Open Access Research

Boundary value problems for fractional differential equations with three-point fractional integral boundary conditions

Weerawat Sudsutad12 and Jessada Tariboon12*

Author Affiliations

1 Department of Mathematics, Faculty of Applied Science, King Mongkut’s University of Technology North Bangkok, Bangkok, 10800, Thailand

2 Centre of Excellence in Mathematics, CHE, Sri Ayutthaya Road, Bangkok, 10400, Thailand

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Advances in Difference Equations 2012, 2012:93  doi:10.1186/1687-1847-2012-93


The electronic version of this article is the complete one and can be found online at: http://www.advancesindifferenceequations.com/content/2012/1/93


Received:9 February 2012
Accepted:8 June 2012
Published:28 June 2012

© 2012 Sudsutad and Tariboon; 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

This article studies a boundary value problem of nonlinear fractional differential equations with three-point fractional integral boundary conditions. Some new existence results are obtained by applying standard fixed point theorems. As an application, we give two examples that illustrate our results.

MSC: 26A33, 34B15.

Keywords:
existence; Caputo fractional derivative; Riemann-Liouville fractional integral; boundary value problem

1 Introduction

Fractional differential equations have recently proved to be valuable tools in the modelling of many phenomena in various field of science and applications, such as physics, mechanics, chemistry, biology, economics, control theory, aerodynamics, engineering, etc. See [1-6]. There has been a significant development in the theory of initial and boundary value problems for nonlinear fractional differential equations; see, for example, [7-15].

Ahmad and co-authors have studied the existence and uniqueness of solutions of nonlinear fractional differential and integro-differential equations for a variety of boundary conditions using standard fixed-point theorems and Leray-Schauder degree theory. Ahmad et al.[16] discusses the existence and uniqueness of solutions of fractional integro-differential equations for fractional nonlocal integral boundary conditions. Ahmad et al.[17] and references therein give details of recent work on the properties of solutions of sequential fractional differential equations. Ahmad et al.[18] considers solutions of fractional differential equations with non-separated type integral boundary conditions. In Ahmad et al.[19], the Krasnoselskii fixed point theorem and the contraction mapping principle are used to prove the existence of solutions of the nonlinear Langevin equation with two fractional orders for a number of different intervals. Ahmad et al.[20] discusses the existence and uniqueness of solutions of nonlinear fractional differential equations with three-point integral boundary conditions.

Cabada et al.[21] have also studied properties of solutions of nonlinear fractional differential equations. They used the properties of the associated Green’s function and the Guo-Krasnosellskii fixed-point theorem to investigate the existence of positive solutions of nonlinear fractional differential equations with integral boundary-value conditions.

Motivated by the papers [16] and [20], this article is concerned with the existence and uniqueness of solutions for a boundary value problem of nonlinear fractional differential equations with three-point fractional integral boundary conditions given by

(1.1)

(1.2)

where <a onClick="popup('http://www.advancesindifferenceequations.com/content/2012/1/93/mathml/M3','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2012/1/93/mathml/M3">View MathML</a> denotes the Caputo fractional derivative of order q, <a onClick="popup('http://www.advancesindifferenceequations.com/content/2012/1/93/mathml/M4','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2012/1/93/mathml/M4">View MathML</a> is the Riemann-Liouville fractional integral of order <a onClick="popup('http://www.advancesindifferenceequations.com/content/2012/1/93/mathml/M5','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2012/1/93/mathml/M5">View MathML</a> is a continuous function and <a onClick="popup('http://www.advancesindifferenceequations.com/content/2012/1/93/mathml/M6','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2012/1/93/mathml/M6">View MathML</a> is such that <a onClick="popup('http://www.advancesindifferenceequations.com/content/2012/1/93/mathml/M7','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2012/1/93/mathml/M7">View MathML</a>. By <a onClick="popup('http://www.advancesindifferenceequations.com/content/2012/1/93/mathml/M8','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2012/1/93/mathml/M8">View MathML</a> we denote the Banach space of all continuous functions from <a onClick="popup('http://www.advancesindifferenceequations.com/content/2012/1/93/mathml/M9','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2012/1/93/mathml/M9">View MathML</a> into ℝ with the norm

