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Impulsive fractional boundary-value problems with fractional integral jump conditions

Chatthai Thaiprayoon1, Jessada Tariboon1* and Sotiris K Ntouyas2

Author Affiliations

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

2 Department of Mathematics, University of Ioannina, Ioannina, 451 10, Greece

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Boundary Value Problems 2014, 2014:17  doi:10.1186/1687-2770-2014-17

The electronic version of this article is the complete one and can be found online at: http://www.boundaryvalueproblems.com/content/2014/1/17


Received:25 July 2013
Accepted:17 December 2013
Published:15 January 2014

© 2014 Thaiprayoon 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 establish the existence and uniqueness of solutions for impulsive fractional boundary-value problems with fractional integral jump conditions. By using a variety of fixed-point theorems, some new existence and uniqueness results are obtained. Illustrative examples of our results are also presented.

MSC: 34A08, 34B37, 34B15, 34B10.

Keywords:
impulsive fractional differential equations; fractional integral jump conditions; fixed-point theorems

1 Introduction

In this paper, we investigate the following boundary-value problem for impulsive fractional differential equations with fractional integral jump conditions:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/17/mathml/M1','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/17/mathml/M1">View MathML</a>

(1.1)

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/17/mathml/M2','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/17/mathml/M2">View MathML</a> is the Caputo fractional derivative of order α, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/17/mathml/M3','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/17/mathml/M3">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/17/mathml/M4','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/17/mathml/M4">View MathML</a> is a continuous function, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/17/mathml/M5','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/17/mathml/M5">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/17/mathml/M6','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/17/mathml/M6">View MathML</a> with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/17/mathml/M7','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/17/mathml/M7">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/17/mathml/M8','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/17/mathml/M8">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/17/mathml/M9','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/17/mathml/M9">View MathML</a> are constants, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/17/mathml/M10','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/17/mathml/M10">View MathML</a> is the Riemann-Liouville fractional integral of order <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/17/mathml/M11','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/17/mathml/M11">View MathML</a> for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/17/mathml/M12','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/17/mathml/M12">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/17/mathml/M13','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/17/mathml/M13">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/17/mathml/M14','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/17/mathml/M14">View MathML</a>, a, b, c are given constants such that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/17/mathml/M15','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/17/mathml/M15">View MathML</a>.

The integral jump conditions are very general and include many conditions as special cases. In particular, if <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/17/mathml/M16','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/17/mathml/M16">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/17/mathml/M17','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/17/mathml/M17">View MathML</a>, then the impulsive fractional integral of equation (1.1) reduces to

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/17/mathml/M18','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/17/mathml/M18">View MathML</a>

Recently, much attention has been paid to the existence of solutions for fractional differential equations due to its wide application in engineering, economics and other fields. A variety of results on initial- and boundary-value problems of fractional differential equations and inclusions can easily be found in the literature on the topic. For some recent results, we can refer to [1-13] and references cited therein.

On the other hand, integer order impulsive differential equations have become important in recent years as mathematical models of phenomena in both the physical and social sciences. There has a significant development in impulsive theory especially in the area of impulsive differential equations with fixed moments, see for instance [14-24].

In this paper we prove some new existence and uniqueness results by using a variety of fixed-point theorems. In Theorem 3.1 we prove an existence and uniqueness result by using Banach’s contraction principle, in Theorem 3.2 we prove an existence and uniqueness result by using Banach’s contraction principle and Hölder’s inequality, in Theorem 3.3 we prove the existence of a solution by using Krasnoselskii’s fixed-point theorem, while in Theorem 3.4 we prove the existence of a solution via Leray-Schauder’s nonlinear alternative. Leray-Schauder’s degree theory is used in proving the existence result in Theorem 3.5.

The rest of the paper is organized as follows: In Section 2 we recall some preliminaries and present a basic lemma which is used to convert the impulsive fractional boundary-value problem (1.1) into an equivalent integral equation. The main results are presented in Section 3, while illustrative examples are contained in Section 4.

