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A study of Riemann-Liouville fractional nonlocal integral boundary value problems

Bashir Ahmad1*, Ahmed Alsaedi1, Afrah Assolami1 and Ravi P Agarwal12

Author Affiliations

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

2 Department of Mathematics, Texas A&M University, Kingsville, TX, 78363-8202, USA

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Boundary Value Problems 2013, 2013:274  doi:10.1186/1687-2770-2013-274


The electronic version of this article is the complete one and can be found online at: http://www.boundaryvalueproblems.com/content/2013/1/274


Received:29 July 2013
Accepted:25 November 2013
Published:13 December 2013

© 2013 Ahmad et al.; licensee Springer.

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

Abstract

In this paper, we discuss the existence and uniqueness of solutions for a Riemann-Liouville type fractional differential equation with nonlocal four-point Riemann-Liouville fractional-integral boundary conditions by means of classical fixed point theorems. An illustration of main results is also presented with the aid of some examples.

MSC: 34A08, 34B10, 34B15.

Keywords:
fractional differential equations; nonlocal fractional-integral boundary conditions; existence; fixed point

1 Introduction

In recent years, boundary value problems of nonlinear fractional differential equations with a variety of boundary conditions have been investigated by many researchers. Fractional differential equations appear naturally in various fields of science and engineering and constitute an important field of research [1-4]. As a matter of fact, fractional derivatives provide an excellent tool for the description of memory and hereditary properties of various materials and processes. This is one of the characteristics of fractional-order differential operators that contributes to the popularity of the subject and has motivated many researchers and modelers to shift their focus from classical models to fractional order models. In consequence, there has been a significant progress in the theoretical analysis like periodicity, asymptotic behavior and numerical methods for fractional differential equations. Some recent work on the topic can be found in [5-20] and the references therein.

Fractional boundary conditions (FBC) involving fractional derivative <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/274/mathml/M1','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/274/mathml/M1">View MathML</a> of order <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/274/mathml/M2','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/274/mathml/M2">View MathML</a> describe an intermediate boundary between the perfect electric conductor (PEC) and the perfect magnetic conductor (PMC), whereas <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/274/mathml/M3','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/274/mathml/M3">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/274/mathml/M4','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/274/mathml/M4">View MathML</a> in FBC correspond to PEC and PMC, respectively. Fractional boundary conditions (FBC) are also matched with impedance boundary conditions (IBC) in the sense that the fractional order <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/274/mathml/M3','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/274/mathml/M3">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/274/mathml/M4','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/274/mathml/M4">View MathML</a> in FBC correspond to the value of impedance <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/274/mathml/M7','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/274/mathml/M7">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/274/mathml/M8','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/274/mathml/M8">View MathML</a>. Recall that the value of the impedance Z varies from 0 for PEC to i∞ for PMC. For more details, see [21].

In [22], the authors recently studied a problem of Riemann-Liouville fractional differential equations with fractional boundary conditions:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/274/mathml/M9','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/274/mathml/M9">View MathML</a>

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/274/mathml/M1','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/274/mathml/M1">View MathML</a> denotes the Riemann-Liouville fractional derivative of order α and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/274/mathml/M11','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/274/mathml/M11">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/274/mathml/M12','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/274/mathml/M12">View MathML</a>.

In this paper, motivated by [22], we study a fully Riemann-Liouville fractional nonlocal integral boundary value problem given by

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/274/mathml/M13','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/274/mathml/M13">View MathML</a>

(1.1)

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/274/mathml/M1','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/274/mathml/M1">View MathML</a> denotes the Riemann-Liouville fractional derivative of order α, f is a given continuous function, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/274/mathml/M15','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/274/mathml/M15">View MathML</a> denotes the Riemann-Liouville integral of order β, and a, A, b, and B are real constants.

The paper is organized as follows. In Section 2, we establish an auxiliary lemma which is needed to define the solutions of the given problem. Section 3 contains main results. In Section 4, we discuss some examples for the illustration of the main results.

2 Preliminaries

Let us recall some basic definitions of fractional theory.

Definition 2.1 The Riemann-Liouville fractional integral of order α for a continuous function <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/274/mathml/M16','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/274/mathml/M16">View MathML</a> is defined as

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/274/mathml/M17','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/274/mathml/M17">View MathML</a>

provided the integral exists.