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2012/1/93/mathml/M10','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2012/1/93/mathml/M10">View MathML</a>

We note that if <a onClick="popup('http://www.advancesindifferenceequations.com/content/2012/1/93/mathml/M11','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2012/1/93/mathml/M11">View MathML</a>, then condition (1.2) reduces to the usual three-point integral condition. In such a case, the boundary condition corresponds to the area under the curve of solutions <a onClick="popup('http://www.advancesindifferenceequations.com/content/2012/1/93/mathml/M12','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2012/1/93/mathml/M12">View MathML</a> from <a onClick="popup('http://www.advancesindifferenceequations.com/content/2012/1/93/mathml/M13','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2012/1/93/mathml/M13">View MathML</a> to <a onClick="popup('http://www.advancesindifferenceequations.com/content/2012/1/93/mathml/M14','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2012/1/93/mathml/M14">View MathML</a>.

2 Preliminaries

In this section, we introduce notations, definitions of fractional calculus and prove a lemma before stating our main results.

Definition 2.1 For a continuous function <a onClick="popup('http://www.advancesindifferenceequations.com/content/2012/1/93/mathml/M15','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2012/1/93/mathml/M15">View MathML</a>, the Caputo derivative of fractional order q is defined as

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2012/1/93/mathml/M16','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2012/1/93/mathml/M16">View MathML</a>

provided that <a onClick="popup('http://www.advancesindifferenceequations.com/content/2012/1/93/mathml/M17','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2012/1/93/mathml/M17">View MathML</a> exists, where <a onClick="popup('http://www.advancesindifferenceequations.com/content/2012/1/93/mathml/M18','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2012/1/93/mathml/M18">View MathML</a> denotes the integer part of the real number q.

Definition 2.2 The Riemann-Liouville fractional integral of order q for a continuous function <a onClick="popup('http://www.advancesindifferenceequations.com/content/2012/1/93/mathml/M19','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2012/1/93/mathml/M19">View MathML</a> is defined as

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2012/1/93/mathml/M20','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2012/1/93/mathml/M20">View MathML</a>

provided that such integral exists.

Definition 2.3 The Riemann-Liouville fractional derivative of order q for a continuous function <a onClick="popup('http://www.advancesindifferenceequations.com/content/2012/1/93/mathml/M19','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2012/1/93/mathml/M19">View MathML</a> is defined by

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2012/1/93/mathml/M22','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2012/1/93/mathml/M22">View MathML</a>

provided that the right-hand side is pointwise defined on <a onClick="popup('http://www.advancesindifferenceequations.com/content/2012/1/93/mathml/M23','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2012/1/93/mathml/M23">View MathML</a>.

Furthermore, we note that the Riemann-Liouville fractional derivative of a constant is usually nonzero which can cause serious problems in real would applications. Actually, the relationship between the two-types of fractional derivative is as follows

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2012/1/93/mathml/M24','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2012/1/93/mathml/M24">View MathML</a>

So, we prefer to use Caputo’s definition which gives better results than those of Riemann-Liouville.

Lemma 2.1[3]

Let<a onClick="popup('http://www.advancesindifferenceequations.com/content/2012/1/93/mathml/M25','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2012/1/93/mathml/M25">View MathML</a>, then the fractional differential equation

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2012/1/93/mathml/M26','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2012/1/93/mathml/M26">View MathML</a>

has solution

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2012/1/93/mathml/M27','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2012/1/93/mathml/M27">View MathML</a>

wherenis the smallest integer greater than or equal toq.

Lemma 2.2[3]

Let<a onClick="popup('http://www.advancesindifferenceequations.com/content/2012/1/93/mathml/M25','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2012/1/93/mathml/M25">View MathML</a>, then

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2012/1/93/mathml/M29','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2012/1/93/mathml/M29">View MathML</a>

for some<a onClick="popup('http://www.advancesindifferenceequations.com/content/2012/1/93/mathml/M30','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2012/1/93/mathml/M30">View MathML</a>, <a onClick="popup('http://www.advancesindifferenceequations.com/content/2012/1/93/mathml/M31','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2012/1/93/mathml/M31">View MathML</a>wherenis the smallest integer greater than or equal to q.