2 Preliminaries

Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/17/mathml/M19','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/17/mathml/M19">View MathML</a> = {<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/17/mathml/M20','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/17/mathml/M20">View MathML</a> is continuous everywhere except for some <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/17/mathml/M21','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/17/mathml/M21">View MathML</a> at which <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/17/mathml/M22','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/17/mathml/M22">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/17/mathml/M23','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/17/mathml/M23">View MathML</a> exist and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/17/mathml/M24','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/17/mathml/M24">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/17/mathml/M25','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/17/mathml/M25">View MathML</a>}. <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/17/mathml/M26','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/17/mathml/M26">View MathML</a> is a Banach space endowed with the norm defined by <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/17/mathml/M27','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/17/mathml/M27">View MathML</a>. Next, we introduce some notations, definitions of fractional calculus [25-27], and we present a preliminary result needed in our proofs later.

Definition 2.1 The Riemann-Liouville fractional integral of order <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/17/mathml/M28','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/17/mathml/M28">View MathML</a> of a function <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/17/mathml/M29','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/17/mathml/M29">View MathML</a> is defined by

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/17/mathml/M30','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/17/mathml/M30">View MathML</a>

where Γ is the Gamma function.

Definition 2.2 The Riemann-Liouville fractional derivative of order <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/17/mathml/M28','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/17/mathml/M28">View MathML</a> of a continuous function <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/17/mathml/M32','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/17/mathml/M32">View MathML</a> is defined by

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/17/mathml/M33','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/17/mathml/M33">View MathML</a>

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/17/mathml/M34','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/17/mathml/M34">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/17/mathml/M35','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/17/mathml/M35">View MathML</a> denotes the integral part of real number α, provided the right-hand side is point-wise defined on <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/17/mathml/M36','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/17/mathml/M36">View MathML</a>.

Definition 2.3 For a continuous function <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/17/mathml/M32','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/17/mathml/M32">View MathML</a>, the Caputo derivative of fractional order α is defined as

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/17/mathml/M38','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/17/mathml/M38">View MathML</a>

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/17/mathml/M34','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/17/mathml/M34">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/17/mathml/M35','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/17/mathml/M35">View MathML</a> denotes the integral part of real number α, provided <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/17/mathml/M41','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/17/mathml/M41">View MathML</a> exists.

Lemma 2.1 ([28])

Let<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/17/mathml/M42','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/17/mathml/M42">View MathML</a>and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/17/mathml/M43','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/17/mathml/M43">View MathML</a>be continuous. A function<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/17/mathml/M44','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/17/mathml/M44">View MathML</a>is a solution of the fractional Cauchy problem

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/17/mathml/M45','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/17/mathml/M45">View MathML</a>

if and only ifxis a solution of the following integral equation:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/17/mathml/M46','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/17/mathml/M46">View MathML</a>

Lemma 2.2Let<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/17/mathml/M47','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/17/mathml/M47">View MathML</a>and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/17/mathml/M15','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/17/mathml/M15">View MathML</a>. The unique solution of the impulsive fractional boundary-value problem (1.1) is given by

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/17/mathml/M49','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/17/mathml/M49">View MathML</a>

(2.1)

Proof For <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/17/mathml/M50','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/17/mathml/M50">View MathML</a>, Riemann-Liouville fractional integrating of order α, from 0 to t, for the first equation of (1.1), we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/17/mathml/M51','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/17/mathml/M51">View MathML</a>

(2.2)

Substituting <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/17/mathml/M52','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/17/mathml/M52">View MathML</a> into (2.2), we get

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/17/mathml/M53','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/17/mathml/M53">View MathML</a>

For <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/17/mathml/M54','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/17/mathml/M54">View MathML</a>, by Lemma 2.1 with the second equation of (1.1), we obtain

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/17/mathml/M55','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/17/mathml/M55">View MathML</a>

If <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/17/mathml/M56','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/17/mathml/M56">View MathML</a> then again from Lemma 2.1, we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/17/mathml/M57','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/17/mathml/M57">View MathML</a>

If <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/17/mathml/M58','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/17/mathml/M58">View MathML</a> then again from Lemma 2.1, we get

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/17/mathml/M59','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/17/mathml/M59">View MathML</a>

In particular, for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/17/mathml/M60','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/17/mathml/M60">View MathML</a>, we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/17/mathml/M61','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/17/mathml/M61">View MathML</a>

(2.3)

From the third equation of (1.1) and (2.3), we get

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/17/mathml/M62','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/17/mathml/M62">View MathML</a>

Therefore, we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/17/mathml/M63','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/17/mathml/M63">View MathML</a>

This completes the proof. □

As in Lemma 2.2, we define an operator <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/17/mathml/M64','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/17/mathml/M64">View MathML</a> by

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/17/mathml/M65','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/17/mathml/M65">View MathML</a>

(2.4)

with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/17/mathml/M15','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/17/mathml/M15">View MathML</a>. It should be noticed that problem (1.1) has solutions if and only if the operator A has fixed points.

