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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

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

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:

(1.1)

where is the Caputo fractional derivative of order α, , is a continuous function, , with , , are constants, is the Riemann-Liouville fractional integral of order for and , , a, b, c are given constants such that .

The integral jump conditions are very general and include many conditions as special cases. In particular, if and , then the impulsive fractional integral of equation (1.1) reduces to

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 = { is continuous everywhere except for some  at which and exist and , }. is a Banach space endowed with the norm defined by . 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 of a function is defined by

where Γ is the Gamma function.

Definition 2.2 The Riemann-Liouville fractional derivative of order of a continuous function is defined by

where , denotes the integral part of real number α, provided the right-hand side is point-wise defined on .

Definition 2.3 For a continuous function , the Caputo derivative of fractional order α is defined as

where , denotes the integral part of real number α, provided exists.

Lemma 2.1 ([28])

Letandbe continuous. A functionis a solution of the fractional Cauchy problem

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

Lemma 2.2Letand. The unique solution of the impulsive fractional boundary-value problem (1.1) is given by

(2.1)

Proof For , Riemann-Liouville fractional integrating of order α, from 0 to t, for the first equation of (1.1), we have

(2.2)

Substituting into (2.2), we get

For , by Lemma 2.1 with the second equation of (1.1), we obtain

If then again from Lemma 2.1, we have

If then again from Lemma 2.1, we get

In particular, for , we have

(2.3)

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

Therefore, we have

This completes the proof. □

As in Lemma 2.2, we define an operator by

(2.4)

with . 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

(3.1)

(3.2)

(3.3)

Theorem 3.1Assume the following.

(H1) There exists a constantsuch that, for eachand.

(H2) There exists a constantsuch thatfor each, .

If

(3.4)

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

Proof We transform the problem (1.1) into a fixed-point problem, , 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 , and choosing , we show that , where . For , we have

which proves that .

Now let . Then, for , we have

Therefore,

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 , we set

(3.5)

(3.6)

(3.7)

Theorem 3.2Assume that the following conditions hold:

(H3) , for each, , where, .

(H4) , for each, with constants, .

Denote.

If

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

Proof For and for each , by Hölder’s inequality, we get

Therefore,

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) whenever; (b) Ais compact and continuous; (c) Bis a contraction mapping. Then there existssuch that.

Theorem 3.3Letbe a continuous function and let (H2) holds. In addition, we assume that:

(H5) , , and.

(H6) There exists a constantsuch that, , for.

Then the impulsive fractional boundary-value problem (1.1) has at least one solution onif

(3.8)

where Φ is defined by equation (3.2).

Proof We define and choose a suitable constant as

where Ω and Ψ are defined by equations (3.1) and (3.3), respectively. We define the operators and on as

For , we find that

Thus, . 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 as

Now we prove the compactness of the operator .

We define , with and consequently we have

which is independent of x and tends to zero as . Thus, is equicontinuous. So is relatively compact on . Hence, by the Arzelá-Ascoli theorem, is compact on . 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 . 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. Suppose thatis a continuous, compact (that is, is a relatively compact subset ofC) map. Then either

(i) Fhas a fixed point in, or

(ii) there is a (the boundary ofUinC) andwith.

Theorem 3.4Assume the following.

(H7) There exist a continuous nondecreasing functionand a functionsuch that

(H8) There exists a continuous nondecreasing functionsuch that

(H9) There exists a constantsuch that

Then the impulsive fractional boundary-value problem (1.1) has at least one solution on.

Proof We show that Amaps bounded sets (balls) into bounded sets in. For a positive number r, let be a bounded ball in . Then for we have

Consequently

Next we show that Amaps bounded sets into equicontinuous sets of. Let , with , , , and . Then we have

Obviously the right-hand side of the above inequality tends to zero independently of as . As A satisfies the above assumptions, therefore it follows by the Arzelá-Ascoli theorem that is completely continuous.

Let x be a solution. Then, for , and following the similar computations as in the first step, we have

Consequently, we have

In view of (H9), there exists such that . Let us set

Note that the operator is continuous and completely continuous. From the choice of U, there is no such that for some . Consequently, by the nonlinear alternative of Leray-Schauder type (Lemma 3.2), we deduce that A has a fixed point 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 constantsandsuch that

(H11) There exist constantsandsuch that

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.

Proof We define an operator as in equation (2.4) and consider the fixed-point problem

(3.9)

We are going to prove that there exists a fixed point satisfying equation (3.9). It is sufficient to show that satisfies

(3.10)

where . We define

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 defined by 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

(3.11)

where I denotes the identity operator. By the nonzero property of the Leray-Schauder degree, for at least one . In order to prove equation (3.10), we assume that for some . Then

Computing directly for , we have

If , 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:

(4.1)

(4.2)

(4.3)

where , .

Set , , , , , , , , , and .

Since and for , then (H1) and (H2) are satisfied with and . We can show that

Hence, by Theorem 3.1, the boundary-value problem (4.1)-(4.3) has a unique solution on .

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

(4.4)

(4.5)

(4.6)

where , .

Set , , , , , , , , , and .

Since , and , then (H3) and (H4) are satisfied with , , , and . We can show that

Hence, by Theorem 3.2, the boundary-value problem (4.4)-(4.6) has a unique solution on .

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

(4.7)

(4.8)

(4.9)

where , .

Set , , , , , , , , , , and .

It is easy to see that . Clearly,

and

Choosing , and , we obtain

which implies that . Hence, by Theorem 3.4, the boundary-value problem (4.7)-(4.9) has at least one solution on .

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

(4.10)

(4.11)

(4.12)

where , , .

Set , , , , , , , , , , , , , , , and .

Since , for , then (H10) and (H11) are satisfied with , , and . We have

and

Hence, by Theorem 3.5, the boundary value-problem (4.10)-(4.12) has at least one solution on .

### Competing interests

The authors declare that they have no competing interests.

### 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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