Lemma 2.3Let<a onClick="popup('http://www.advancesindifferenceequations.com/content/2012/1/93/mathml/M32','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2012/1/93/mathml/M32">View MathML</a>, <a onClick="popup('http://www.advancesindifferenceequations.com/content/2012/1/93/mathml/M33','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2012/1/93/mathml/M33">View MathML</a>. Then for<a onClick="popup('http://www.advancesindifferenceequations.com/content/2012/1/93/mathml/M34','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2012/1/93/mathml/M34">View MathML</a>, the problem

(2.1)

(2.2)

has a unique solution

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2012/1/93/mathml/M37','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2012/1/93/mathml/M37">View MathML</a>

(2.3)

Proof By applying Lemma 2.2, we may reduce (2.1) to an equivalent integral equation

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2012/1/93/mathml/M38','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2012/1/93/mathml/M38">View MathML</a>

(2.4)

for some <a onClick="popup('http://www.advancesindifferenceequations.com/content/2012/1/93/mathml/M39','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2012/1/93/mathml/M39">View MathML</a>.

From <a onClick="popup('http://www.advancesindifferenceequations.com/content/2012/1/93/mathml/M40','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2012/1/93/mathml/M40">View MathML</a>, it follows <a onClick="popup('http://www.advancesindifferenceequations.com/content/2012/1/93/mathml/M41','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2012/1/93/mathml/M41">View MathML</a>. Using the Riemann-Liouville integral of order p for (2.4), we have

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2012/1/93/mathml/M42','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2012/1/93/mathml/M42">View MathML</a>

The second condition of (2.2) implies that

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2012/1/93/mathml/M43','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2012/1/93/mathml/M43">View MathML</a>

Thus,

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2012/1/93/mathml/M44','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2012/1/93/mathml/M44">View MathML</a>

Substituting the values of <a onClick="popup('http://www.advancesindifferenceequations.com/content/2012/1/93/mathml/M45','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2012/1/93/mathml/M45">View MathML</a> and <a onClick="popup('http://www.advancesindifferenceequations.com/content/2012/1/93/mathml/M46','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2012/1/93/mathml/M46">View MathML</a> in (2.4), we obtain the solution (2.3). □

In the following, for the sake of convenience, set

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2012/1/93/mathml/M47','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2012/1/93/mathml/M47">View MathML</a>

(2.5)

3 Main results

Now we are in the position to establish the main results.

Theorem 3.1Assume that there exists a constant<a onClick="popup('http://www.advancesindifferenceequations.com/content/2012/1/93/mathml/M48','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2012/1/93/mathml/M48">View MathML</a>such that

(<a onClick="popup('http://www.advancesindifferenceequations.com/content/2012/1/93/mathml/M49','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2012/1/93/mathml/M49">View MathML</a>) <a onClick="popup('http://www.advancesindifferenceequations.com/content/2012/1/93/mathml/M50','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2012/1/93/mathml/M50">View MathML</a>, for each<a onClick="popup('http://www.advancesindifferenceequations.com/content/2012/1/93/mathml/M51','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2012/1/93/mathml/M51">View MathML</a>, and all<a onClick="popup('http://www.advancesindifferenceequations.com/content/2012/1/93/mathml/M52','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2012/1/93/mathml/M52">View MathML</a>.

If<a onClick="popup('http://www.advancesindifferenceequations.com/content/2012/1/93/mathml/M53','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2012/1/93/mathml/M53">View MathML</a>, where Λ is defined by (2.5), then the BVP (1.1)-(1.2) has a unique solution on<a onClick="popup('http://www.advancesindifferenceequations.com/content/2012/1/93/mathml/M54','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2012/1/93/mathml/M54">View MathML</a>.