3 Main results

We are in a position to establish our main results. In the following subsections we prove existence as well as existence and uniqueness results for the impulsive fractional BVP (1.1) by using a variety of fixed-point theorems.

3.1 Existence and uniqueness results via Banach’s fixed-point theorem

In this subsection we give first an existence and uniqueness result for the impulsive fractional BVP (1.1) by using Banach’s fixed-point theorem.

For convenience, we set

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/17/mathml/M67','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/17/mathml/M67">View MathML</a>

(3.1)

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/17/mathml/M68','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/17/mathml/M68">View MathML</a>

(3.2)

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/17/mathml/M69','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/17/mathml/M69">View MathML</a>

(3.3)

Theorem 3.1Assume the following.

(H1) There exists a constant<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/17/mathml/M70','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/17/mathml/M70">View MathML</a>such that<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/17/mathml/M71','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/17/mathml/M71">View MathML</a>, for each<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/17/mathml/M72','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/17/mathml/M72">View MathML</a>and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/17/mathml/M73','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/17/mathml/M73">View MathML</a>.

(H2) There exists a constant<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/17/mathml/M74','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/17/mathml/M74">View MathML</a>such that<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/17/mathml/M75','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/17/mathml/M75">View MathML</a>for each<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/17/mathml/M73','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/17/mathml/M73">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/17/mathml/M77','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/17/mathml/M77">View MathML</a>.

If

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/17/mathml/M78','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/17/mathml/M78">View MathML</a>

(3.4)

then impulsive fractional boundary-value problem (1.1) has a unique solution in<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/17/mathml/M79','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/17/mathml/M79">View MathML</a>.

Proof We transform the problem (1.1) into a fixed-point problem, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/17/mathml/M80','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/17/mathml/M80">View MathML</a>, where the operator A is defined by equation (2.4). Using Banach’s contraction principle, we shall show that A has a fixed point.

Setting <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/17/mathml/M81','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/17/mathml/M81">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/17/mathml/M82','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/17/mathml/M82">View MathML</a> and choosing <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/17/mathml/M83','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/17/mathml/M83">View MathML</a>, we show that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/17/mathml/M84','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/17/mathml/M84">View MathML</a>, where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/17/mathml/M85','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/17/mathml/M85">View MathML</a>. For <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/17/mathml/M86','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/17/mathml/M86">View MathML</a>, we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/17/mathml/M87','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/17/mathml/M87">View MathML</a>

which proves that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/17/mathml/M84','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/17/mathml/M84">View MathML</a>.

Now let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/17/mathml/M89','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/17/mathml/M89">View MathML</a>. Then, for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/17/mathml/M72','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/17/mathml/M72">View MathML</a>, we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/17/mathml/M91','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/17/mathml/M91">View MathML</a>

Therefore,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/17/mathml/M92','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/17/mathml/M92">View MathML</a>

As follows from equation (3.4), A is a contraction. As a consequence of Banach’s fixed-point theorem, we have A has a fixed point which is a unique solution of the impulsive fractional boundary-value problem (1.1). This completes the proof. □

Now we give another existence and uniqueness result for impulsive fractional BVP (1.1) by using Banach’s fixed-point theorem and Hölder’s inequality. In addition, for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/17/mathml/M93','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/17/mathml/M93">View MathML</a>, we set

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/17/mathml/M94','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/17/mathml/M94">View MathML</a>

(3.5)

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/17/mathml/M95','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/17/mathml/M95">View MathML</a>

(3.6)

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/17/mathml/M96','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/17/mathml/M96">View MathML</a>

(3.7)

Theorem 3.2Assume that the following conditions hold:

(H3) <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/17/mathml/M97','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/17/mathml/M97">View MathML</a>, for each<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/17/mathml/M98','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/17/mathml/M98">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/17/mathml/M99','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/17/mathml/M99">View MathML</a>, where<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/17/mathml/M100','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/17/mathml/M100">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/17/mathml/M101','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/17/mathml/M101">View MathML</a>.

