotes the Caputo fractional derivative of order α, f is a given continuous function, and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/124/mathml/M3','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/124/mathml/M3">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/124/mathml/M4','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/124/mathml/M4">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/124/mathml/M5','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/124/mathml/M5">View MathML</a> (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/124/mathml/M6','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/124/mathml/M6">View MathML</a>) are suitably chosen real constants.

The main aim of the present study is to obtain some existence results for the problem (1.1). As a first step, we transform the given problem to a fixed point problem and show the existence of fixed points for the transformed problem which in turn correspond to the solutions of the actual problem. The methods used to prove the existence results are standard; however, their exposition in the framework of the problem (1.1) is new.

2 Preliminaries

Let us recall some basic definitions of fractional calculus [1,2].

Definition 2.1 For <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/124/mathml/M7','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/124/mathml/M7">View MathML</a>-times absolutely continuous function <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/124/mathml/M8','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/124/mathml/M8">View MathML</a>, the Caputo derivative of fractional order q is defined as

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/124/mathml/M9','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/124/mathml/M9">View MathML</a>

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/124/mathml/M10','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/124/mathml/M10">View MathML</a> denotes the integer part of the real number q.

Definition 2.2 The Riemann-Liouville fractional integral of order q is defined as

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/124/mathml/M11','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/124/mathml/M11">View MathML</a>

provided the integral exists.

To define the solution of the boundary value problem (1.1), we need the following lemma, which deals with a linear variant of the problem (1.1).

Lemma 2.3For a given<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/124/mathml/M12','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/124/mathml/M12">View MathML</a>, the unique solution of the linear fractional boundary value problem

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/124/mathml/M13','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/124/mathml/M13">View MathML</a>

(2.1)

is given by

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/124/mathml/M14','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/124/mathml/M14">View MathML</a>

(2.2)

where

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/124/mathml/M15','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/124/mathml/M15">View MathML</a>

(2.3)

Proof It is well known [2] that the solution of the fractional differential equation in (2.1) can be written as

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/124/mathml/M16','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/124/mathml/M16">View MathML</a>

(2.4)

Using <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/124/mathml/M17','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/124/mathml/M17">View MathML</a> (b is a constant), <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/124/mathml/M18','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/124/mathml/M18">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/124/mathml/M19','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/124/mathml/M19">View MathML</a>, (2.4) gives

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/124/mathml/M20','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/124/mathml/M20">View MathML</a>

(2.5)

Using the integral boundary conditions of the problem (2.1) together with (2.3), (2.4), and (2.5) yields

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/124/mathml/M21','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/124/mathml/M21">View MathML</a>

Substituting the values of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/124/mathml/M22','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/124/mathml/M22">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/124/mathml/M23','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/124/mathml/M23">View MathML</a> in (2.4), we get (2.2). This completes the proof. □

Remark 2.4 Notice that the solution (2.2) is independent of the parameter <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/124/mathml/M24','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/124/mathml/M24">View MathML</a>, which distinguishes the present work from the one containing the fractional differential equation of (2.1) with the boundary conditions of the form:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/124/mathml/M25','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/124/mathml/M25">View MathML</a>

(2.6)

In case <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/124/mathml/M26','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/124/mathml/M26">View MathML</a>, the boundary conditions in (2.1) coincide with (2.6) and consequently the corresponding solutions become identical.

3 Main results

Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/124/mathml/M27','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/124/mathml/M27">View MathML</a> denote the Banach space of all continuous functions from <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/124/mathml/M28','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/124/mathml/M28">View MathML</a> into ℝ endowed with the usual supremum norm.

In view of Lemma 2.3, we define an operator <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/124/mathml/M29','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/124/mathml/M29">View MathML</a> by

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/124/mathml/M30','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/124/mathml/M30">View MathML</a>

(3.1)

Observe that the problem (1.1) has solutions if and only if the operator equation <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/124/mathml/M31','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/124/mathml/M31">View MathML</a> has fixed points.

In the sequel, we use the following notation:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/124/mathml/M32','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/124/mathml/M32">View MathML</a>

(3.2)

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/124/mathml/M33','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/124/mathml/M33">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/124/mathml/M34','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/124/mathml/M34">View MathML</a> with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/124/mathml/M35','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/124/mathml/M35">View MathML</a> (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/124/mathml/M36','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/124/mathml/M36">View MathML</a>) given by (2.3).

Our first result is based on the Leray-Schauder nonlinear alternative [15].