Definition 2.2 For a continuous function <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/274/mathml/M18','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/274/mathml/M18">View MathML</a>, the Riemann-Liouville derivative of fractional order <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/274/mathml/M19','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/274/mathml/M19">View MathML</a> is defined as

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/274/mathml/M20','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/274/mathml/M20">View MathML</a>

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/274/mathml/M21','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/274/mathml/M21">View MathML</a>, where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/274/mathml/M22','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/274/mathml/M22">View MathML</a> denotes the integer part of the real number α.

Lemma 2.1For<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/274/mathml/M23','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/274/mathml/M23">View MathML</a>, the solution of<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/274/mathml/M24','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/274/mathml/M24">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/274/mathml/M25','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/274/mathml/M25">View MathML</a>subject to the boundary conditions given by (1.1) is

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/274/mathml/M26','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/274/mathml/M26">View MathML</a>

(2.1)

where

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/274/mathml/M27','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/274/mathml/M27">View MathML</a>

(2.2)

Proof For arbitrary constants <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/274/mathml/M28','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/274/mathml/M28">View MathML</a>, it is well known that the general solution of the equation <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/274/mathml/M24','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/274/mathml/M24">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/274/mathml/M23','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/274/mathml/M23">View MathML</a>, can be written as

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/274/mathml/M31','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/274/mathml/M31">View MathML</a>

(2.3)

From (2.3), we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/274/mathml/M32','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/274/mathml/M32">View MathML</a>

(2.4)

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/274/mathml/M33','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/274/mathml/M33">View MathML</a>

(2.5)

where ϱ denotes ξ or η. Applying the given boundary conditions, we get

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/274/mathml/M34','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/274/mathml/M34">View MathML</a>

(2.6)

Solving the system of equations (2.6) for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/274/mathml/M35','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/274/mathml/M35">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/274/mathml/M36','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/274/mathml/M36">View MathML</a>, we find that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/274/mathml/M37','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/274/mathml/M37">View MathML</a>

Substituting these values in (2.3), we get

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/274/mathml/M38','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/274/mathml/M38">View MathML</a>

(2.7)

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/274/mathml/M39','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/274/mathml/M39">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/274/mathml/M40','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/274/mathml/M40">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/274/mathml/M41','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/274/mathml/M41">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/274/mathml/M42','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/274/mathml/M42">View MathML</a> and δ are given by (2.2). This completes the proof. □

3 Existence results

Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/274/mathml/M43','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/274/mathml/M43">View MathML</a> denote the Banach space of all continuous real-valued functions defined on <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/274/mathml/M44','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/274/mathml/M44">View MathML</a> with the norm <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/274/mathml/M45','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/274/mathml/M45">View MathML</a>. For <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/274/mathml/M25','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/274/mathml/M25">View MathML</a>, define <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/274/mathml/M47','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/274/mathml/M47">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/274/mathml/M48','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/274/mathml/M48">View MathML</a>, and let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/274/mathml/M49','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/274/mathml/M49">View MathML</a> be the space of all functions <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/274/mathml/M50','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/274/mathml/M50">View MathML</a> such that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/274/mathml/M51','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/274/mathml/M51">View MathML</a> which turns out to be a Banach space when endowed with the norm <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/274/mathml/M52','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/274/mathml/M52">View MathML</a>.

Let us define an operator <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/274/mathml/M53','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/274/mathml/M53">View MathML</a> as

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/274/mathml/M54','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/274/mathml/M54">View MathML</a>

(3.1)

Observe that problem (1.1) has solutions only if the operator has fixed points.

To establish the first existence result, we need the following fixed point theorem.

Theorem 3.1 ([23])

LetEbe a Banach space. Let<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/274/mathml/M56','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/274/mathml/M56">View MathML</a>be a completely continuous operator, and let the set<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/274/mathml/M57','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/274/mathml/M57">View MathML</a>be bounded. Then the operatorThas a fixed point inE.

Theorem 3.2Assume that there exists a constant<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/274/mathml/M58','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/274/mathml/M58">View MathML</a>such that<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/274/mathml/M59','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/274/mathml/M59">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/274/mathml/M60','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/274/mathml/M60">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/274/mathml/M61','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/274/mathml/M61">View MathML</a>. Then problem (1.1) has at least one solution in the space<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/274/mathml/M62','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/274/mathml/M62">View MathML</a>.