(H4) <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/17/mathml/M102','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/17/mathml/M102">View MathML</a>, for each<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/17/mathml/M103','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/17/mathml/M103">View MathML</a>, with constants<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/17/mathml/M104','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/17/mathml/M104">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/17/mathml/M105','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/17/mathml/M105">View MathML</a>.

Denote<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/17/mathml/M106','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/17/mathml/M106">View MathML</a>.

If

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/17/mathml/M107','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/17/mathml/M107">View MathML</a>

then the impulsive fractional boundary-value problem (1.1) has a unique solution.

Proof For <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/17/mathml/M89','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/17/mathml/M89">View MathML</a> and for each <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/17/mathml/M98','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/17/mathml/M98">View MathML</a>, by Hölder’s inequality, we get

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/17/mathml/M110','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/17/mathml/M110">View MathML</a>

Therefore,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/17/mathml/M111','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/17/mathml/M111">View MathML</a>

It follows that A is a contraction mapping. Hence Banach’s fixed-point theorem implies that A has a unique fixed point, which is the unique solution of the impulsive fractional boundary-value problem (1.1). This completes the proof. □

3.2 Existence result via Krasnoselskii’s fixed-point theorem

Lemma 3.1 (Krasnoselskii’s fixed point theorem) [29]

LetMbe a closed, bounded, convex and nonempty subset of a Banach spaceX. LetA, Bbe the operators such that (a) <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/17/mathml/M112','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/17/mathml/M112">View MathML</a>whenever<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/17/mathml/M113','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/17/mathml/M113">View MathML</a>; (b) Ais compact and continuous; (c) Bis a contraction mapping. Then there exists<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/17/mathml/M114','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/17/mathml/M114">View MathML</a>such that<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/17/mathml/M115','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/17/mathml/M115">View MathML</a>.

Theorem 3.3Let<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/17/mathml/M116','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/17/mathml/M116">View MathML</a>be a continuous function and let (H2) holds. In addition, we assume that:

(H5) <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/17/mathml/M117','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/17/mathml/M117">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/17/mathml/M118','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/17/mathml/M118">View MathML</a>, and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/17/mathml/M119','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/17/mathml/M119">View MathML</a>.

(H6) There exists a constant<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/17/mathml/M120','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/17/mathml/M120">View MathML</a>such that<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/17/mathml/M121','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/17/mathml/M121">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/17/mathml/M122','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/17/mathml/M122">View MathML</a>, for<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/17/mathml/M77','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/17/mathml/M77">View MathML</a>.

Then the impulsive fractional boundary-value problem (1.1) has at least one solution on<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/17/mathml/M124','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/17/mathml/M124">View MathML</a>if

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/17/mathml/M125','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/17/mathml/M125">View MathML</a>

(3.8)

where Φ is defined by equation (3.2).

Proof We define <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/17/mathml/M126','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/17/mathml/M126">View MathML</a> and choose a suitable constant <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/17/mathml/M127','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/17/mathml/M127">View MathML</a> as

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/17/mathml/M128','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/17/mathml/M128">View MathML</a>

where Ω and Ψ are defined by equations (3.1) and (3.3), respectively. We define the operators and on <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/17/mathml/M131','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/17/mathml/M131">View MathML</a> as

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/17/mathml/M132','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/17/mathml/M132">View MathML</a>

For <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/17/mathml/M133','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/17/mathml/M133">View MathML</a>, we find that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/17/mathml/M134','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/17/mathml/M134">View MathML</a>

Thus, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/17/mathml/M135','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/17/mathml/M135">View MathML</a>. It follows from the assumption (H2) together with (3.8) that is a contraction mapping. Continuity of f implies that the operator is continuous. Also, is uniformly bounded on <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/17/mathml/M139','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/17/mathml/M139">View MathML</a> as

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/17/mathml/M140','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/17/mathml/M140">View MathML</a>

Now we prove the compactness of the operator .

