Abstract and Applied Analysis

Volume 2012 (2012), Article ID 356132, 18 pages

doi:10.1155/2012/356132

Research Article

On Impulsive Boundary Value Problems of Fractional Differential Equations with Irregular Boundary Conditions

Guotao Wang,¹ Bashir Ahmad,² and Lihong Zhang¹

¹School of Mathematics and Computer Science, Shanxi Normal University, Linfen, Shanxi 041004, China

²Department of Mathematics, Faculty of Science, King Abdulaziz University, P.O. Box 80203, Jeddah 21589, Saudi Arabia

Received 25 February 2012; Accepted 4 September 2012

Academic Editor: Yong H. Wu

Copyright © 2012 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.

We study nonlinear impulsive differential equations of fractional order with irregular boundary conditions. Some existence and uniqueness results are obtained by applying standard fixed-point theorems. For illustration of the results, some examples are discussed.

1. Introduction

Boundary value problems of nonlinear fractional differential equations have recently been studied by several researchers. Fractional differential equations appear naturally in various fields of science and engineering and constitute an important field of research. As a matter of fact, fractional derivatives provide an excellent tool for the description of memory and hereditary properties of various materials and processes [1–4]. Some recent work on boundary value problems of fractional order can be found in [5–23] and the references therein. In [24], some existence and uniqueness results were obtained for an irregular boundary value problem of fractional differential equations.

Dynamical systems with impulse effect are regarded as a class of general hybrid systems. Impulsive hybrid systems are composed of some continuous variable dynamic systems along with certain reset maps that define impulsive switching among them. It is the switching that resets the modes and changes the continuous state of the system. There are three classes of impulsive hybrid systems, namely, impulsive differential systems [25, 26], sampled data or digital control system [27, 28], and impulsive switched system [29, 30]. Applications of such systems include air traffic management [31], automotive control [32, 33], real-time software verification [34], transportation systems [35, 36], manufacturing [37], mobile robotics [38], and process industry [39]. In fact, hybrid systems have a central role in embedded control systems that interact with the physical world. Using hybrid models, one may represent time and event-based behaviors more accurately so as to meet challenging design requirements in the design of control systems for problems such as cut-off control and idle speed control of the engine. For more details, see [40] and the references therein.

The theory of impulsive differential equations of integer order has found its extensive applications in realistic mathematical modelling of a wide variety of practical situations and has emerged as an important area of investigation. The impulsive differential equations of fractional order have also attracted a considerable attention and a variety of results can be found in the papers [41–50].

In this paper, motivated by [24], we study a nonlinear impulsive hybrid system of fractional differential equations with irregular boundary conditions given by

where is the Caputo fractional derivative, , where and denote the right and the left limits of at , respectively. have a similar meaning for .

Here, we remark that irregular boundary value problems for ordinary and partial differential equations occur in scientific and engineering disciplines and have been addressed by many authors, for instance, see [24] and the references.

The paper is organized as follows. Section 2 deals with some definitions and preliminary results, while the main results are presented in Section 3.

2. Preliminaries

Let us fix with and introduce the spaces:

with the norm , and

with the norm . Obviously, and are Banach spaces.

Definition 2.1. A function with its Caputo derivative of order existing on is a solution of (1.1) if it satisfies (1.1).

To prove the existence of solutions of problem (1.1), we need the following fixed-point theorems.

Theorem 2.2 (see [51]). Let be a Banach space. Assume that is an open bounded subset of with and let be a completely continuous operator such that

Then has a fixed point in .

Lemma 2.3 (see [1]). For , the general solution of fractional differential equation is

where ( denotes integer part of ).

Lemma 2.4 (see [1]). Let . Then

for some .

Lemma 2.5. For a given , a function is a solution of the following impulsive irregular boundary value problem

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

where

Proof. Let be a solution of (2.6). Then, by Lemma 2.4, we have

for some . Differentiating (2.9), we get

If , then

for some . Thus,

Using the impulse conditions

we find that

Consequently, we obtain

By a similar process, we get

Applying the boundary conditions and , we find that

Substituting the value of in (2.9) and (2.16), we obtain (2.7). Conversely, assume that is a solution of the impulsive fractional integral equation (2.7), then by a direct computation, it follows that the solution given by (2.7) satisfies (2.6). This completes the proof.

Remark 2.6. With , the first five terms of the solution (2.7) correspond to the solution for the problem without impulses [24].

3. Main Results

Define an operator by

Notice that problem (1.1) has a solution if and only if the operator has a fixed point.

For the sake of convenience, we set the following notations:

Theorem 3.1. Assume that

there exists a nonnegative function such that

where is a nonnegative constant; there exist positive constants and such that

Then problem (1.1) has at least one solution.

Proof. As a first step, we show that the operator is completely continuous. Observe that continuity of follows from the continuity of and .

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

which implies

On the other hand, for any , we get

Hence, for with , we have

This implies that is equicontinuous on all and hence, by the Arzela-Ascoli theorem, the operator is completely continuous.

Next, we prove that . For that, let us choose and define a ball . For any , by the assumptions and , we have

Thus,

where and are given by (3.2). This implies . Hence, is completely continuous. Therefore, by the Schauder fixed-point theorem, the operator has at least one fixed point. Consequently, problem (1.1) has at least one solution in .

Remark 3.2. For in (), if , we can take , then the conclusion of Theorem 3.1 holds.

Theorem 3.3. Suppose that there exist a nonnegative functions and a nonnegative number such that for . Furthermore, the assumption holds. Then problem (1.1) has at least one solution.

Proof. The proof is similar to that of Theorem 3.1, so we omit it.

Theorem 3.4. Suppose that

Then problem (1.1) has at least one solution.

Proof. By Theorem 3.1, we know that the operator is completely continuous. In view of (3.11), we can find a constant such that and for , where satisfy

Let . Take such that , which means . Then, as in the proof of Theorem 3.1, we have

which, in view of (3.12), implies that , . Therefore, by Theorem 2.2, the operator has at least one fixed point. Thus we conclude that problem (1.1) has at least one solution .

Theorem 3.5. Assume that

there exist positive constants such that

for .

Then problem (1.1) has a unique solution if

Proof. For , we have

which, by (3.15), yields . So, is a contraction. Therefore, by the Banach contraction mapping principle, problem (1.1) has a unique solution.

Example 3.6. Consider the following fractional impulsive irregular boundary value problem

where and .

Observe that

Clearly, , and the conditions of Theorem 3.1 hold for . Thus, by Theorem 3.1, problem (3.17) has at least one solution. In a similar way, for , the impulsive irregular fractional boundary value problem (3.17) has at least one solution by means of Theorem 3.3.

Example 3.7. Consider the impulsive fractional irregular boundary value problem given by

where and .

It can easily be verified that all the assumptions of Theorem 3.4 are satisfied. Thus, by the conclusion of Theorem 3.4, we deduce that the problem (3.19) has at least one solution.

Example 3.8. Consider

Here , and . With

we find that

Thus, all the conditions of Theorem 3.5 are satisfied. Consequently, the conclusion of Theorem 3.5 applies and the fractional order impulsive irregular boundary value problem (3.20) has a unique solution on .

Acknowledgment

The research of G. Wang and L. Zhang was supported by the Natural Science Foundation for Young Scientists of Shanxi Province (2012021002-3), China.

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