Proof Transform the BVP (1.1)-(1.2) into a fixed point problem. In view of Lemma 2.3, we consider the operator <a onClick="popup('http://www.advancesindifferenceequations.com/content/2012/1/93/mathml/M55','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2012/1/93/mathml/M55">View MathML</a> defined by

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2012/1/93/mathml/M56','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2012/1/93/mathml/M56">View MathML</a>

Obviously, the fixed points of the operator F are solution of the problem (1.1)-(1.2). We shall use the Banach fixed point theorem to prove that F has a fixed point. We will show that F is a contraction.

Let <a onClick="popup('http://www.advancesindifferenceequations.com/content/2012/1/93/mathml/M57','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2012/1/93/mathml/M57">View MathML</a>. Then, for each <a onClick="popup('http://www.advancesindifferenceequations.com/content/2012/1/93/mathml/M51','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2012/1/93/mathml/M51">View MathML</a> we have

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2012/1/93/mathml/M59','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2012/1/93/mathml/M59">View MathML</a>

By using the property of beta function, <a onClick="popup('http://www.advancesindifferenceequations.com/content/2012/1/93/mathml/M60','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2012/1/93/mathml/M60">View MathML</a>, we have

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2012/1/93/mathml/M61','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2012/1/93/mathml/M61">View MathML</a>

Thus

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2012/1/93/mathml/M62','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2012/1/93/mathml/M62">View MathML</a>

Therefore, F is a contraction. Hence, by Banach fixed point theorem, we get that F has a fixed point which is a solution of the problem (1.1)-(1.2). □

The following result is based on Schaefer’s fixed point theorem.

Theorem 3.2Assume that:

(<a onClick="popup('http://www.advancesindifferenceequations.com/content/2012/1/93/mathml/M63','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2012/1/93/mathml/M63">View MathML</a>) The function<a onClick="popup('http://www.advancesindifferenceequations.com/content/2012/1/93/mathml/M64','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2012/1/93/mathml/M64">View MathML</a>is continuous.

(<a onClick="popup('http://www.advancesindifferenceequations.com/content/2012/1/93/mathml/M65','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2012/1/93/mathml/M65">View MathML</a>) There exists a constant<a onClick="popup('http://www.advancesindifferenceequations.com/content/2012/1/93/mathml/M66','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2012/1/93/mathml/M66">View MathML</a>such that<a onClick="popup('http://www.advancesindifferenceequations.com/content/2012/1/93/mathml/M67','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2012/1/93/mathml/M67">View MathML</a>for each<a onClick="popup('http://www.advancesindifferenceequations.com/content/2012/1/93/mathml/M68','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2012/1/93/mathml/M68">View MathML</a>and all<a onClick="popup('http://www.advancesindifferenceequations.com/content/2012/1/93/mathml/M69','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2012/1/93/mathml/M69">View MathML</a>.

Then the BVP (1.1)-(1.2) has at least one solution on<a onClick="popup('http://www.advancesindifferenceequations.com/content/2012/1/93/mathml/M54','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2012/1/93/mathml/M54">View MathML</a>.

Proof We shall use Schaefer’s fixed point theorem to prove that F has a fixed point. We divide the proof into four steps.

Step I. Continuity of F.

Let <a onClick="popup('http://www.advancesindifferenceequations.com/content/2012/1/93/mathml/M71','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2012/1/93/mathml/M71">View MathML</a> be a sequence such that <a onClick="popup('http://www.advancesindifferenceequations.com/content/2012/1/93/mathml/M72','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2012/1/93/mathml/M72">View MathML</a> in <a onClick="popup('http://www.advancesindifferenceequations.com/content/2012/1/93/mathml/M73','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2012/1/93/mathml/M73">View MathML</a>. Then for each <a onClick="popup('http://www.advancesindifferenceequations.com/content/2012/1/93/mathml/M51','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2012/1/93/mathml/M51">View MathML</a>

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2012/1/93/mathml/M75','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2012/1/93/mathml/M75">View MathML</a>

Since f is continuous function, then <a onClick="popup('http://www.advancesindifferenceequations.com/content/2012/1/93/mathml/M76','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2012/1/93/mathml/M76">View MathML</a> as <a onClick="popup('http://www.advancesindifferenceequations.com/content/2012/1/93/mathml/M77','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2012/1/93/mathml/M77">View MathML</a>. This means that F is continuous.