We define <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/17/mathml/M142','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/17/mathml/M142">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/17/mathml/M143','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/17/mathml/M143">View MathML</a> with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/17/mathml/M144','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/17/mathml/M144">View MathML</a> and consequently we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/17/mathml/M145','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/17/mathml/M145">View MathML</a>

which is independent of x and tends to zero as <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/17/mathml/M146','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/17/mathml/M146">View MathML</a>. Thus, is equicontinuous. So is relatively compact on <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/17/mathml/M139','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/17/mathml/M139">View MathML</a>. Hence, by the Arzelá-Ascoli theorem, is compact on <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/17/mathml/M139','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/17/mathml/M139">View MathML</a>. Thus all the assumptions of Lemma 3.1 are satisfied. So the conclusion of Lemma 3.1 implies that the impulsive fractional boundary-value problem (1.1) has at least one solution on <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/17/mathml/M124','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/17/mathml/M124">View MathML</a>. The proof is completed. □

3.3 Existence result via Leray-Schauder’s Nonlinear Alternative

Lemma 3.2 (Nonlinear alternative for single valued maps) [30]

LetEbe a Banach space, Ca closed, convex subset ofE, Uan open subset ofCand<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/17/mathml/M153','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/17/mathml/M153">View MathML</a>. Suppose that<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/17/mathml/M154','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/17/mathml/M154">View MathML</a>is a continuous, compact (that is, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/17/mathml/M155','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/17/mathml/M155">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/2014/1/17/mathml/M156','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/17/mathml/M156">View MathML</a>, or

(ii) there is a<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/17/mathml/M157','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/17/mathml/M157">View MathML</a> (the boundary ofUinC) and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/17/mathml/M158','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/17/mathml/M158">View MathML</a>with<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/17/mathml/M159','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/17/mathml/M159">View MathML</a>.

Theorem 3.4Assume the following.

(H7) There exist a continuous nondecreasing function<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/17/mathml/M160','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/17/mathml/M160">View MathML</a>and a function<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/17/mathml/M161','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/17/mathml/M161">View MathML</a>such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/17/mathml/M162','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/17/mathml/M162">View MathML</a>

(H8) There exists a continuous nondecreasing function<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/17/mathml/M163','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/17/mathml/M163">View MathML</a>such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/17/mathml/M164','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/17/mathml/M164">View MathML</a>

(H9) There exists a constant<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/17/mathml/M165','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/17/mathml/M165">View MathML</a>such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/17/mathml/M166','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/17/mathml/M166">View MathML</a>

Then the impulsive fractional boundary-value problem (1.1) has at least one solution on<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/17/mathml/M124','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/17/mathml/M124">View MathML</a>.

Proof We show that Amaps bounded sets (balls) into bounded sets in<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/17/mathml/M26','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/17/mathml/M26">View MathML</a>. For a positive number r, let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/17/mathml/M169','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/17/mathml/M169">View MathML</a> be a bounded ball in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/17/mathml/M170','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/17/mathml/M170">View MathML</a>. Then for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/17/mathml/M171','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/17/mathml/M171">View MathML</a> we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/17/mathml/M172','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/17/mathml/M172">View MathML</a>

Consequently

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/17/mathml/M173','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/17/mathml/M173">View MathML</a>

Next we show that Amaps bounded sets into equicontinuous sets of<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/17/mathml/M26','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/17/mathml/M26">View MathML</a>. Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/17/mathml/M175','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/17/mathml/M175">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/17/mathml/M143','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/17/mathml/M143">View MathML</a> with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/17/mathml/M177','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/17/mathml/M177">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/17/mathml/M178','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/17/mathml/M178">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/17/mathml/M179','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/17/mathml/M179">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/17/mathml/M180','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/17/mathml/M180">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/17/mathml/M86','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/17/mathml/M86">View MathML</a>. Then we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/17/mathml/M182','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/17/mathml/M182">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/2014/1/17/mathml/M86','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/17/mathml/M86">View MathML</a> as <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/17/mathml/M184','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/17/mathml/M184">View MathML</a>. As A satisfies the above assumptions, therefore it follows by the Arzelá-Ascoli theorem that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/17/mathml/M185','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/17/mathml/M185">View MathML</a> is completely continuous.