Proof As a first step, we show that the operator is completely continuous. The continuity of follows from the continuity of f. Let ℋ be a bounded set in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/274/mathml/M62','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/274/mathml/M62">View MathML</a>. Hence ℋ is bounded on <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/274/mathml/M43','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/274/mathml/M43">View MathML</a>. Then, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/274/mathml/M67','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/274/mathml/M67">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/274/mathml/M68','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/274/mathml/M68">View MathML</a>, we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/274/mathml/M69','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/274/mathml/M69">View MathML</a>

which implies that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/274/mathml/M70','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/274/mathml/M70">View MathML</a>. Hence <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/274/mathml/M71','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/274/mathml/M71">View MathML</a> is uniformly bounded. Also, for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/274/mathml/M72','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/274/mathml/M72">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/274/mathml/M73','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/274/mathml/M73">View MathML</a>, we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/274/mathml/M74','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/274/mathml/M74">View MathML</a>

Thus <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/274/mathml/M75','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/274/mathml/M75">View MathML</a> and hence <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/274/mathml/M71','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/274/mathml/M71">View MathML</a> is equicontinuous. So, by the Arzela-Ascoli theorem, is completely continuous. Next, we consider the set

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/274/mathml/M78','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/274/mathml/M78">View MathML</a>

and show that V is bounded. For <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/274/mathml/M79','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/274/mathml/M79">View MathML</a>, we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/274/mathml/M80','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/274/mathml/M80">View MathML</a>

This implies that the set V is bounded independently of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/274/mathml/M81','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/274/mathml/M81">View MathML</a>. Therefore, Theorem 3.1 applies and problem (1.1) has at least one solution on <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/274/mathml/M82','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/274/mathml/M82">View MathML</a>. This completes the proof. □

Theorem 3.3Assume that there exists a constant<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/274/mathml/M83','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/274/mathml/M83">View MathML</a>such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/274/mathml/M84','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/274/mathml/M84">View MathML</a>

then problem (1.1) has a unique solution in<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/274/mathml/M62','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/274/mathml/M62">View MathML</a>if<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/274/mathml/M86','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/274/mathml/M86">View MathML</a>, where

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/274/mathml/M87','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/274/mathml/M87">View MathML</a>

(3.2)

Proof For every <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/274/mathml/M25','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/274/mathml/M25">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/274/mathml/M89','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/274/mathml/M89">View MathML</a>, we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/274/mathml/M90','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/274/mathml/M90">View MathML</a>

By the definition of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/274/mathml/M91','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/274/mathml/M91">View MathML</a>, we obtain

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/274/mathml/M92','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/274/mathml/M92">View MathML</a>

It follows that is a contraction. Hence, by the Banach contraction theorem, problem (1.1) has a unique solution in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/274/mathml/M62','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/274/mathml/M62">View MathML</a>. This completes the proof. □

Our next existence result is based on Leray-Schauder nonlinear alternative [24].

Lemma 3.1 (Leray-Schauder’s nonlinear alternative type)

LetEbe a Banach space, Mbe a closed, convex subset ofE, Ube an open subset ofCand<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/274/mathml/M95','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/274/mathml/M95">View MathML</a>. Suppose that<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/274/mathml/M96','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/274/mathml/M96">View MathML</a>is a continuous, compact (that is, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/274/mathml/M97','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/274/mathml/M97">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/2013/1/274/mathml/M98','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/274/mathml/M98">View MathML</a>or (ii) there are<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/274/mathml/M99','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/274/mathml/M99">View MathML</a>and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/274/mathml/M100','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/274/mathml/M100">View MathML</a>with<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/274/mathml/M101','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/274/mathml/M101">View MathML</a>.

Theorem 3.4Let<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/274/mathml/M102','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/274/mathml/M102">View MathML</a>be a continuous function. Furthermore, assume that:

(A1) There exist a function<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/274/mathml/M103','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/274/mathml/M103">View MathML</a>and a nondecreasing function<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/274/mathml/M104','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/274/mathml/M104">View MathML</a>such that<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/274/mathml/M105','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/274/mathml/M105">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/274/mathml/M106','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/274/mathml/M106">View MathML</a>;

(A2) There exists a constant<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/274/mathml/M58','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/274/mathml/M58">View MathML</a>such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/274/mathml/M108','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/274/mathml/M108">View MathML</a>

whereνis given by (3.2).