Lemma 3.1 (Nonlinear alternative for single valued maps)

LetEbe a Banach space, Ca closed, convex subset ofE, Uan open subset ofC, and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/124/mathml/M37','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/124/mathml/M37">View MathML</a>. Suppose that<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/124/mathml/M38','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/124/mathml/M38">View MathML</a>is a continuous, compact (that is, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/124/mathml/M39','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/124/mathml/M39">View MathML</a>is a relatively compact subset ofC) map. Then either

(i) Fhas a fixed point in<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/124/mathml/M40','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/124/mathml/M40">View MathML</a>, or

(ii) there is a<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/124/mathml/M41','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/124/mathml/M41">View MathML</a> (the boundary ofUinC) and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/124/mathml/M42','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/124/mathml/M42">View MathML</a>with<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/124/mathml/M43','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/124/mathml/M43">View MathML</a>.

Theorem 3.2Let<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/124/mathml/M44','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/124/mathml/M44">View MathML</a>be a jointly continuous function. Assume that:

(<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/124/mathml/M45','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/124/mathml/M45">View MathML</a>) there exist a function<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/124/mathml/M46','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/124/mathml/M46">View MathML</a>and a nondecreasing function<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/124/mathml/M47','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/124/mathml/M47">View MathML</a>such that<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/124/mathml/M48','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/124/mathml/M48">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/124/mathml/M49','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/124/mathml/M49">View MathML</a>;

(<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/124/mathml/M50','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/124/mathml/M50">View MathML</a>) there exists a constant<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/124/mathml/M51','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/124/mathml/M51">View MathML</a>such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/124/mathml/M52','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/124/mathml/M52">View MathML</a>

Then the boundary value problem (1.1) has at least one solution on<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/124/mathml/M28','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/124/mathml/M28">View MathML</a>.

Proof Consider the operator <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/124/mathml/M54','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/124/mathml/M54">View MathML</a> defined by (3.1). We show that Fmaps bounded sets into bounded sets in<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/124/mathml/M55','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/124/mathml/M55">View MathML</a>. For a positive number r, let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/124/mathml/M56','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/124/mathml/M56">View MathML</a> be a bounded set in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/124/mathml/M55','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/124/mathml/M55">View MathML</a>. Then

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/124/mathml/M58','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/124/mathml/M58">View MathML</a>

Next, we show that Fmaps bounded sets into equicontinuous sets of<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/124/mathml/M55','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/124/mathml/M55">View MathML</a>. Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/124/mathml/M60','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/124/mathml/M60">View MathML</a> with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/124/mathml/M61','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/124/mathml/M61">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/124/mathml/M62','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/124/mathml/M62">View MathML</a>, where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/124/mathml/M63','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/124/mathml/M63">View MathML</a> is a bounded set of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/124/mathml/M55','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/124/mathml/M55">View MathML</a>. Then we obtain

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/124/mathml/M65','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/124/mathml/M65">View MathML</a>

Obviously, the right-hand side of the above inequality tends to zero independently of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/124/mathml/M62','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/124/mathml/M62">View MathML</a> as <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/124/mathml/M67','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/124/mathml/M67">View MathML</a>. As ℱ satisfies the above assumptions, therefore, it follows by the Arzelá-Ascoli theorem that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/124/mathml/M68','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/124/mathml/M68">View MathML</a> is completely continuous.

Let x be a solution. Then for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/124/mathml/M69','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/124/mathml/M69">View MathML</a>, using the computations in proving that ℱ is bounded, we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/124/mathml/M70','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/124/mathml/M70">View MathML</a>

Consequently, we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/124/mathml/M71','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/124/mathml/M71">View MathML</a>

In view of (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/124/mathml/M50','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/124/mathml/M50">View MathML</a>), there exists M such that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/124/mathml/M73','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/124/mathml/M73">View MathML</a>. Let us set

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/124/mathml/M74','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/124/mathml/M74">View MathML</a>

Note that the operator <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/124/mathml/M75','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/124/mathml/M75">View MathML</a> is continuous and completely continuous. From the choice of U, there is no <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/124/mathml/M76','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/124/mathml/M76">View MathML</a> such that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/124/mathml/M77','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/124/mathml/M77">View MathML</a> for some <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/124/mathml/M42','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/124/mathml/M42">View MathML</a>. Consequently, by the nonlinear alternative of Leray-Schauder type (Lemma 3.1), we deduce that ℱ has a fixed point <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/124/mathml/M79','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/124/mathml/M79">View MathML</a> which is a solution of the problem (1.1). This completes the proof. □

In the special case when <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/124/mathml/M80','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/124/mathml/M80">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/124/mathml/M81','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/124/mathml/M81">View MathML</a> (κ and N are suitable constants) in the statement of Theorem 3.2, we have the following corollary.