Let x be a solution. Then, for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/17/mathml/M171','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/17/mathml/M171">View MathML</a>, and following the similar computations as in the first step, we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/17/mathml/M187','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/17/mathml/M187">View MathML</a>

Consequently, we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/17/mathml/M188','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/17/mathml/M188">View MathML</a>

In view of (H9), there exists <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/17/mathml/M189','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/17/mathml/M189">View MathML</a> such that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/17/mathml/M190','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/17/mathml/M190">View MathML</a>. Let us set

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/17/mathml/M191','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/17/mathml/M191">View MathML</a>

Note that the operator <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/17/mathml/M192','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/17/mathml/M192">View MathML</a> is continuous and completely continuous. From the choice of U, there is no <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/17/mathml/M193','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/17/mathml/M193">View MathML</a> such that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/17/mathml/M194','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/17/mathml/M194">View MathML</a> for some <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/17/mathml/M158','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/17/mathml/M158">View MathML</a>. Consequently, by the nonlinear alternative of Leray-Schauder type (Lemma 3.2), we deduce that A has a fixed point <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/17/mathml/M196','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/17/mathml/M196">View MathML</a> which is a solution of the problem (1.1). This completes the proof. □

3.4 Existence result via Leray-Schauder degree

Theorem 3.5Assume the following.

(H10) There exist constants<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/17/mathml/M197','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/17/mathml/M197">View MathML</a>and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/17/mathml/M198','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/17/mathml/M198">View MathML</a>such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/17/mathml/M199','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/17/mathml/M199">View MathML</a>

(H11) There exist constants<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/17/mathml/M200','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/17/mathml/M200">View MathML</a>and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/17/mathml/M120','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/17/mathml/M120">View MathML</a>such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/17/mathml/M202','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/17/mathml/M202">View MathML</a>

where Ω and Φ are given by equations (3.1) and (3.2), respectively.

Then the impulsive fractional boundary-value problem (1.1) has at least one solution on<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/17/mathml/M124','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/17/mathml/M124">View MathML</a>.

Proof We define an operator <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/17/mathml/M64','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/17/mathml/M64">View MathML</a> as in equation (2.4) and consider the fixed-point problem

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/17/mathml/M205','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/17/mathml/M205">View MathML</a>

(3.9)

We are going to prove that there exists a fixed point <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/17/mathml/M206','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/17/mathml/M206">View MathML</a> satisfying equation (3.9). It is sufficient to show that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/17/mathml/M207','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/17/mathml/M207">View MathML</a> satisfies

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/17/mathml/M208','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/17/mathml/M208">View MathML</a>

(3.10)

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/17/mathml/M209','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/17/mathml/M209">View MathML</a>. We define

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/17/mathml/M210','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/17/mathml/M210">View MathML</a>

As shown in Theorem 3.4, we find that the operator A is continuous, uniformly bounded, and equicontinuous. Then, by the Arzelá-Ascoli theorem, a continuous map <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/17/mathml/M211','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/17/mathml/M211">View MathML</a> defined by <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/17/mathml/M212','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/17/mathml/M212">View MathML</a> is completely continuous. If equation (3.10) is true, then the following Leray-Schauder degrees are well defined and by the homotopy invariance of topological degree, it follows that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/17/mathml/M213','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/17/mathml/M213">View MathML</a>

(3.11)

where I denotes the identity operator. By the nonzero property of the Leray-Schauder degree, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/17/mathml/M214','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/17/mathml/M214">View MathML</a> for at least one <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/17/mathml/M215','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/17/mathml/M215">View MathML</a>. In order to prove equation (3.10), we assume that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/17/mathml/M194','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/17/mathml/M194">View MathML</a> for some <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/17/mathml/M217','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/17/mathml/M217">View MathML</a>. Then

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/17/mathml/M218','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/17/mathml/M218">View MathML</a>

Computing directly for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/17/mathml/M219','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/17/mathml/M219">View MathML</a>, we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/17/mathml/M220','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/17/mathml/M220">View MathML</a>

If <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/17/mathml/M221','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/17/mathml/M221">View MathML</a>, inequality (3.10) holds. This completes the proof. □

4 Examples

In this section we give examples to illustrate our results.