Then boundary value problem (1.1) has at least one solution.

Proof First we shall show that the operator defined by (3.1) maps bounded sets into bounded ones in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/274/mathml/M110','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/274/mathml/M110">View MathML</a>. For <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/274/mathml/M111','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/274/mathml/M111">View MathML</a>, let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/274/mathml/M112','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/274/mathml/M112">View MathML</a> be a bounded set in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/274/mathml/M110','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/274/mathml/M110">View MathML</a>. Then, for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/274/mathml/M114','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/274/mathml/M114">View MathML</a>, we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/274/mathml/M115','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/274/mathml/M115">View MathML</a>

where ν is given by (3.2).

Next, we shall show that the operator maps bounded sets into equicontinuous sets. Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/274/mathml/M117','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/274/mathml/M117">View MathML</a> with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/274/mathml/M118','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/274/mathml/M118">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/274/mathml/M114','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/274/mathml/M114">View MathML</a>. Then we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/274/mathml/M120','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/274/mathml/M120">View MathML</a>

which tends to zero independently of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/274/mathml/M114','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/274/mathml/M114">View MathML</a> as <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/274/mathml/M122','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/274/mathml/M122">View MathML</a>. Thus is completely continuous. Now let u be a solution of problem (1.1), then for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/274/mathml/M124','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/274/mathml/M124">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/274/mathml/M100','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/274/mathml/M100">View MathML</a>, we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/274/mathml/M126','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/274/mathml/M126">View MathML</a>

(3.3)

which can be rewritten as

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/274/mathml/M127','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/274/mathml/M127">View MathML</a>

(3.4)

By assumption (A2), there exists M such that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/274/mathml/M128','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/274/mathml/M128">View MathML</a>. Let us set

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/274/mathml/M129','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/274/mathml/M129">View MathML</a>

Note that the operator <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/274/mathml/M130','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/274/mathml/M130">View MathML</a> is completely continuous and by the definition of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/274/mathml/M131','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/274/mathml/M131">View MathML</a>, there is no <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/274/mathml/M132','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/274/mathml/M132">View MathML</a> such that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/274/mathml/M133','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/274/mathml/M133">View MathML</a> for some <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/274/mathml/M100','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/274/mathml/M100">View MathML</a>. In consequence, by Lemma 3.1, we conclude that has at least one fixed point <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/274/mathml/M136','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/274/mathml/M136">View MathML</a>, which is a solution of problem (1.1). □

4 Examples

Example 4.1 Consider the following fractional integral boundary value problem:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/274/mathml/M137','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/274/mathml/M137">View MathML</a>

(4.1)

Since

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/274/mathml/M138','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/274/mathml/M138">View MathML</a>

therefore, Theorem 3.2 applies and problem (4.1) has at least one solution on <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/274/mathml/M139','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/274/mathml/M139">View MathML</a>.

Example 4.2 Consider the problem

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/274/mathml/M140','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/274/mathml/M140">View MathML</a>

(4.2)

Here <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/274/mathml/M141','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/274/mathml/M141">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/274/mathml/M142','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/274/mathml/M142">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/274/mathml/M143','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/274/mathml/M143">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/274/mathml/M144','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/274/mathml/M144">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/274/mathml/M145','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/274/mathml/M145">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/274/mathml/M146','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/274/mathml/M146">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/274/mathml/M147','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/274/mathml/M147">View MathML</a>. Clearly,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/274/mathml/M148','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/274/mathml/M148">View MathML</a>

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/274/mathml/M149','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/274/mathml/M149">View MathML</a> (ν is given by (3.2)) and in consequence, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/274/mathml/M150','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/274/mathml/M150">View MathML</a>. Thus, all the assumptions of Theorem 3.3 are satisfied. Therefore, by the conclusion of Theorem 3.3, there exists a unique solution for problem (4.2).

Competing interests

The authors declare that they have no competing interests.

Authors’ contributions

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

Acknowledgements

This research was partially supported by the Deanship of Scientific Research (DSR), King Abdulaziz University, Jeddah, Saudi Arabia.

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