Abstract and Applied Analysis

Volume 2014 (2014), Article ID 932747, 10 pages

http://dx.doi.org/10.1155/2014/932747

Research Article

A New Impulsive Multi-Orders Fractional Differential Equation Involving Multipoint Fractional Integral Boundary Conditions

Guotao Wang,¹ Sanyang Liu,¹ Dumitru Baleanu,2,3,4 and Lihong Zhang⁵

¹Department of Applied Mathematics, Xidian University, Xian, Shaanxi 710071, China

²Department of Chemical and Materials Engineering, Faculty of Engineering, King Abdulaziz University, P.O. Box 80204, Jeddah 21589, Saudi Arabia

³Department of Mathematics and Computer Science, Çankaya University, 06530 Ankara, Turkey

⁴Institute of Space Sciences, P.O. Box MG-23, 76900 Magurele-Bucharest, Romania

⁵School of Mathematics and Computer Science, Shanxi Normal University, Linfen, Shanxi 041004, China

Received 6 April 2014; Accepted 24 May 2014; Published 16 June 2014

Academic Editor: J. C. Prajapati

Copyright © 2014 Guotao Wang et al. 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.

A new impulsive multi-orders fractional differential equation is studied. The existence and uniqueness results are obtained for a nonlinear problem with fractional integral boundary conditions by applying standard fixed point theorems. An example for the illustration of the main result is presented.

1. Introduction

Nowadays, fractional differential equations have attracted a lot of attention due to its wide range of applications in many practical problems such as in physics, engineering, economics, and so on; see [1–5].

Impulsive differential equations have extensively been studied in the past two decades. Indeed impulsive differential equations are used to describe the dynamics of processes in which sudden, discontinuous jumps occur. Such processes are naturally seen in harvesting, earthquakes, diseases, and so forth. Recently, fractional impulsive differential equations have attracted the attention of many researchers. For the general theory and applications of such equations we refer the interested reader to see [6–24] and the references therein.

In this paper, we investigate a new impulsive nonlinear differential equation involving multi-orders fractional derivatives and deviating argument. Precisely, we consider the following multipoint fractional integral boundary value problem:

where is the Caputo fractional derivative of order and is fractional Riemann-Liouville integral of order , , , , deviating argument , , , , and , where and denote the right and the left limits of at , respectively. have a similar meaning for .

The paper is organized as follows. Section 2 gives some definitions and necessary lemmas, while the main results are presented in Section 3.

2. Preliminaries

Let us fix , , and with and introduce a Banach space:

with the norm .

For the reader’s convenience, we present some necessary definitions from fractional calculus theory and lemmas.

Definition 1. The Riemann-Liouville fractional integral of order for a function is defined as

provided the integral exists.

Definition 2. The Caputo fractional derivative of order for a function is defined by

where denotes the integer part of real number .

Lemma 3. For a given , a function is a solution of the following impulsive boundary value problem:

if and only if is a solution of the impulsive fractional integral equation

where

Proof. Let be a solution of (5). For any , we have

for some . Differentiating (8), we get

If , then

for some . Thus,

Using the impulse conditions

we find that

Consequently,

By a similar process, we can get

The boundary condition implies . For , we have

Applying the boundary condition , then

Substituting the value of in (8) and (15), we obtain (6). Conversely, assume that is a solution of the impulsive fractional integral equation (6); then by a direct computation, it follows that the solution given by (6) satisfies (5). This completes the proof.

3. Main Results

Define an operator by

Notice that problem (1) has a solution if and only if the operator has a fixed point.

For convenience, we will give some notations:

Theorem 4. Assume the following.

(H1)There exists a nonnegative function such that

where are nonnegative constants.(H2)There exist positive constants and such that

Then problem (1) has at least one solution.

Proof. Firstly, we will prove that is a completely continuous operator. Obviously, the continuity of functions , , and ensures the continuity of operator .

Let be bounded. Then, there exist positive constants such that , and for all . Thus, for any , we have

which implies

On the other hand, for any , , we have

Hence, for with and , we have

This implies that is equicontinuous on all . Consequently, Arzela-Ascoli theorem ensures the operator is a completely continuous operator.

Next, we will show that the operator maps into . For that, let us choose and define a ball . For any , by the conditions (H ₁) and (H ₂), we have

Thus,

This implies . Hence, we conclude that is completely continuous. It follows from the Schauder fixed point theorem that the operator has at least one fixed point. That is, problem (1) has at least one solution in .

Theorem 5. Suppose that there exist a nonnegative function and nonnegative constants such that

for , and . Furthermore, the assumption holds. Then problem (1) has a unique solution.

Proof. For , we have

As , we have . Therefore, is a contraction. It follows from the Banach contraction mapping principle that problem (1) has a unique solution.

Example 6. For , , , , , , , , , , and , we consider the following impulsive multi-orders fractional differential equation:

Observe that

Clearly, , , , , and and the conditions of Theorem 4 hold. Thus, by Theorem 4, the impulsive multi-orders fractional boundary value problem (30) has at least one solution.

Conflict of Interests

The authors declare that they have no conflict of interests.

Authors’ Contribution

All authors have equal contributions.

Acknowledgments

This work is supported by the NNSF of China (no. 61373174) and the Natural Science Foundation for Young Scientists of Shanxi Province, China (no. 20120211002-3).

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