Example 4.1 Consider the following impulsive fractional boundary-value problem:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/17/mathml/M222','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/17/mathml/M222">View MathML</a>

(4.1)

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/17/mathml/M223','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/17/mathml/M223">View MathML</a>

(4.2)

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/17/mathml/M224','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/17/mathml/M224">View MathML</a>

(4.3)

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/17/mathml/M225','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/17/mathml/M225">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/17/mathml/M226','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/17/mathml/M226">View MathML</a>.

Set <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/17/mathml/M227','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/17/mathml/M227">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/17/mathml/M228','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/17/mathml/M228">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/17/mathml/M229','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/17/mathml/M229">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/17/mathml/M230','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/17/mathml/M230">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/17/mathml/M231','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/17/mathml/M231">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/17/mathml/M232','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/17/mathml/M232">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/17/mathml/M233','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/17/mathml/M233">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/17/mathml/M234','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/17/mathml/M234">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/17/mathml/M235','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/17/mathml/M235">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/17/mathml/M236','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/17/mathml/M236">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/17/mathml/M237','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/17/mathml/M237">View MathML</a>.

Since <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/17/mathml/M238','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/17/mathml/M238">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/17/mathml/M239','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/17/mathml/M239">View MathML</a> for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/17/mathml/M240','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/17/mathml/M240">View MathML</a>, then (H1) and (H2) are satisfied with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/17/mathml/M241','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/17/mathml/M241">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/17/mathml/M242','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/17/mathml/M242">View MathML</a>. We can show that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/17/mathml/M243','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/17/mathml/M243">View MathML</a>

Hence, by Theorem 3.1, the boundary-value problem (4.1)-(4.3) has a unique solution on <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/17/mathml/M244','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/17/mathml/M244">View MathML</a>.

Example 4.2 Consider the following impulsive fractional boundary-value problem:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/17/mathml/M245','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/17/mathml/M245">View MathML</a>

(4.4)

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/17/mathml/M246','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/17/mathml/M246">View MathML</a>

(4.5)

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/17/mathml/M247','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/17/mathml/M247">View MathML</a>

(4.6)

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/17/mathml/M248','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/17/mathml/M248">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/17/mathml/M249','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/17/mathml/M249">View MathML</a>.

Set <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/17/mathml/M250','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/17/mathml/M250">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/17/mathml/M251','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/17/mathml/M251">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/17/mathml/M252','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/17/mathml/M252">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/17/mathml/M253','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/17/mathml/M253">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/17/mathml/M254','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/17/mathml/M254">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/17/mathml/M255','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/17/mathml/M255">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/17/mathml/M256','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/17/mathml/M256">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/17/mathml/M257','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/17/mathml/M257">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/17/mathml/M235','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/17/mathml/M235">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/17/mathml/M236','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/17/mathml/M236">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/17/mathml/M260','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/17/mathml/M260">View MathML</a>.

Since <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/17/mathml/M261','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/17/mathml/M261">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/17/mathml/M262','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/17/mathml/M262">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/17/mathml/M263','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/17/mathml/M263">View MathML</a>, then (H3) and (H4) are satisfied with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/17/mathml/M264','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/17/mathml/M264">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/17/mathml/M265','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/17/mathml/M265">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/17/mathml/M266','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/17/mathml/M266">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/17/mathml/M267','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/17/mathml/M267">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/17/mathml/M268','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/17/mathml/M268">View MathML</a>. We can show that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/17/mathml/M269','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/17/mathml/M269">View MathML</a>

Hence, by Theorem 3.2, the boundary-value problem (4.4)-(4.6) has a unique solution on <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/17/mathml/M270','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/17/mathml/M270">View MathML</a>.

Example 4.3 Consider the following impulsive fractional boundary-value problem:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/17/mathml/M271','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/17/mathml/M271">View MathML</a>

(4.7)

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/17/mathml/M272','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/17/mathml/M272">View MathML</a>

(4.8)

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/17/mathml/M273','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/17/mathml/M273">View MathML</a>

(4.9)

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/17/mathml/M274','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/17/mathml/M274">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/17/mathml/M275','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/17/mathml/M275">View MathML</a>.