Corollary 3.3Let<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/124/mathml/M82','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/124/mathml/M82">View MathML</a>be a continuous function. Assume that there exist constants<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/124/mathml/M83','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/124/mathml/M83">View MathML</a>, whereωis given by (3.2) and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/124/mathml/M84','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/124/mathml/M84">View MathML</a>such that<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/124/mathml/M85','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/124/mathml/M85">View MathML</a>for all<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/124/mathml/M86','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/124/mathml/M86">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/124/mathml/M87','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/124/mathml/M87">View MathML</a>. Then the boundary value problem (1.1) has at least one solution.

Next, we prove an existence and uniqueness result by means of Banach’s contraction mapping principle.

Theorem 3.4Suppose that<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/124/mathml/M88','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/124/mathml/M88">View MathML</a>is a continuous function and satisfies the following assumption:

(<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/124/mathml/M89','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/124/mathml/M89">View MathML</a>) <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/124/mathml/M90','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/124/mathml/M90">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/124/mathml/M91','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/124/mathml/M91">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/124/mathml/M92','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/124/mathml/M92">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/124/mathml/M93','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/124/mathml/M93">View MathML</a>.

Then the boundary value problem (1.1) has a unique solution provided

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/124/mathml/M94','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/124/mathml/M94">View MathML</a>

(3.3)

whereωis given by (3.2).

Proof With <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/124/mathml/M95','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/124/mathml/M95">View MathML</a>, we define <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/124/mathml/M96','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/124/mathml/M96">View MathML</a>, where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/124/mathml/M97','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/124/mathml/M97">View MathML</a> and ω is given by (3.2). Then we show that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/124/mathml/M98','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/124/mathml/M98">View MathML</a>. For <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/124/mathml/M62','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/124/mathml/M62">View MathML</a>, we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/124/mathml/M100','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/124/mathml/M100">View MathML</a>

Using <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/124/mathml/M101','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/124/mathml/M101">View MathML</a>, the above expression yields

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/124/mathml/M102','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/124/mathml/M102">View MathML</a>

where we used (3.2). Now, for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/124/mathml/M103','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/124/mathml/M103">View MathML</a> and for each <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/124/mathml/M86','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/124/mathml/M86">View MathML</a>, we obtain

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/124/mathml/M105','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/124/mathml/M105">View MathML</a>

Note that ω depends only on the parameters involved in the problem. As <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/124/mathml/M106','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/124/mathml/M106">View MathML</a>, therefore, ℱ is a contraction. Hence, by Banach’s contraction mapping principle, the problem (1.1) has a unique solution on <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/124/mathml/M107','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/124/mathml/M107">View MathML</a>. □

Now, we prove the existence of solutions of (1.1) by applying Krasnoselskii’s fixed point theorem [16].

Theorem 3.5 (Krasnoselskii’s fixed point theorem)

LetMbe a closed, bounded, convex, and nonempty subset of a Banach spaceX. LetA, Bbe the operators such that (i) <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/124/mathml/M108','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/124/mathml/M108">View MathML</a>whenever<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/124/mathml/M109','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/124/mathml/M109">View MathML</a>; (ii) Ais compact and continuous; (iii) Bis a contraction mapping. Then there exists<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/124/mathml/M110','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/124/mathml/M110">View MathML</a>such that<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/124/mathml/M111','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/124/mathml/M111">View MathML</a>.

Theorem 3.6Let<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/124/mathml/M112','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/124/mathml/M112">View MathML</a>be a jointly continuous function satisfying the assumption (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/124/mathml/M89','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/124/mathml/M89">View MathML</a>). In addition we assume that:

(<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/124/mathml/M114','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/124/mathml/M114">View MathML</a>) <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/124/mathml/M115','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/124/mathml/M115">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/124/mathml/M116','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/124/mathml/M116">View MathML</a>, and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/124/mathml/M117','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/124/mathml/M117">View MathML</a>.