Step II. F maps bounded sets into bounded sets in <a onClick="popup('http://www.advancesindifferenceequations.com/content/2012/1/93/mathml/M78','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2012/1/93/mathml/M78">View MathML</a>.

So, let us prove that for any <a onClick="popup('http://www.advancesindifferenceequations.com/content/2012/1/93/mathml/M79','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2012/1/93/mathml/M79">View MathML</a>, there exists a positive constant l such that for each <a onClick="popup('http://www.advancesindifferenceequations.com/content/2012/1/93/mathml/M80','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2012/1/93/mathml/M80">View MathML</a>, we have <a onClick="popup('http://www.advancesindifferenceequations.com/content/2012/1/93/mathml/M81','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2012/1/93/mathml/M81">View MathML</a>. Indeed, we have for any <a onClick="popup('http://www.advancesindifferenceequations.com/content/2012/1/93/mathml/M82','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2012/1/93/mathml/M82">View MathML</a>

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2012/1/93/mathml/M83','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2012/1/93/mathml/M83">View MathML</a>

which in view of (<a onClick="popup('http://www.advancesindifferenceequations.com/content/2012/1/93/mathml/M65','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2012/1/93/mathml/M65">View MathML</a>) gives

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2012/1/93/mathml/M85','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2012/1/93/mathml/M85">View MathML</a>

Hence, we deduce that

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2012/1/93/mathml/M86','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2012/1/93/mathml/M86">View MathML</a>

Thus,

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2012/1/93/mathml/M87','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2012/1/93/mathml/M87">View MathML</a>

Step III. We prove that <a onClick="popup('http://www.advancesindifferenceequations.com/content/2012/1/93/mathml/M88','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2012/1/93/mathml/M88">View MathML</a> is equicontinuous with <a onClick="popup('http://www.advancesindifferenceequations.com/content/2012/1/93/mathml/M89','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2012/1/93/mathml/M89">View MathML</a> defined as in Step II.

Let <a onClick="popup('http://www.advancesindifferenceequations.com/content/2012/1/93/mathml/M90','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2012/1/93/mathml/M90">View MathML</a> and <a onClick="popup('http://www.advancesindifferenceequations.com/content/2012/1/93/mathml/M82','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2012/1/93/mathml/M82">View MathML</a>, then

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2012/1/93/mathml/M92','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2012/1/93/mathml/M92">View MathML</a>

Actually, as <a onClick="popup('http://www.advancesindifferenceequations.com/content/2012/1/93/mathml/M93','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2012/1/93/mathml/M93">View MathML</a>, the right-hand side of the above inequality tends to zero. As a consequence of Steps I to III together with the Arzela-Ascoli theorem, we get that <a onClick="popup('http://www.advancesindifferenceequations.com/content/2012/1/93/mathml/M94','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2012/1/93/mathml/M94">View MathML</a> is completely continuous.

Step IV. A priori bounds.

We show that the set

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2012/1/93/mathml/M95','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2012/1/93/mathml/M95">View MathML</a>

is bounded.

Let <a onClick="popup('http://www.advancesindifferenceequations.com/content/2012/1/93/mathml/M96','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2012/1/93/mathml/M96">View MathML</a>. Then <a onClick="popup('http://www.advancesindifferenceequations.com/content/2012/1/93/mathml/M97','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2012/1/93/mathml/M97">View MathML</a> for some <a onClick="popup('http://www.advancesindifferenceequations.com/content/2012/1/93/mathml/M98','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2012/1/93/mathml/M98">View MathML</a>. Thus, for each <a onClick="popup('http://www.advancesindifferenceequations.com/content/2012/1/93/mathml/M51','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2012/1/93/mathml/M51">View MathML</a> we have

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2012/1/93/mathml/M100','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2012/1/93/mathml/M100">View MathML</a>