Set <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/17/mathml/M276','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/17/mathml/M276">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/17/mathml/M277','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/17/mathml/M277">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/17/mathml/M229','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/17/mathml/M229">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/17/mathml/M230','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/17/mathml/M230">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/17/mathml/M280','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/17/mathml/M280">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/17/mathml/M232','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/17/mathml/M232">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/17/mathml/M282','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/17/mathml/M282">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/17/mathml/M283','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/17/mathml/M283">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/17/mathml/M284','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/17/mathml/M284">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/17/mathml/M285','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/17/mathml/M285">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/17/mathml/M286','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/17/mathml/M286">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/17/mathml/M287','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/17/mathml/M287">View MathML</a>.

It is easy to see that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/17/mathml/M288','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/17/mathml/M288">View MathML</a>. Clearly,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/17/mathml/M289','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/17/mathml/M289">View MathML</a>

and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/17/mathml/M290','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/17/mathml/M290">View MathML</a>

Choosing <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/17/mathml/M291','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/17/mathml/M291">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/17/mathml/M292','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/17/mathml/M292">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/17/mathml/M293','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/17/mathml/M293">View MathML</a>, we obtain

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/17/mathml/M294','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/17/mathml/M294">View MathML</a>

which implies that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/17/mathml/M295','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/17/mathml/M295">View MathML</a>. Hence, by Theorem 3.4, the boundary-value problem (4.7)-(4.9) has at least one solution on <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/17/mathml/M296','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/17/mathml/M296">View MathML</a>.

Example 4.4 Consider the following impulsive fractional boundary-value problem:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/17/mathml/M297','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/17/mathml/M297">View MathML</a>

(4.10)

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/17/mathml/M298','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/17/mathml/M298">View MathML</a>

(4.11)

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/17/mathml/M299','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/17/mathml/M299">View MathML</a>

(4.12)

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/17/mathml/M300','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/17/mathml/M300">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/17/mathml/M301','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/17/mathml/M301">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/17/mathml/M302','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/17/mathml/M302">View MathML</a>.

Set <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/17/mathml/M303','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/17/mathml/M303">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/17/mathml/M304','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/17/mathml/M304">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/17/mathml/M305','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/17/mathml/M305">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/17/mathml/M306','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/17/mathml/M306">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/17/mathml/M307','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/17/mathml/M307">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/17/mathml/M308','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/17/mathml/M308">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/17/mathml/M309','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/17/mathml/M309">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/17/mathml/M310','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/17/mathml/M310">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/17/mathml/M311','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/17/mathml/M311">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/17/mathml/M233','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/17/mathml/M233">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/17/mathml/M313','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/17/mathml/M313">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/17/mathml/M314','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/17/mathml/M314">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/17/mathml/M315','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/17/mathml/M315">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/17/mathml/M316','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/17/mathml/M316">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/17/mathml/M284','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/17/mathml/M284">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/17/mathml/M318','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/17/mathml/M318">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/17/mathml/M286','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/17/mathml/M286">View MathML</a>.

Since <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/17/mathml/M320','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/17/mathml/M320">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/17/mathml/M321','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/17/mathml/M321">View MathML</a> for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/17/mathml/M322','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/17/mathml/M322">View MathML</a>, then (H10) and (H11) are satisfied with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/17/mathml/M323','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/17/mathml/M323">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/17/mathml/M324','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/17/mathml/M324">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/17/mathml/M325','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/17/mathml/M325">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/17/mathml/M326','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/17/mathml/M326">View MathML</a>. We have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/17/mathml/M327','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/17/mathml/M327">View MathML</a>

and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/17/mathml/M328','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/17/mathml/M328">View MathML</a>

Hence, by Theorem 3.5, the boundary value-problem (4.10)-(4.12) has at least one solution on <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/17/mathml/M329','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/17/mathml/M329">View MathML</a>.

Competing interests

The authors declare that they have no competing interests.

Authors’ contributions

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

Authors’ information

The third author is a Member of Nonlinear Analysis and Applied Mathematics (NAAM)-Research Group at King Abdulaziz University, Jeddah, Saudi Arabia.

Acknowledgements

We would like to thank the reviewers for their valuable comments and suggestions on the manuscript. This research of CT and JT is supported by King Mongkut’s University of Technology North Bangkok, Thailand.

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