Abstract and Applied Analysis

Volume 2012 (2012), Article ID 387629, 15 pages

doi:10.1155/2012/387629

Research Article

Some Existence Results for Impulsive Nonlinear Fractional Differential Equations with Closed Boundary Conditions

Hilmi Ergören¹ and Adem Kiliçman²

¹Department of Mathematics, Faculty of Sciences, Yuzuncu Yil University, 65080 Van, Turkey

²Department of Mathematics and Institute for Mathematical Research, University Putra Malaysia, 43400 Serdang, Malaysia

Received 30 September 2012; Accepted 22 October 2012

Academic Editor: Beata Rzepka

Copyright © 2012 Hilmi Ergören and Adem Kiliçman. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract.

We investigate some existence results for the solutions to impulsive fractional differential equations having closed boundary conditions. Our results are based on contracting mapping principle and Burton-Kirk fixed point theorem.

1. Introduction

This paper considers the existence and uniqueness of the solutions to the closed boundary value problem (BVP), for the following impulsive fractional differential equation:

where is Caputo fractional derivative, , ,

with

and has a similar meaning for , where

, , , and are real constants with .

The boundary value problems for nonlinear fractional differential equations have been addressed by several researchers during last decades. That is why, the fractional derivatives serve an excellent tool for the description of hereditary properties of various materials and processes. Actually, fractional differential equations arise in many engineering and scientific disciplines such as, physics, chemistry, biology, electrochemistry, electromagnetic, control theory, economics, signal and image processing, aerodynamics, and porous media (see [1–7]). For some recent development, see, for example, [8–14].

On the other hand, theory of impulsive differential equations for integer order has become important and found its extensive applications in mathematical modeling of phenomena and practical situations in both physical and social sciences in recent years. One can see a noticeable development in impulsive theory. For instance, for the general theory and applications of impulsive differential equations we refer the readers to [15–17].

Moreover, boundary value problems for impulsive fractional differential equations have been studied by some authors (see [18–20] and references therein). However, to the best of our knowledge, there is no study considering closed boundary value problems for impulsive fractional differential equations.

Here, we notice that the closed boundary conditions in (1.1) include quasi-periodic boundary conditions and interpolate between periodic and antiperiodic boundary conditions.

Motivated by the mentioned recent work above, in this study, we investigate the existence and uniqueness of solutions to the closed boundary value problem for impulsive fractional differential equation (1.1). Throughout this paper, in Section 2, we present some notations and preliminary results about fractional calculus and differential equations to be used in the following sections. In Section 3, we discuss some existence and uniqueness results for solutions of BVP (1.1), that is, the first one is based on Banach’s fixed point theorem, the second one is based on the Burton-Kirk fixed point theorem. At the end, we give an illustrative example for our results.

2. Preliminaries

Let us set , , , and introduce the set of functions:

, and there exist and , with and

, , and there exist and , with which is a Banach space with the norm where .

The following definitions and lemmas were given in [4].

Definition 2.1. The fractional (arbitrary) order integral of the function of order is defined by

where is the Euler gamma function.

Definition 2.2. For a function given on the interval , Caputo fractional derivative of order is defined by

where the function has absolutely continuous derivatives up to order .

Lemma 2.3. Let , then the differential equation

has solutions

The following lemma was given in [4, 10].

Lemma 2.4. Let , then

for some , , .

The following theorem is known as Burton-Kirk fixed point theorem and proved in [21].

Theorem 2.5. Let be a Banach space and , two operators satisfying:

(a) is a contraction, and (b) is completely continuous. Then either

(i) the operator equation has a solution, or(ii) the set is unbounded for .

Theorem 2.6 (see [22], Banach’s fixed point theorem). Let be a nonempty closed subset of a Banach space , then any contraction mapping of into itself has a unique fixed point.

Next we prove the following lemma.

Lemma 2.7. Let and let be continuous. A function is a solution of the fractional integral equation:

if and only if is a solution of the fractional BVP

where

Proof. Let be the solution of (2.7). If , then Lemma 2.4 implies that

for some .

If , then Lemma 2.4 implies that

for some . Thus we have

Observing that

then we have

hence, for ,

If , then Lemma 2.4 implies that

for some . Thus we have

Similarly we observe that

thus we have

Hence, for ,

By a similar process, if , then again from Lemma 2.4 we get

Now if we apply the conditions:

we have

In view of the relations (2.8), when the values of and are replaced in (2.9) and (2.20), the integral equation (2.7) is obtained.

Conversely, assume that satisfies the impulsive fractional integral equation (2.6), then by direct computation, it can be seen that the solution given by (2.6) satisfies (2.7). The proof is complete.

3. Main Results

Definition 3.1. A function with its -derivative existing on is said to be a solution of (1.1), if satisfies the equation on and satisfies the conditions:

For the sake of convenience, we define

The followings are main results of this paper.

Theorem 3.2. Assume that

(A1) the function is continuous and there exists a constant such that , for all , and ,(A2), are continuous, and there exist constants and such that , for each and .

Moreover, consider the following:

Then, BVP (1.1) has a unique solution on .

Proof. Define an operator by

Now, for and for each , we obtain

Therefore, by (3.3), the operator is a contraction mapping. In a consequence of Banach’s fixed theorem, the BVP (1.1) has a unique solution. Now, our second result relies on the Burton-Kirk fixed point theorem.

Theorem 3.3. Assume that (A1)-(A2) hold, and

(A3) there exist constants , , such that , , for each and .

Then the BVP (1.1) has at least one solution on .

Proof. We define the operators by

It is obvious that is contraction mapping for

Now, in order to check that is completely continuous, let us follow the sequence of the following steps.

Step 1 ( is continuous). Let be a sequence such that in . Then for , we have

Since is continuous function, we get

Step 2 ( maps bounded sets into bounded sets in ). Indeed, it is enough to show that for any , there exists a positive constant such that for each , we have . By (A3), we have for each ,

Step 3 ( maps bounded sets into equicontinuous sets in ). Let with and let be a bounded set of as in Step 2, and let . Then

where

This implies that is equicontinuous on all the subintervals , . Therefore, by the Arzela-Ascoli Theorem, the operator is completely continuous.

To conclude the existence of a fixed point of the operator , it remains to show that the set

is bounded.

Let , for each ,

Hence, from (A3), we have

Consequently, we conclude the result of our theorem based on the Burton-Kirk fixed point theorem.

4. An Example

Consider the following impulsive fractional boundary value problem:

Here, , , , , , , . Obviously, , , , , . Further,

Since the assumptions of Theorem 3.2 are satisfied, the closed boundary value problem (4.1) has a unique solution on . Moreover, it is easy to check the conclusion of Theorem 3.3.

Acknowledgments

The authors would like to express their sincere thanks and gratitude to the reviewer(s) for their valuable comments and suggestions for the improvement of this paper. The second author also gratefully acknowledges that this research was partially supported by the University Putra Malaysia under the Research University Grant Scheme  05-01-09-0720RU.

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