This implies by (<a onClick="popup('http://www.advancesindifferenceequations.com/content/2012/1/93/mathml/M65','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2012/1/93/mathml/M65">View MathML</a>) that for all <a onClick="popup('http://www.advancesindifferenceequations.com/content/2012/1/93/mathml/M51','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2012/1/93/mathml/M51">View MathML</a>, we get

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2012/1/93/mathml/M103','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2012/1/93/mathml/M103">View MathML</a>

Hence, we deduce that

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2012/1/93/mathml/M104','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2012/1/93/mathml/M104">View MathML</a>

This shows that the set E is bounded. As a consequence of Schaefer’s fixed point theorem, we conclude that F has a fixed point which is a solution of the problem (1.1)-(1.2). □

4 Examples

In this section, in order to illustrate our results, we consider two examples.

Example 4.1 Consider the following three-point fractional integral boundary value problem

(4.1)

(4.2)

Set <a onClick="popup('http://www.advancesindifferenceequations.com/content/2012/1/93/mathml/M107','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2012/1/93/mathml/M107">View MathML</a>, <a onClick="popup('http://www.advancesindifferenceequations.com/content/2012/1/93/mathml/M108','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2012/1/93/mathml/M108">View MathML</a>, <a onClick="popup('http://www.advancesindifferenceequations.com/content/2012/1/93/mathml/M109','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2012/1/93/mathml/M109">View MathML</a>, <a onClick="popup('http://www.advancesindifferenceequations.com/content/2012/1/93/mathml/M110','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2012/1/93/mathml/M110">View MathML</a> and <a onClick="popup('http://www.advancesindifferenceequations.com/content/2012/1/93/mathml/M111','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2012/1/93/mathml/M111">View MathML</a>. Since <a onClick="popup('http://www.advancesindifferenceequations.com/content/2012/1/93/mathml/M112','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2012/1/93/mathml/M112">View MathML</a>, then, (<a onClick="popup('http://www.advancesindifferenceequations.com/content/2012/1/93/mathml/M49','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2012/1/93/mathml/M49">View MathML</a>) is satisfied with <a onClick="popup('http://www.advancesindifferenceequations.com/content/2012/1/93/mathml/M114','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2012/1/93/mathml/M114">View MathML</a>. We can show that

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2012/1/93/mathml/M115','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2012/1/93/mathml/M115">View MathML</a>

Hence, by Theorem 3.1, the boundary value problem (4.1)-(4.2) has a unique solution on <a onClick="popup('http://www.advancesindifferenceequations.com/content/2012/1/93/mathml/M54','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2012/1/93/mathml/M54">View MathML</a>.

Example 4.2 Consider the following three-point fractional integral boundary value problem

(4.3)

(4.4)

Set <a onClick="popup('http://www.advancesindifferenceequations.com/content/2012/1/93/mathml/M119','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2012/1/93/mathml/M119">View MathML</a>, <a onClick="popup('http://www.advancesindifferenceequations.com/content/2012/1/93/mathml/M120','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2012/1/93/mathml/M120">View MathML</a>, <a onClick="popup('http://www.advancesindifferenceequations.com/content/2012/1/93/mathml/M121','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2012/1/93/mathml/M121">View MathML</a>, <a onClick="popup('http://www.advancesindifferenceequations.com/content/2012/1/93/mathml/M122','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2012/1/93/mathml/M122">View MathML</a> and <a onClick="popup('http://www.advancesindifferenceequations.com/content/2012/1/93/mathml/M123','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2012/1/93/mathml/M123">View MathML</a>. Clearly <a onClick="popup('http://www.advancesindifferenceequations.com/content/2012/1/93/mathml/M124','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2012/1/93/mathml/M124">View MathML</a>.

Hence, all the conditions of Theorem 3.2 are satisfied and consequently the problem (4.1)-(4.2) has at least one solution.

Competing interests

The authors declare that they have no competing interests.

Authors’ contributions

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

Acknowledgements

The authors thank the referees for several useful remarks and interesting comments. This research is supported by the Centre of Excellence in Mathematics, the Commission on Higher Education, Thailand.

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