Then the problem (1.1) has at least one solution on<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/124/mathml/M28','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/124/mathml/M28">View MathML</a>if

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/124/mathml/M119','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/124/mathml/M119">View MathML</a>

(3.4)

Proof Letting <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/124/mathml/M120','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/124/mathml/M120">View MathML</a>, we choose a real number <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/124/mathml/M121','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/124/mathml/M121">View MathML</a> satisfying the inequality

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/124/mathml/M122','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/124/mathml/M122">View MathML</a>

and consider <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/124/mathml/M123','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/124/mathml/M123">View MathML</a>. We define the operators <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/124/mathml/M124','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/124/mathml/M124">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/124/mathml/M125','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/124/mathml/M125">View MathML</a> on <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/124/mathml/M126','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/124/mathml/M126">View MathML</a> as

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/124/mathml/M127','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/124/mathml/M127">View MathML</a>

For <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/124/mathml/M128','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/124/mathml/M128">View MathML</a>, we find that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/124/mathml/M129','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/124/mathml/M129">View MathML</a>

Thus, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/124/mathml/M130','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/124/mathml/M130">View MathML</a>. It follows from the assumption (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/124/mathml/M89','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/124/mathml/M89">View MathML</a>) together with (3.4) that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/124/mathml/M125','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/124/mathml/M125">View MathML</a> is a contraction mapping. Continuity of f implies that the operator <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/124/mathml/M124','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/124/mathml/M124">View MathML</a> is continuous. Also, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/124/mathml/M124','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/124/mathml/M124">View MathML</a> is uniformly bounded on <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/124/mathml/M126','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/124/mathml/M126">View MathML</a> as

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/124/mathml/M136','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/124/mathml/M136">View MathML</a>

Now, we prove the compactness of the operator <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/124/mathml/M124','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/124/mathml/M124">View MathML</a>.

In view of (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/124/mathml/M45','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/124/mathml/M45">View MathML</a>), we define <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/124/mathml/M139','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/124/mathml/M139">View MathML</a>, and consequently, for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/124/mathml/M140','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/124/mathml/M140">View MathML</a>, we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/124/mathml/M141','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/124/mathml/M141">View MathML</a>

which is independent of x. Thus, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/124/mathml/M124','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/124/mathml/M124">View MathML</a> is equicontinuous. Hence, by the Arzelá-Ascoli theorem, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/124/mathml/M124','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/124/mathml/M124">View MathML</a> is compact on <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/124/mathml/M126','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/124/mathml/M126">View MathML</a>. Thus, all the assumptions of Theorem 3.5 are satisfied. So, the conclusion of Theorem 3.5 implies that the boundary value problem (1.1) has at least one solution on <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/124/mathml/M28','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/124/mathml/M28">View MathML</a>. □

4 Examples

Example 4.1 Consider the following boundary value problem:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/124/mathml/M146','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/124/mathml/M146">View MathML</a>

(4.1)

Here, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/124/mathml/M147','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/124/mathml/M147">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/124/mathml/M148','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/124/mathml/M148">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/124/mathml/M149','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/124/mathml/M149">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/124/mathml/M150','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/124/mathml/M150">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/124/mathml/M151','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/124/mathml/M151">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/124/mathml/M152','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/124/mathml/M152">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/124/mathml/M153','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/124/mathml/M153">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/124/mathml/M154','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/124/mathml/M154">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/124/mathml/M155','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/124/mathml/M155">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/124/mathml/M156','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/124/mathml/M156">View MathML</a>, and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/124/mathml/M157','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/124/mathml/M157">View MathML</a>

Clearly,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/124/mathml/M158','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/124/mathml/M158">View MathML</a>

Clearly, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/124/mathml/M159','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/124/mathml/M159">View MathML</a> and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/124/mathml/M160','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/124/mathml/M160">View MathML</a>

Thus, all the conditions of Corollary 3.3 are satisfied and consequently the problem (4.1) has at least one solution.

Example 4.2 Consider the following fractional boundary value problem:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/124/mathml/M161','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/124/mathml/M161">View MathML</a>

(4.2)

where α, p, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/124/mathml/M3','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/124/mathml/M3">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/124/mathml/M4','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/124/mathml/M4">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/124/mathml/M5','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/124/mathml/M5">View MathML</a>, (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/124/mathml/M165','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/124/mathml/M165">View MathML</a>) η, σ are the same as given in (4.1) and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/124/mathml/M166','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/124/mathml/M166">View MathML</a>. Clearly, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/124/mathml/M167','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/124/mathml/M167">View MathML</a> and thus, for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/124/mathml/M168','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/124/mathml/M168">View MathML</a>, all the conditions of Theorem 3.4 are satisfied. Hence, the boundary value problem (4.2) has a unique solution on <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/124/mathml/M28','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/124/mathml/M28">View MathML</a>.

Competing interests

The authors declare that they have no competing interests.

Authors’ contributions

Each of the authors, BA and AA contributed to each part of this work equally and read and approved the final version of the manuscript.

Acknowledgements

The authors thank the reviewers for their useful comments that led to the improvement of the original manuscript. This research was partially supported by the Deanship of Scientific Research (DSR), King Abdulaziz University, Jeddah, Saudi Arabia.

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